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Jul 1, 2015 - MWCNTs/water, Al2O3/radiator coolant, Al2O3/R141b, Al, CNTs/Engine Oil, and Cu/Therminol 66, and suits the

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International Journal of Heat and Mass Transfer 90 (2015) 121–130

Contents lists available at ScienceDirect

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

A new correlation for predicting the thermal conductivity of nanofluids; using dimensional analysis Samir Hassani a, R. Saidur b, Saad Mekhilef c,⇑, Arif Hepbasli d a

Department of Mechanical Engineering, Faculty of Engineering, University of Malaya, 50603 Kuala Lumpur, Malaysia Center of Research Excellence in Renewable Energy (CoRERE), King Fahd University of Petroleum & Minerals (KFUPM), Dhahran 31261, Saudi Arabia c Power Electronics and Renewable Energy Research Laboratory (PEARL), Department of Electrical Engineering, University of Malaya, Kuala Lumpur, Malaysia d Department of Energy Systems Engineering, Faculty of Engineering, Yasßar University, 35100 Izmir, Turkey b

a r t i c l e

i n f o

Article history: Received 6 February 2015 Received in revised form 13 June 2015 Accepted 14 June 2015 Available online 1 July 2015 Keywords: Correlation Nanofluids Thermal conductivity Vaschy–Buckingham theorem Nonlinear regression analysis

a b s t r a c t Thermal conductivity of nanofluids is a key thermophysical property, which depends on concentration and size of nanoparticles, temperature and thermophysical properties of the base fluid. Over last decades, several works have been done on the thermal conductivity of nanofluids while a number of numerical and theoretical models have been proposed. However, most of these models were not able to predict appropriately the thermal conductivity for a variety of nanofluids. In the present paper, using the Vaschy–Buckingham theorem, new correlations for predicting the thermal conductivity of nanofluids were developed based on the existing experimental data. The new correlation proposed took into account the Brownian motion, the variation of volume fraction, the temperature and the size distribution of nanoparticles. The expression developed successfully predicts the thermal conductivity of a variety of nanofluids, TiO2, Al2O3, Al, Cu, Fe, MWCNTs/EG, Al2O3, SiO2/methanol, TiO2, Al2O3, CuO, MWCNTs/water, Al2O3/radiator coolant, Al2O3/R141b, Al, CNTs/Engine Oil, and Cu/Therminol 66, and suits the data with a mean and standard deviation of 2.74%, 3.63%, respectively. The correlation was derived from 196 values of nanofluids thermal conductivity, 86% of them are correlated within a mean deviation of ±5%, while 98% of them belong to an interval of ±10%. Moreover, the proposed correlation has been tested on 284 values of thermal conductivity of different nanofluids, and the predicted values have been found in excellent agreement with the experimental ones with a mean deviation of 3%. The mean deviation between the correlated and the tested point found to be 2.94%. The present correlation will be a good tool for engineers in preparing the nanofluid for different applications in heat exchangers and thermal solar collectors. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction A nanofluid is an engineered colloidal suspension of nanometer-sized particles, named nanoparticles, in the base fluid. The nanoparticles used in nanofluids are typically made of metals, oxides, carbides, or carbon nanotubes. The common base fluids include water, ethylene glycol and oil. Nanofluids have remarkable thermophysical properties that make them potentially useful in many heat transfer applications [1], including electronic cooling systems, fuel cells, engine cooling/vehicle thermal management, solar thermal collectors, domestic refrigerators, chillers, and heat exchangers. Over the last few years, nanofluids have attracted the interest of a growing number ⇑ Corresponding author. E-mail addresses: [email protected] (S. Hassani), [email protected] (R. Saidur), [email protected] (S. Mekhilef), [email protected] (A. Hepbasli). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.06.040 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.

of scientists due to the significant research effort devoted to this subject, and the main findings have been summarized in the recent review papers written by Wen et al. [2], Mahian et al. [3], Saidur et al. [4], Aybar et al. [5], Haddad et al. [6], and Younes et al. [7]. Nanofluids exhibit the enhanced thermal conductivity and the convective heat transfer coefficient compared to the base fluid. The development of models for predicting the thermophysical properties of nanofluids, such as the thermal conductivity, attracted the interest of several researchers. There are several analytical models that allows engineers and scientists to estimate the thermal conductivity values of nanofluids. Former models, such as Maxwell [8], Hamilton and Crosser [9], and Bruggemen [10], which are based on the classical theory of composites and mixtures containing particles of the order of millimeter or micrometer, fail dramatically in predicting the abnormal thermal conductivity of nanoparticle suspensions. This is essentially due to the nature of the models, which take into consideration only the effect of the

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S. Hassani et al. / International Journal of Heat and Mass Transfer 90 (2015) 121–130

Nomenclature cpf dref dp kb kf kn kp Pr Re T Tb T fr

specific heat of the base fluid, J kg1 K1 reference diameter diameter of the nanoparticle, m Boltzmann constant = 1:3807  1023 , J/K thermal conductivity of the base fluid, W m1 K1 thermal conductivity of the nanofluid, W m1 K1 thermal conductivity of the nanoparticle, W m1 K1 Prandtl number Reynolds number temperature, K boiling point temperature of the base fluid, K freezing point of the base fluid, K

nanoparticle concentration. For example, Murshed et al. [11] found that the results obtained from the Hamilton and Crosser, and Bruggemen models differed from their experimental data by 17% for a 5% particle volumetric concentration. Thereby, new insights and mathematical models on the thermal conductivity have been proposed. They mainly considered the nanolayering of the liquid at the liquid/nanoparticle interface [12–14], and/or Brownian motion of the nanoparticles [15–18], and/or the effects of nanoparticle clustering [19–21]. In addition, other models which were in the form of empirical correlations, based on the experimental data, considered the variation of the temperature, the volume concentration and the diameter of nanoparticles [22–26]. Although numerous models have been suggested, some of them were inconsistent with the experimental data. Moreover, it was not clear which is the best model in predicting the thermal conductivity of nanofluids [5]. Some of the models proposed contain empirical constants that are either heavily dependent on the experimental observation, or poorly defined [5,23]. Therefore, an accurate and adaptable model or correlation is needed to be developed and it is also essential to determine a model, which must take into account the experimental observations available in the literatures. To develop a correlation having a wide-range applications with different base fluids, it is indispensable to generate suitable dimensionless parameters for obtaining a wide spectrum of database. Most of the authors, who have proposed empirical correlations to predict the thermal conductivity of nanofluids, have developed their correlations from their own experimental data, or using limited databases for a certain base fluid. Therefore, the major contribution of the present work to the literature is to develop a hybrid correlation for predicting the thermal conductivity of nanofluids with large varieties of nanoparticles and base fluid. In this study, the Vaschy–Buckingham theorem [27,28] was applied to form dimensionless parameters based on the nanoparticles and base fluid properties. The generalized correlation is established in a power law form of dimensionless parameters based on the extensive experimental data available in the literature. The correlation has been verified by comparing the predicted values with the database and the existing correlations reported by the previous researchers. The role of some physical parameters in enhancing thermal conductivity of nanofluids with different base fluids are also proposed from the generalized correlation. 2. Existing correlations for predicting the thermal conductivity of nanofluids Based on the existing literature on thermal conductivity of nanofluids, a few correlations found to be available on this subject.

Greek symbols a Thermal diffusivity, m2 s1 qp density of the nanoparticles, kg m3 v Br Brownian velocity, m s1 lf dynamic viscosity, kg m1 s1 mf kinematic viscosity, m2 s1 / volume fraction Abbreviations EG ethylene glycol TH66 Therminol 66 RC radiator coolant

The validity of most existing correlations are limited to a certain extent. For example, most of them are valid only for oxide or metallic nanoparticles suspended in water or ethylene glycol. We have noted that none of these correlations could be used as a general reference to predict the thermal conductivity for a large variety of nanofluids. This limitation is essentially due to the shortage of the experimental databases used to drive these correlations. A summary of selective correlations on the thermal conductivity of nanofluids and their validity range are listed in Table 1. 3. Methodology 3.1. Database to develop the correlation As reported in the previous section, the main objective of the present work is to develop a new correlation, which is able to predict the thermal conductivity for a wide variety of nanofluids. As reported previously, several models on the thermal conductivity have been suggested to predict the experimental results. However, none of them was successful in developing a general model with accurate prediction results for different types of nanofluids with several alternative base fluids. The present correlation is derived from a wide variety of experimental data of the thermal conductivity of nanofluids. Several alternative base fluids, such as water, ethylene glycol, methanol, engine oil (EO), Therminol 66, ethylene glycol based coolant used in car radiator (RC), R141a, containing different types of nanoparticles, like TiO2, Al, Al2O3, Fe, Cu, SiO2, CuO, CNTs and MWCNTs were considered in developing this correlation. The experimental database used to develop the present correlation is obtained from the open literature [22,30,40–57], as shown in Table 2. There are total 196 data points for six base fluids; water, ethylene glycol, methanol, radiator coolant, R141b, engine oil and Therminol 66 and 62. 3.2. Correlation development for effective thermal conductivity The thermal conductivity of nanofluid depends on several parameters related either, to the medium (base fluid) such as, the temperature, the thermal conductivity, the specific heat, the viscosity, the density, or to the physical properties of nanoparticles, such as the Brownian velocity, the volume fraction, nanoparticle’s thermal conductivity, the density and the size. The Brownian velocity is introduced due to the Brownian motion of nanoparticles within the base fluid. Therefore, the resulting relationship between the thermal conductivity and the selected variables is represented in a functional form, as given by Eq. (1):

kn ¼ f ð/; kf ; kp ; v Br ; cpf ; dref ; dp ; T; mf ; T b Þ

ð1Þ

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S. Hassani et al. / International Journal of Heat and Mass Transfer 90 (2015) 121–130 Table 1 A summary of selected correlations on thermal conductivity of nanofluids. References Koo and Kleinstreuer [29]

Correlations h i h i qffiffiffiffiffiffiffiffi k þ2k 2/ðk k Þ kn ¼ kpp þ2kf f þ/ðkf fkppÞ kf þ 5  104 b/qf cpf qkbdTp f ðT; /Þ

Remarks The term f ðT; /Þ is obtained based only on the experimental data of Das et al. [30] for CuO (dp ¼ 28:6 nm)-water nanofluid. The range of validity of f ðT; /Þ is; 1% < / < 4% and 300 < T < 325 K

p

f ðT; /Þ ¼ ð6:04/ þ 0:4705ÞT þ ð1722:3/  134:63Þ b ¼ 0:0011ð100/Þ0:7272 for CuO / > 1%

Prasher et al. [31]

b ¼ 0:0017ð100/Þ0:0841 for Al2O3 / > 1% h i ½k ð1þ2aÞþ2k þ2/½k ð1aÞk  kn ¼ ð1 þ ARem Pr0:333 /Þ ½kpp ð1þ2aÞþ2kmm /½kppð1aÞkmm kf qffiffiffiffiffiffiffiffiffiffi     bT a ¼ 2Rdbpkm ; km ¼ kf 1 þ 14 Re  Pr ; Re ¼ m1f p18k q dp

The thermal interface resistance Rb is not clearly defined [32,33]. The authors have assumed 2

Rb  0:77  108 km W1 for water,

p

Rb  1:2  10

8

2

km W1 for EG, and 2

Vajjha and Das [35]

kn ¼

h

kp þ2kf 2/ðkf kp Þ kp þ2kf þ/ðkf kp Þ

h i i qffiffiffiffiffiffiffiffi kf þ 5  104 b/qf cpf qkbdTp f ðT; /Þ p

f ðT; /Þ ¼ ð2:8217  102 / þ 3:917  103 Þ TT0 þ ð3:0669  102 /  3:91123  103 Þ b ¼ 9:881ð100/Þ0:9446 for CuO b ¼ 8:4407ð100/Þ1:07304 for Al2O3 b ¼ 8:4407ð100/Þ1:07304 for ZnO Chon et al. [22]

kn ¼ ð1 þ 64:7:/0:746 l

 0:369  0:746 df dp

kp kf

The correlation was obtained using Buckingham–Pi theorem and valid only for Al2O3–water nanofluids. The mean-free path lf was taken as 0.17 nm for water. The authors did not provide the precise value of molecular diameter of a base fluid. In the literature, one can found several values for molecular diameter of water; df ¼ 0:384 nm [36], df ¼ 0:272 nm [37].The correlation is valid for nanoparticle sizes ranging between 11 and 150 nm. For temperature, the related validity range is 294–344 k. Only two points of volume fractions have been investigated; 1% and 4%

Pr 0:9955 Re1:2321 Þkf

qk T

f b Pr ¼ q af f ; Re ¼ 3pl 2l f f f

Azmi et al. [25]

 1:37       dp 0:0336 ap 0:01737 / T 0:2777 kn ¼ kf 0:8938 1 þ 100 1 þ 70 1 þ 150 af / in %, T in °C and dp in nm

Corcione [23]



 10  0:03 kp kn ¼ kf 1 þ 4:4Re0:4 Pr 0:66 TTfr /0:66 k f

2q k T

Re ¼ plf 2 db f p

Patel et al. [24]



 0:273    0:234 k T 0:547 100 kn ¼ kf 1 þ 0:135 kp /0:467 20 dp f

Patel et al. [38]

  kp /df kn ¼ kf 1 þ kf ð1/Þd p

The main question is how to build up a correlation, which is able to combine all the aforementioned physical variables in only one expression? The answer to this question is the Vaschy– Buckingham theorem (Buckingham p-theorem) [27,28], which is one of the basic theorems for the dimensional analysis. The theorem states that if there is a physical equation with n number of physical variables which depend on k of fundamental units, then there is an equivalent equation involving n  k dimensionless parameters p1 , p2 ; . . . pnk constructed from the original variables.

Rb  1:9  108 km W1 for oil. For the empirical constants A and m, the authors assumed A ¼ 40; 000 for all types of nanofluids. However, the second constant m has been found to depend on the type of base fluid and diameter of the nanoparticles. Because of the need for curve-fitting parameters, A and m, Prasher’s model lacks generality [34] This model is the improved version of Koo and Kleinstreuer model [29]. The new empirical correlations for b and f ðT; /Þ are limited only for 60:40 EG/water, and water based fluid. The range of validity of the empirical correlations is; 298 < T < 363 K, and concentration range of 1% < / < 6% for CuO, 1% < / < 10% for Al2O3, and 1% < / < 7% for ZnO

The correlation is only for oxide nanoparticles incorporated in water based fluid. The correlation is valid for a volume concentration less than 4.0%, diameters in the range of 20—150 nm, and temperature of 293—343 K. [26] The correlation is for oxide and metal nanoparticles suspended in water or ethylene glycol based nanofluids. The correlation is based on experimental data with 1.86% standard deviation of error. The ranges of the nanoparticle diameter, volume fraction and temperature are 10  150nm, 0:002  0:09 and 294  324K, respectively. The correlation is based on a large experimental data for water, ethylene glycol and transformer oil based nanofluids only. The correlation is valid for nanofluids with spherical nanoparticles shapes of 10—150 nm diameter, thermal conductivity of 20  400W=mK, base fluid having thermal conductivity of 0:1—0:7 W=m K, volume fraction of 0:1 < / < 3% and temperature range from 293 to 323 K The model supposed to be a general tool to predict the thermal conductivity of CNT-Nanofluids. However, the model is not able to predict well at higher temperature of nanofluids [39]

By applying the Vaschy–Buckingham theorem to the physical parameters aforementioned, seven dimensionless p-groups are generated with four repetitive variables, kf , dp , v Br , and T. The resulting p-groups are given in Table 3. A generalized empirical correlation for the dimensionless thermal conductivity of the nanofluids, kn , normalized by the base fluid thermal conductivity, kf , is developed from the remaining p-group, as given in Eq. (2). The coefficients and exponents of the correlations are obtained by applying a non-linear regression analysis for the experimental data mentioned in Table 2

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S. Hassani et al. / International Journal of Heat and Mass Transfer 90 (2015) 121–130

Table 2 Database for the proposed correlations. References

Nanoparticles

dp (nm)

Base fluid

Volume fraction (%)

Temperature (K)

Number of data

[41] [41] [43] [43] [48] [44] [56] [55] [42] [22] [22] [22] [46] [22] [30] [50] [45] [52] [52] [52] [54] [55] [51] [53] [43] [43] [47] [56] [57] [40] [49] [49]

TiO2 TiO2 Al Al2O3 Cu Fe CNTs MWCNTs TiO2 Al2O3 Al2O3 Al2O3 Al2O3 Al2O3 CuO CuO CuO Ag Ag Ag MWCNTs MWCNTs Al2O3 Al2O3 Al Al CNTs CNTs Cu Cu Al2O3 SiO2

15 10 80 80 200 10 20 25 34 47 47 150 13 11 33 29 18 60 60 60 40 20 13 13 80 80 25 25 20 160 45 15

Ethylene glycol

1

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