Journal of Nanofluids Vol. 4, pp. 302–309, 2015 (www.aspbs.com/jon)
Copyright © 2015 by American Scientific Publishers All rights reserved. Printed in the United States of America
Effect of Nanoparticle Size, Morphology and Concentration on Specific Heat Capacity and Thermal Conductivity of Nanofluids S. A. Angayarkanni, Vijutha Sunny, and John Philip∗
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SMARTS, Metallurgy and Materials Group, Indira Gandhi Center for Atomic Research, Kalpakkam 603102, Tamilnadu, India
The effect of particle size, particle morphology and volume fraction of nanoparticles on the temperature dependent specific heat capacity of metal oxide nanofluids is investigated using differential scanning calorimeter. The stable colloidal suspensions of kerosene based magnetite (Fe3 O4 ), polyalphaolefin (PAO) based alumina (Al2 O3 ) spheres and alumina nanorods are used in the present studies. The nanoparticle concentrations and size of Fe3 O4 nanoparticles are varied from 5 to 25 wt% and 3.6 to 8.6 nm, respectively. The results show that the specific heat capacity decreases with increase in volume fraction and particle size in kerosene based Fe3 O4 nanofluids but enhances in the case of PAO based Al2 O3 nanofluids. These results suggest that PAO molecules strongly modify the interfacial thermal characteristics of Al2 O3 nanoparticles that in turn increases the heat capacity of PAO based Al2 O3 nanofluids. For kerosene based nanofluids, the Cp data was in reasonable agreement with theoretical model for specific heat, which is derived by assuming thermal equilibrium between the particles and the surrounding fluid (Model II) using classical and statistical mechanics but showed large deviation from the mixing model (Model I). Our study shows that the Cp decreases with increase in the aspect ratio of nanoparticles due to reduced surface atomic contributions. We also compare the effect of particle size and surface morphologies on the thermal conductivity enhancement of nanofluids. KEYWORDS: Nanofluids, Heat Capacity, Thermal Properties, Aggregation, Heat Transfer.
1. INTRODUCTION Nanofluids have been a topic of intense research during the last one decade due to their interesting thermophysical properties and anticipated applications in heat transfer.1–12 Several factors affect the heat transfer properties of nanofluids that include thermal conductivity, viscosity, heat capacity, density and diffusion coefficient. For effective utilization of nanofluids for practical heat transfer applications, a synergetic balance of these thermophysical properties is a prerequisite. Though the thermal properties of nanofluids are investigated intensely, the studies on heat capacity of pure nanofluids are scarce.13–15 A knowledge of specific heat capacity and thermal conductivity is important in order to understand the heat transfer properties of nanofluids. Nanofluids with higher heat capacities are necessary to enhance heat transfer efficiency at lower operating costs. Thermal diffusivity (D), thermal conductivity (k)
∗
Author to whom correspondence should be addressed. Email:
[email protected] Received: 27 March 2015 Accepted: 15 April 2015
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and specific heat capacity Cp are inter-related as Cp = k/D, where is the density. Unlike solids and gases, the accurate determination of heat transfer parameters in nanofluids is complex. So far, thermal conductivity studies have been the main focus of nanofluid research.16 An increase of thermal conductivity is necessary but not a sufficient condition for achieving high performance in heat exchange equipments. The specific heat capacity determines the rate at which the substance will heat up or cool down.17 As the heat capacity of a material is directly related to the atomic structure, measurements of heat capacity as a function of temperature could provide structural properties of nanostructured materials.18 In ultrafine particles, both bulk and surface phonon mode contributes to the vibrational specific heat. As the surface area to volume ratio increases, the surface mode contribution plays a prominent role over the bulk modes. Studies show that the specific heat in nanocrystals increases upto 20% under an increase of atomic-level strain due to changes in the vibrational and configurational entropy due to large anharmonic atomic vibrations at grain boundaries and disordered lattice sites.19 2169-432X/2015/4/302/008
doi:10.1166/jon.2015.1167
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Effect of Nanoparticle Size, Morphology and Concentration on Specific Heat Capacity and Thermal Conductivity
There are several studies which report the strong and weak dependence of Cp on the volume fraction of nanoparticles. Yang et al. reported that at low volume fraction and moderate temperature change, the heat capacity of nanofluid doesn’t change much when compared to the base fluid.20 Several studies report a decrease in the heat capacity with increase in volume fraction.14 16 21 Yang et al.22 reported an increase in specific heat capacity with the increase in temperature. Unlike large thermal conductivity contrast between the suspended particles and base fluids, the Cp contrast between them is not significant and hence significant variation of Cp on addition of particle loading is not anticipated. The objective of the present work is to systematically study the effect of volume fraction, properties of dispersed particle (density, thermal conductivity, morphology etc.) on the specific heat capacity of nanofluids for a better understanding of their utility in heat transfer applications such as concentrating solar power and waste heat recovery.
2. EXPERIMENTAL DETAILS
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2.2. Measurements of Specific Heat Capacity In differential scanning calorimetry (DSC), the measured heat flow is directly proportional to the specific heat capacity. The Cp can be calculated from the DSC signal with blank corrections and by keeping a sample and a blank. The specific heat measurements are done using a differential scanning calorimetry as per the ASTM standard (E 1269-05)26 27 by comparing the heat flow of the sample with that of a sapphire reference. The instrument was initially calibrated with indium, zinc and aluminum. The Cp was determined from the heat flow data using the Stare software of Mettler Toledo TGA/DSC instrument. The specific heat is measured by the direct comparison with a standard to reduce the effects of instrumental variation.28
3. RESULTS AND DISCUSSION 3.1. XRD, DLS, TEM and TGA Figures 1(a) and (b) show the XRD pattern of Al2 O3 (sphere) and Fe3 O4 nanoparticles, respectively. The diffraction peaks of (120), (031), (051), (151), and (002) were indexed to an orthorhombic structure of Al2 O3 (JCP DS card no: 21-1307). The diffraction peaks of (220), (311), (400), (422), (511), and (440) were indexed to a cubic spinel structure of Fe3 O4 (JCP DS card no: 89-3854). An inverse spinel structure consists of oxide ions that form a cubic close-packed arrangement in which 1/3 of tetrahedral interstices and 2/3 of octahedral interstices coordinate with oxygen. The parallel alignment of Fe2+ and Fe3+ ions spin in adjacent octahedral sites leads to a net magnetization and thus a ferri-magnetic behavior.29 303
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2.1. Materials and Methods For the present study, kerosene based Fe3 O4 and PAO based Al2 O3 nanofluids are used. The Fe3 O4 nanofluids are prepared in our laboratory using coprecipitation approach. The synthesis procedure is briefly discussed below. Magnetic fluids composed of magnetite (Fe3 O4 ) nanoparticles are prepared by a chemical coprecipitation technique,23–25 coated with oleic acid and dispersed in kerosene. The freshly prepared iron salt solutions of 0.2 M FeCl2 · 4H2 O and 0.4 M FeCl3 · 6H2 O were mixed at 1:1 ratio at a constant stirring speed of 1000 rpm. After the addition of ammonia solution, the pH of the solution reached a value of 10. When the solution turned to black, 20 mL of oleic acid was added and the dispersion pH adjusted to 9.5 with the temperature increased to 70 C. At the same pH, temperature and stirring speed, the solution was kept for 30 min to finish the coating process. After this step, the temperature was increased to 79 C in order to eliminate the excess ammonia and the protonation of adsorbed and un-adsorbed ammonium oleate as oleic acid. The surfactant coated particles were washed with triply distilled water at 60 C, until the pH became 7 to remove the ionic impurities and later dispersed in hexane. The surfactantcoated magnetite nanoparticles were dried at 35 C for 48 hours in inert atmosphere and the dried particles were dispersed in kerosene. The Fe3 O4 nanoparticles with different particle size are synthesized by changing the solvent polarity during chemical coprecipitation. The Fe3 O4 nanoparticles having different sizes are synthesized in a mixed solvent of ethanol and water at four different ratios of 20:80; 40:60, 50:50 and 60:40. The Al2 O3 nanoparticles dispersed in PAO were provided by Sasol and used as such. The Al2 O3 nanorods (NR) and nanospheres (NS) dispersed in polyalphaolefin
(PAO) were stabilized using a surfactant solsperse 21000 (Lubrizol Chemical). Fe3 O4 nanoparticles of average particle size 8 nm and Al2 O3 nanospheres (hereafter referred as AlNS) of size 10 nm were used to study the effect of particle volume fraction on the specific heat capacity. The effects of particle morphology on specific heat capacity of nanofluids were studied using Al2 O3 nanospheres of size 10 nm and nanorods with an aspect ratio of 8. The effect of particle size on specific heat capacity of nanofluids was studied using Fe3 O4 nanoparticle of size varying from 2.3–8.5 nm. The Al2 O3 and Fe3 O4 nanoparticles were characterized for their phase identity by X-ray diffraction using Rigaku Ultima IV X-ray diffractometer. The crystallite size (d) was calculated using the Scherrer formula, d = k/ cos , where, k = 0 89, = 1 5418 Å, is the peak width at half height (FWHM) of the highest peak and is the half of the diffraction angle. The size distribution of nanoparticles was determined by dynamic light scattering using a Zetasizer-Nano (Malvern Instrument). The thermogravimetric analysis (TGA) of nanofluids was carried out using a Mettler Toledo TGA/DSC system under inert atmosphere from 40–100 C at a heating rate of 2 C/min.
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2θ (degrees)
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Figure 2(a) shows the hydrodynamic diameter of the PAO based Al2 O3 and kerosene based Fe3 O4 nanofluids. The average hydrodynamic sizes are found to be 10 and ∼8 nm for Al2 O3 and Fe3 O4 nanoparticles, respectively.
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(a) Al2O3/PAO Fe3O4/kerosene
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Fig. 1. XRD patterns of (a) Al2 O3 nanoparticles and (b) Fe3 O4 Nanoparticles.
(b)
20
The size distribution is found to be narrow for both Al2 O3 and Fe3 O4 nanofluids. The time dependent hydrodynamic measurements carried out in these systems showed a time independent hydrodynamic size, which indicate the good stability of the dispersions used.30 Figure 2(b) shows the hydrodynamic size distribution of Fe3 O4 nanoparticles of different sizes synthesized with different solvent dielectric constants. The hydrodynamic diameters of particles prepared with ethanol water ratio of 20:80; 40:60, 50:50 and 60:40 are found to be 3.6, 4.8, 6.5 and 8 6 ± 0 8 nm, respectively. Figure 3 shows the TEM images of the (a) Fe3 O4 31 (b) Al2 O3 nano spheres1 and (c) Al2 O3 nano rods1 nanoparticles. The thermogravimetric curve for PAO based Al2 O3 nanofluid (Fig. 4(a)) shows a single step weight loss at ≈468 C (∼88 wt%) due to the decomposition of PAO. The TGA curve for the magnetite nanoparticles shows a two-step weight loss between 199–310 C (∼4 wt%) and 310–400 C (∼12 wt%) (Fig. 4(b)). The first step (Solid arrow) is due to the removal of loosely bound or free surfactant and the second step (Dotted arrow) should be due to the removal of strongly bound surfactant molecules.23 3.2. Effect of Particle Volume Fraction, Particle Size and Particle Shape on Specific Heat Capacity of Nanofluids 3.2.1. Effect of Volume Fraction of Nanoparticles on the Specific Heat Capacity of Nanofluids The variation of absolute specific heat capacity with temperature for different concentrations and the ratio of the specific heat capacity of nanofluid with respect to the Cp of base fluid for kerosene based nanofluids are shown in Figures 5(a) and (b), respectively. Figure 5 shows a marginal increase in the specific heat capacity with increase in temperature. As the nanoparticle concentration increases, the specific heat capacity of the kerosene based Fe3 O4 nanofluid is found to decrease. In general, the specific heat capacity of nanofluids can be explained using two models. The first model I, which is similar to the mixing theory for ideal gas mixtures,32 calculates the specific heat of nanofluids (Cp nf by averaging the Cp of the base fluid (Cp bf , the Cp of nanoparticle (Cp np ) and its volume fraction (). According to model I, Cp nf = Cp np + 1 − Cp bf
10
0 1
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Size (nm) Fig. 2. (a) Volume averaged hydrodynamic size distribution of Fe3 O4 and Al2 O3 (sphere) nanoparticles. (b) Volume averaged size distribution of Fe3 O4 nanoparticles of different size, prepared with different ethanol water ratio.
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(1)
This equation has been used for the evaluation of heat transfer behavior of nanofluids in many experiments33 34 and also in theoretical simulations.35–37 In Model II, Cp nf is derived by assuming thermal equilibrium between particles and the surrounding fluid, using classical and statistical mechanics.16 According to model II, cp nf =
cp n + 1 − cp f n + 1 − f
(2)
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TEM images of (a) Fe3 O4 (b) Al2 O3 nano spheres and (c) Al2 O3 nano rods nanoparticles.
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Temperature [°C] Fig. 4. Thermogravimetric curves for (a) PAO based Al2 O3 and (b) oleic acid coated Fe3 O4 nanoparticle. J. Nanofluids, 4, 302–309, 2015
Fig. 5. (a) The variation of specific heat capacity with temperature for different concentration (wt%) of kerosene based Fe3 O4 nanofluids containing nanoparticles of average size ∼8 nm. b) Specific heat capacity ratio Cp nf /Cp bf as a function of temperature for different concentration of Fe3 O4 nanoparticles.
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where mnf and Vnf are the mass and volume of nanofluid. The specific heat of the bulk and nanosized Fe3 O4 are 0.6512 and 0.79 J/g · C, respectively.38 The specific heat capacity of pure kerosene is 2.01 J/g · C.39 According to the theoretical model I, the effective specific heat of the nanofluid should be lower than the specific heat of pure kerosene. Here, the lower specific heat of nanoparticles in the base fluid results in a decrease in specific heat capacity of the nanofluid with increase in particle concentration. An earlier study showed a similar decrease in the heat capacity of nanofluid with increase in volume concentration and an increase with temperature in kerosene based iron oxide nanofluids.40 Such a behavior was also observed in CuO nanofluids where the constrained liquid layering at the surface of nanoparticle free boundary is attributed as the cause for the variations in the Gibbs free energy and change in specific heat capacity. Studies on the effect of volume fraction on the specific heat
capacity of water-based Al2 O3 nanofluid showed that the specific heat capacity decreases with increase in particle concentration.16 Studies also conclude that the solid–liquid interfacial conformation plays a critical role in the heat capacity changes because of the changes in the phonon vibration mode at interface.15 Murshed et al.41 reported that the thermal diffusivity of nanofluids increases significantly with increasing volume fraction of nanoparticles in the case of TiO2 , Al2 O3 and aluminum nanoparticles dispersed in ethylene glycol and engine oil. Their study revealed that the particle size and shape can influence the effective thermal diffusivity of nanofluids. Zhang et al.42 observed an increase in the thermal diffusivity with particle concentration in Au/toluene, Al2 O3 /water and CNT/water nanofluids. Figure 6 shows the variation of Cp nf of kerosene based Fe3 O4 nanofluids along with theoretical fit using Model I and Model II at different temperatures. From the Figure 6 shows that the experimental data fits better with the Model II than Model I. Similar agreement with model II was
Cp (J/g . ºC)
where n and f are the particle and fluid densities, respectively and the nanofluid density nf is given by m nf = nf = n + 1 − f (3) Vnf
Cp, nf /Cp, bf
Fig. 3.
Effect of Nanoparticle Size, Morphology and Concentration on Specific Heat Capacity and Thermal Conductivity
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for the unusual enhancement of the specific heat capacity are: (1) Enhanced specific heat capacity of nanoparticle due to higher specific surface energy (compared with bulk material); (2) Additional thermal storage mechanisms due to interfacial interactions (e.g., such as interfacial thermal resistance and capacitance) between nanoparticle and the adhering liquid molecules because of the extremely high specific surface area of the nanoparticle; and (3) The existence of semi-solid liquid layer adhering to the nanoparticles, which are likely to enhance the specific heat capacity due to the smaller inter-molecular spacing similar to the nanoparticle lattice structure on the surface (compared to the higher inter-molecular spacing in the bulk liquid).
observed by Zhou et al.16 in water based Al2 O3 nanofluid and Higano et al.40 in kerosene based Fe3 O4 nanofluid. Figures 7(a) and (b) shows the variation of specific heat capacity with temperature for different concentration of PAO based Al2 O3 nanofluids and the variation of normalized specific heat capacity, respectively. Surprisingly, the specific heat capacity of Al2 O3 /PAO increases upto 4 wt% nanoparticle concentration. Further increase in Al2 O3 concentration led to a decrease in the Cp nf but larger than the Cp bf . These results suggest that PAO molecules strongly modify the interfacial thermal characteristics of Al2 O3 nanoparticles that increases the heat capacity of PAO based Al2 O3 nanofluids. In general, the mechanisms considered PAO 2 wt% Al2O3 4 wt% Al2O3 6 wt% Al2O3 8 wt% Al2O3
Cp(J/(g . ºC))
2.0 1.8 1.6
10 wt% Al2O3 12 wt% Al2O3
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1.4 1.2 (b)
Cp, nf/Cp, bf
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Fig. 6. Variation of Cp as a function of volume fraction of kerosene based Fe3 O4 nanofluid at different temperatures along with Model I (dotted line) and Model II (solid line) fits. The corresponding weight % is shown in the top x-axis.
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Temperature (ºC) Fig. 7. (a) Specific heat capacity of alumina nanofluids with different weight percentage of Al2 O3 nanoparticles in PAO and (b) Specific heat capacity ratio (Cp nf /Cp bf ) of alumina nanofluid with respect to PAO.
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Nelson et al.43 reported a specific heat capacity enhancement of 50% in 0.6% exfoliated graphite nanoparticle fibers suspended in polyalphaolefin. The specific heat capacity of Al2 O3 nanoparticle was found to increase up to 25% compared with the bulk value of Al2 O3 .18 A volumetric heat capacity (Cv ) enhancement with increase in temperature in Al2 O3 and ZnO dispersed in 60:40 by mass of ethylene glycol/water and in water based SiO2 nanofluids is reported.32 The enhancement of specific heat capacity of the silica nanocomposite is attributed to the high surface energy of nanoparticles that induces a phase transformation in the solvent material (liquid ordering) in the vicinity of the nanoparticle. Such ordering of liquids on nanoparticles is expected to have semi-solid properties that gives rise to an increase in specific heat capacity of nanofluids.13 44 The heat capacities of ionic salts are shown to increase with molar mass due to an increase in number of translational, vibrational and rotational modes.45 3.2.2. Effect of Nanoparticle Size on the Specific Heat Capacity of Nanofluids The variation of specific heat capacity and the normalized specific heat capacity of nanofluids with temperature with respect to particle size is shown in Figures 8(a) and (b). From the Figure 8 it is clearly seen that the normalized specific heat capacity decreases with increase in particle size. A similar trend was reported by Xiong et al.46 for Ag nanoparticles of size ranging from 5 to 50 nm suggesting that the enhanced Cp in smaller sized nanoparticle is caused by the larger atomic thermal vibrational energies of surface atoms. Wang et al.15 reported that the specific heat capacity of nanoparticle increases with decrease in size of the particle in the case of CuO nanoparticle of size ranging from 3.4 to 50 nm. The major contribution to the specific heat capacity above ambient temperature is determined by the vibrational degrees of freedom, and the peculiarities of surface phonon spectra of nanoparticles are responsible for the anomalous behavior of specific heat capacity. The vibrational states J. Nanofluids, 4, 302–309, 2015
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Fig. 8. (a) Specific heat capacity of kerosene based Fe3 O4 nanofluids (35 wt%) with different average particle sizes 3.6, 4.8, 6.5 and 8.6 nm and (b) the corresponding specific heat capacity ratio Cp nf /Cp bf .
This means more heat is required to raise the temperature of surface atoms by one degree. Therefore, the observed larger specific heat capacity of nanospheres than that of nanorods is in good agreement with surface atomic consideration. This study hence shows that the particle morphology can also plays a major role in the specific heat capacity of nanofluids. The observed reduction in Cp with increasing aspect ratio of nanoparticles is in sharp contrast with the increase in thermal conductivity in several nanofluids.51–59 The key findings from our thermal conductivity and Cp studies along with the literature are listed in Table I. The schematic representation of the effect of particle size and morphology on the specific heat capacity and thermal conductivity of the nanofluid is shown in Figure 10.
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2.1
Cp(J/g . ºC)
3.2.3. Effect of Shape of Nanostructures on the Specific Heat Capacity of Nanofluids Figure 9 shows the variation of Cp as a function of temperature for PAO based Al2 O3 with different particle morphologies. A concentration of 0.01 wt% was used in the case of spherical Al2 O3 and rod shaped Al2 O3 in PAO. For both the nanofluids, Cp increases with temperature. At the same concentration (0.01 wt%), the specific heat capacity of nanofluid containing spherical particles was more compared to that of rod shaped Al2 O3 nanoparticles. In the present case, the aspect ratio of the nanorods was ∼8 (80 × 10 nm2 ), while the nanospheres diameter was 10 nm. The surface area to volume ratio for nanorods is 425 × 106 while that of nanospheres (diameter 10 nm) is 600 × 106 . Therefore, the surface atoms are more in the case of nanospheres compared to that of nanorods.
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Fig. 9. Variation of Cp as a function of temperature for PAO based Al2 O3 nanofluids with spherical and rod shaped particles of the same concentration.
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with lower frequencies make a larger contribution to heat capacity.47 Studies on effect of nanoparticle size on specific heat capacity by Dudda et al.48 showed 8, 12 , 19 and 27% enhancement of specific heat capacity for the SiO2 nanoparticle size of 5, 10, 30 and 60 nm, respectively. Novotny et al.49 reported a decrease in the specific heat capacity with increase in the nanoparticle size which is attributed to the absence of low frequency modes and the increase in the specific heat capacity at low temperatures was attributed to the additional vibrational mode of surface phonon. More heat is required to raise the temperature of surface atoms because of larger vibrational amplitude of surface atoms, compared to the bulk atoms.50 A study on high-energy ball milled Ruthenium (Ru) and AluminumRuthenium (AlRu) nanoparticle of size ranging from 7–12 nm, with large atomic-level strain, showed a specific heat increase of ∼20%, which was attributed to the large changes in the vibrational and configurational part of the entropy.19
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Table I. The key findings from our thermal conductivity and Cp studies (present work). Cpnf /Cp bf Particle Effect of particle size
Cu Fe3 O4 /kerosene (present work) Al2 O3 /water
k/kf
Smaller d
Larger d
Smaller d
Larger d
1.04 (5 nm)46 0.99 (3.6 nm)
1 (50 nm)46 0.84 (8.6 nm)
– 1.05 (2.8 nm)60 61
– 1.25 (9.5 nm)60 61
–
–
– –
– –
1.036 (12 nm)62 1.14 (13 nm)11 1.088 (12 nm)62 1.11 (10 nm)63
Al2 O3 /EG SiO2 /water
Cp nf /Cp bf
Effect of particle morphology
Al2 O3 /PAO (present work) TiO2 /water Fe3 O4 /kerosene
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Fe3 O4 /kerosene (present work) Al2 O3 /PAO (present
(282 nm)62 (24 nm)11 (282 nm)62 (60 nm)63
k/kf
Spherical
Non-spherical
Spherical
Non-spherical
1.98 (0.01 wt%)
1.81 (0.01 wt%)
1.051
1.21
– –
– –
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1.3364 3.33
Cp nf /Cp bf
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1.115 1.19 1.132 1.13
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Low (wt%)
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1.06 (5 wt%)30
1.125 (25 wt%)30
1.109 (2 wt%)
0.99 (12 wt%)
1.051
1.151
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(b) Larger k
Larger Cp
Larger D (nm)
Smaller k
Smaller D (nm)
Smaller Cp
work)
Non- Spherical
Spherical
Smaller Cp
Larger Cp
Smaller k
Larger k
(d)
(c)
Fig. 10. Schematic representation of the effect of (a) smaller d, (b) larger d, (c) spherical particle and (d) non spherical particle on the specific heat capacity and thermal conductivity of the nanofluid.
4. CONCLUSIONS We studied the thermophysical properties of nanofluids with different types of nanoparticles as a function of concentration, shapes, size and base fluids. The present work concludes that the nanoparticle concentration, shape and size play a major role in specific heat capacity of nanofluid. In the Fe3 O4 nanofluids, the specific heat capacity is found to decrease with increasing particle concentration while an increase in the specific heat 308
capacity is observed in Al2 O3 /PAO based nanofluid. This suggests that PAO molecules strongly modify the interfacial thermal characteristics of Al2 O3 nanoparticles. The size dependent studies on specific heat capacity suggest that the Cp increases with decreasing dispersed nanoparticle size. Our study shows that the particle morphology also plays a major role in specific heat capacity of nanofluids and the Cp decreases with increasing aspect ratio of nanoparticles due to reduced surface atomic contributions. Symbol Symbol D k Cp Cp nf Cp bf n f d mnf Vnf
Details Thermal diffusivity Thermal conductivity Specific heat capacity Specific heat capacity of nanofluid Specific heat capacity of basefluid Density Density of the nanoparticle Density of the base fluid Wavelength of X-ray source Crystallite size Full width at half maximum Half of the diffraction angle Volume fraction of the nanoparticle Mass of the nanofluid Volume of the nanofluid.
Acknowledgments: The authors thank Dr. T. Jayakumar and Dr. P. R. Vasudeva Rao for fruitful J. Nanofluids, 4, 302–309, 2015
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Effect of Nanoparticle Size, Morphology and Concentration on Specific Heat Capacity and Thermal Conductivity
discussions. John Philip thanks the Board of Research Nuclear Sciences (BRNS) for support through a research grant for the advanced nanofluid development program.
References and Notes
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