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
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