Effect of particle size on the thermal conductivity of nanofluids containing metallic nanoparticles
 Pramod Warrier^{1} and
 Amyn Teja^{1}Email author
DOI: 10.1186/1556276X6247
© Warrier and Teja; licensee Springer. 2011
Received: 19 October 2010
Accepted: 22 March 2011
Published: 22 March 2011
Abstract
A oneparameter model is presented for the thermal conductivity of nanofluids containing dispersed metallic nanoparticles. The model takes into account the decrease in thermal conductivity of metal nanoparticles with decreasing size. Although literature data could be correlated well using the model, the effect of the size of the particles on the effective thermal conductivity of the nanofluid could not be elucidated from these data. Therefore, new thermal conductivity measurements are reported for six nanofluids containing silver nanoparticles of different sizes and volume fractions. The results provide strong evidence that the decrease in the thermal conductivity of the solid with particle size must be considered when developing models for the thermal conductivity of nanofluids.
Introduction
Recent interest in nanofluids stems from the work of Choi et al. [1] and Eastman et al. [2], who reported large enhancements in the thermal conductivity of common heat transfer fluids when small amounts of metallic and other nanoparticles were dispersed in these fluids. Others [3–9] have also reported large thermal conductivity enhancements in nanofluids containing metal nanoparticles, although the effect of particle size, in particular, was not studied explicitly in these experiments. In our previous work [10–15], we have reported data for the thermal conductivity of nanofluids containing metal oxide nanoparticles, and critically reviewed [15] these and other data to determine the effect of temperature, base fluid properties, and particle size on the thermal conductivity of the nanofluids. These studies have led us to the conclusion that the temperature dependence of the nanofluid thermal conductivity arises predominantly from the temperaturethermal conductivity behavior of the base fluid, and that the effective thermal conductivity of nanofluids decreases with decreasing size of dispersed particles below a critical particle size. We have also presented a model [15] based on the geometric mean of the thermal conductivity of the two phases to predict the thermal conductivity of the heterogeneous nanofluid. The model incorporated the size dependence of the thermal conductivity of semiconductor and insulator particles using the phenomenological relationship proposed by Liang and Li [16]. The resulting 'modified geometric mean model' was able to predict the thermal conductivity of nanofluids containing semiconductor and insulator particles dispersed in a variety of base fluids over an extended temperature range. In the present work, we propose a similar geometric mean model that incorporates the size dependence of the thermal conductivity of metallic particles.
Previous experimental studies of nanofluids containing metallic particles employed very low volume fractions (<1%) of these particles. As a result, any size dependence of the thermal conductivity of the nanofluid was not apparent from these measurements and the data could be correlated using the bulk thermal conductivities of the solid and base fluid. We have now measured the thermal conductivity of nanofluids containing several volume fractions of silver nanoparticles of three sizes, and fitted the data with a model that incorporates the size dependence of the thermal conductivity of the solid phase. We show that such a model provides a better representation of the data than models that assume a constant (bulk) thermal conductivity for metallic particles of different sizes.
The thermal conductivity of metallic nanoparticles
Properties of metals at 298.15 K [17]
k_{b}/W m^{1}K^{1}  μ_{F}/eV  n_{e}10^{28}/m^{3}  λ_{e,b}/nm  

Silver  424  5.51  5.85  49.10 
Copper  398  7  8.45  35.97 
Gold  315  5.5  5.9  36.14 
Geometric mean model for the thermal conductivity of nanofluids
where k_{eff}(L,T, φ) is the effective thermal conductivity of the nanofluid as a function of particle size (L), temperature (T), and particle volume fraction (φ), k_{l}(T) is the thermal conductivity of the base fluid as a function of temperature, and k_{P}(L,T) is the thermal conductivity of the particle as a function of particle size and temperature. In this work, we calculate k_{P}(L,T) using Equations 3 and 5 as discussed in "The thermal conductivity of metallic nanoparticles" section. Equation 6 is used to fit measurements of the thermal conductivity of nanofluids with n as the adjustable parameter.
Thermal conductivity of nanofluids
Evaluation of the modified geometric mean thermal conductivity model
Size indep.  Size dep.  

Particle  Fluid  φ/% v/v  T /K  L /nm  Data Ref.  AAD  N  AAD  n 
Ag  Water  14 × 10^{1}  298  15  [3]  0.40  0.38  0.40  0.55 
Ag + citrate  Water  1 × 10^{3}  303333  70  [4]  2.99  1.00  3.25  1.00 
Cu  EG  13 × 10^{1}  298  10  [2]  5.24  0.60  5.40  0.82 
Cu  Water  2.57.5  298  100  [5]  2.15  0.06  2.10  0.08 
Cu  PFTE  225 × 10^{1}  298  26  [6]  3.47  0.14  3.45  0.19 
Cu  EG  35 × 10^{1}  278323  7.5  [7]  7.07  0.39  6.75  0.61 
Cu  Water  530 × 10^{2}  298  42.5  [8]  1.61  0.81  1.56  0.92 
Cu  Water  29 × 10^{3}  298  25  [9]  6.27  0.77  6.24  0.93 
Au + thiolate  Toluene  511 × 10^{3}  299333  3.5  [4]  0.77  0.81  2.60  1.00 
Au + citrate  Water  1.32.6 × 10^{3}  303333  15  [4]  5.19  1.00  5.25  1.00 
Experimental
The thermal conductivity of each nanofluid was measured using a liquid metal transient hotwire device. The transient hotwire method has been used successfully in our laboratory to measure the thermal conductivity of electrically conducting liquids [23] and nanofluids [10–14] over a broad range of temperatures. Our transient hotwire device consists of a glass capillary, filled with mercury, and suspended vertically in the nanoparticle dispersion in a cylindrical glass cell. The glass capillary insulates the mercury 'wire' from the electrically conducting dispersion, and prevents current leakage when a voltage is applied to the 'wire'. The 'wire' is heated by application of a voltage and its resistance is measured using a Wheatstone bridge circuit with the 'wire' forming one arm of the circuit. The temperature change of the wire is computed from the resistance change of the mercury 'wire' with time. The data are used to calculate the effective thermal conductivity of the nanofluid via an analytical solution of Fourier's equation for a linear heat source of infinite length in an infinite medium. This solution predicts a linear relationship between the temperature change of the wire and the natural log of time, and this is used to confirm that the primary mode of heat transfer during the measurement is conduction. Corrections to the temperature are included for the insulating layer around the wire, the finite dimensions of the wire, the finite volume of the fluid, and heat loss due to radiation. The thermal conductivity is obtained from the slope of the corrected temperaturetime line using the length of the mercury 'wire' in the calculation. An effective length of the wire that corrects for nonuniform capillary thickness and end effects is obtained by calibration with two reference fluids. In the present study, water and dimethyl phthalate were used as the reference fluids [24] and their properties were obtained from the literature [25]. Additional details of the apparatus and method are available elsewhere [23]. The experiment was performed five times for each sample and condition, and a data point reported in this work thus represents an average of five measurements with an estimated error of ±2%.
Results
Thermal conductivity of nanofluids consisting of silver nanoparticles dispersed in ethylene glycol
T/K  φ/% v/v  d/nm  k_{EG}/W m^{1}K^{1}[25]  k_{P}/W m^{1}K^{1}  k_{eff}/W m^{1}K^{1}  Standard deviation in k_{eff} 

299.3  1  20  0.2544  123.49  0.2700  0.0052 
299.9  1  3050  0.2544  191.32  0.2701  0.0025 
298.4  1  80  0.2544  263.50  0.2798  0.0023 
300.8  2  20  0.2544  123.49  0.3048  0.0029 
300.9  2  3050  0.2544  191.32  0.2907  0.0023 
300.5  2  80  0.2544  263.50  0.3089  0.0033 
Conclusions
A phenomenological model is presented for the thermal conductivity of metallic nanofluids that takes account of the size dependence of the thermal conductivity of metallic particles. The model was able to fit literature data for nanofluids using one adjustable parameter, although values of the fitted parameter were higher than expected. The thermal conductivity of nanofluids containing three sizes of silver nanoparticles dispersed in EG was measured and the data were fitted using our model. The results are in agreement with our previous work on nanofluids containing semiconductor or insulator particles, and appear to confirm that the thermal conductivity of silver nanofluids decreases with decreasing particle size.
Abbreviations
 CNT:

carbon nanotubes
 EG:

ethylene glycol
 PVP:

polyvinylpyrrolidone.
Declarations
Authors’ Affiliations
References
 Choi SUS, Zhang ZG, Yu W, Lockwood FE, Grulke EA: Anomalous thermal conductivity enhancement in nanotube suspensions. Appl Phys Lett 2001, 79: 2252. 10.1063/1.1408272View Article
 Eastman JA, Choi SUS, Li S, Yu W, Thompson W: Anomalously increased effective thermal conductivities of ethylene glycolbased nanofluids containing copper nanoparticles. Appl Phys Lett 2001, 78: 718. 10.1063/1.1341218View Article
 Kang HU, Kim SH, Oh JM: Estimation of thermal conductivity of nanofluid using experimental effective particle volume. Exp Heat Transfer 2006, 19: 181. 10.1080/08916150600619281View Article
 Patel HE, Das SK, Sundararajan T, Nair AS, George B, Pradeep T: Thermal conductivity of naked and monolayer protected metal nanoparticles based nanofluids: manifestation of anomalous enhancement and chemical effects. Appl Phys Lett 2003, 83: 2931. 10.1063/1.1602578View Article
 Xuan Y, Li Q: Heat transfer enhancement of nanofluids. Int J Heat Fluid Flow 2000, 21: 58. 10.1016/S0142727X(99)000673View Article
 Li Q, Xuan Y: Enhanced heat transfer behaviors of new heat carrier for spacecraft thermal management. J Spacecraft Rockets 2006, 43: 687. 10.2514/1.15554View Article
 Yu W, Xie H, Chen L, Li Y: Investigation on the thermal transport properties of ethylene glycolbased nanofluids containing copper nanoparticles. Powder Technol 2010, 197: 218. 10.1016/j.powtec.2009.09.016View Article
 Jana S, SalehiKhojin A, Zhong WH: Enhancement of fluid thermal conductivity by the addition of single and hybrid nanoadditives. Thermochim Acta 2007, 462: 45. 10.1016/j.tca.2007.06.009View Article
 Li XF, Zhu DS, Wang XJ, Wang N, Gao JW, Li H: Thermal conductivity enhancement dependent pH and chemical surfactant for CuH2O nanofluids. Thermochim Acta 2008, 469: 98. 10.1016/j.tca.2008.01.008View Article
 Beck MP, Sun T, Teja AS: The thermal conductivity of alumina nanoparticles dispersed in ethylene glycol. Fluid Phase Equilibr 2007, 260: 275. 10.1016/j.fluid.2007.07.034View Article
 Beck MP, Yuan Y, Warrier P, Teja AS: The effect of particle size on the thermal conductivity of nanofluids. J Nanopart Res 2009, 11: 1129. 10.1007/s1105100895002View Article
 Beck MP, Yuan Y, Warrier P, Teja AS: The thermal conductivity of alumina nanofluids in water, ethylene glycol, and ethylene glycol + water mixtures. J Nanopart Res 2009, 12: 1469. 10.1007/s1105100997169View Article
 Beck MP, Yuan Y, Warrier P, Teja AS: The thermal conductivity of aqueous nanofluids containing ceria nanoparticles. J Appl Phys 2010, 107: 066101. 10.1063/1.3330506View Article
 Beck MP, Yuan Y, Warrier P, Teja AS: The limiting behavior of the thermal conductivity of nanoparticles and nanofluids. J Appl Phys 2010, 107: 114319. 10.1063/1.3330506View Article
 Warrier P, Yuan Y, Beck MP, Teja AS: Heat Transfer in Nanoparticle Suspensions: Modeling the Thermal Conductivity of Nanofluids. AICHE J 2010, 56: 3243. 10.1002/aic.12228View Article
 Liang LH, Li B: Sizedependent thermal conductivity of nanoscale semiconducting systems. Phys Rev B 2006, 73: 153303. 10.1103/PhysRevB.73.153303View Article
 Zhang ZM: Nano/Microscale Heat Transfer. McGraw Hill Nanoscience and Nanotechnology Series, New York; 2007.
 Nath P, Chopra KL: Thermal conductivity of copper films. Thin Solid Films 1974, 20: 53. 10.1016/00406090(74)900339View Article
 Landau LD, Lifshitz EM: Electrodynamics of Continuous Media. Oxford: Pergamon Press; 1960. Translated by J. B. Sykes and J. S. Bell
 Turian RM, Sung DJ, Hsu FL: Thermal conductivity of granular coals, coalwater mixtures and multisolid/liquid suspensions. Fuel 1991, 70: 1157. 10.1016/00162361(91)902375View Article
 Nan CW: Physics of inhomogeneous inorganic materials. Prog Mater Sci 1993, 37: 1. 10.1016/00796425(93)900045View Article
 Maxwell JC: A Treatise on Electricity and Magnetism. London: Oxford University Press; 1892.
 Bleazard JG, Teja AS: Thermal conductivity of electrically conducting liquids by the transient hotwire method. J Chem Eng Data 1995, 40: 732. 10.1021/je00020a003View Article
 Marsh KN, (Ed): Recommended Reference Materials for the Realization of Physicochemical Properties. Boston: Blackwell Scientific Publications; 1987.
 Rowley RL, Wilding WV, Oscarson JL, Yang Y, Giles NF:DIPPR® Data Compilation of Pure Chemical Properties. Provo, Utah: Brigham Young University; 2010. Design Institute for Physical Properties. [http://dippr.byu.edu] Design Institute for Physical Properties.
 Prasher R, Evans W, Meakin P, Fish J, Phelan P, Keblinski P: Effect of aggregation on thermal conduction in colloidal nanofluids. Appl Phys Lett 2006, 89: 143119. 10.1063/1.2360229View Article
 Kumar S, Murthy JY: A numerical technique for computing effective thermal conductivity of fluidparticle mixtures. Numer Heat Transf B Fundam 2005, 47: 555. 10.1080/10407790590928937View Article
 Gao L, Zhou XF: Differential effective medium theory for thermal conductivity in nanofluids. Phys Lett A 2006, 348: 355. 10.1016/j.physleta.2005.08.069View Article
 Eapen J, Li J, Yip S: Beyond the Maxwell limit: Thermal conduction in nanofluids with percolating fluid structures. Phys Rev E 2007, 76: 062501. 10.1103/PhysRevE.76.062501View Article
Copyright
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.