Introduction

The number of important and continuous advances in nanotechnology during the last years have led to the emergence of a new generation of heat transfer fluids called nanofluids, in which nanometer-sized particles with high thermal conductivity are dispersed and suspended in a base liquid. Nanofluids have great potential to improve the heat transfer and energy efficiency in a variety of applications. Thermal properties of nanofluids, including thermal conductivity, viscosity, specific heat capacity, convective heat transfer coefficient, and critical heat flux have been extensively studied [1–31].

One of the most outstanding features of nanofluids is the increase in thermal conductivity compared to that of the base fluid, even in the case of very small particle volume fractions. Experimental studies show that thermal conductivity of nanofluids depends on many factors such as particle material, particle size and shape, particle volume fraction, agglomeration of particles, base fluid material, pH value, temperature, and additives.

There are two kinds of methods for thermal conductivity measurement [32], namely transient and steady state. We performed our measurements with the steady-state coaxial cylinders method, using a Setaram C80D microcalorimeter equipped with calorimetric vessels, also developed by Setaram, suitable for thermal conductivities of liquids. This method has the drawback that it is time consuming, though it permits a good temperature control and a very accurate measurement of the heat flow which passes through the sample; this is the key measurement that, with a good calibration method, gets accurate and reliable experimental thermal conductivity data. This method is particularly suitable for studying nanofluids because the measurement is made with very small temperature gradients and with practical absence of natural convection [33].

General measurements of thermal conductivity are often characterized by poor reproducibility and errors of about 2–5 % [34–36]. Specifically in the case of nanofluids, there are large discrepancies and inconsistencies in bibliography, showing the complexity of the thermal transport mechanism in this type of fluids. Identical samples being measured with different methods usually result in large differences among values [37]. The main source of discrepancies seems to be the particles and conglomerates size distribution, which depends largely on the way the nanofluids are made. In this work, nanofluids were prepared by a well-established two-step method [38] and the nanoparticle size distribution was measured using a light scattering technique.

The specific heat basically represents the thermal storage capacity of a system, but it is also useful to calculate other related quantities like dynamic thermal conductivity and diffusivity. Furthermore, it is a key property for describing the heat transfer performance of flowing nanofluids in terms of the convective heat transfer coefficient and the Nusselt number. Research work on the specific heat capacity at constant pressure, c p, of nanofluids is limited [39–52] compared to other properties such as thermal conductivity and viscosity. It depends on the specific heat capacity of nanoparticles and base fluid, particle volume fraction, and temperature. In order to measure the specific heat capacity of nanoparticles and nanofluids, we used a Setaram Micro DSC II microcalorimeter which provides very accurate values of specific heat capacity.

This paper presents thermal conductivity and specific heat capacity measurements of nanoparticles of CuO dispersed in water and ethylene glycol (EG) as a function of the particle volume concentration, at temperatures between 298 and 338 K. Some molecular theories on heat transfer in nanofluids applied to our systems are also discussed.

Experimental

The nanoparticles used in this work were CuO (NanoArk®, 97.5 %, 23–37 nm APS powder) supplied by Alfa Aesar. Bi-distilled and deionised water and ethylene glycol (Fluka, puriss. p.a. ≥99.5 %) were used as the base fluids. The nanoparticle volume fractions of nanofluids were calculated from the masses of nanoparticles powder, using the true density provided by the supplier (6,500 kg m−3), and that of the base fluid. A two-step method was used to prepare the nanofluids. Nanoparticles were first dispersed into the base fluid in a flask of 30 mL, approximately, and the mixture was dispersed ultrasonically using a tip sonicator (Hielscher UP100H) for 1 h at a power of 60 W, to break up the agglomerates. Nanofluids made up following this protocol were found to be appropriate to obtain reliable thermal conductivity and specific heat capacity measurements, with good repeatability during long periods of time. An increase of the sonication time did not improve the results. These facts were confirmed by light scattering measurements using a Malvern AutoSizer Lo-C for particle size measurements (Malvern Instruments Ltd., UK). Copper oxide nanoparticles size distributions in water are presented in Fig. 1. They show that the sonication process is very effective for removing larger aggregates and narrowing the main peak. Furthermore, it can be seen that the size distributions do not vary significantly over several days. Similar behavior is found for EG nanofluids.

Fig. 1
figure 1

Nanoparticles size distribution for water containing 1 vol% CuO. Without sonication (thick line), at 1st day (dashed line) and at 4th day (dashed–dotted line)

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The method for measuring thermal conductivities k, using a differential heat-flow microcalorimeter C80D provided with a set of vessels from Setaram (France), was described in detail elsewhere [33, 38]. It can be obtained as a function of the calorimeter signal \( \Delta \dot{Q} \), by the expression:

$$ k = \frac{{A - \Delta \dot{Q}}}{{B \times \Delta \dot{Q} + C}} $$
(1)

where the constants A, B, and C depend on the thermal response of the calorimeter components with temperature [53], but not on the liquid measured. A suitable calibration was performed, using three liquids of well-known thermal conductivity: distilled and deionised water [35, 54], glycerol anhydrous (Fluka ≥99.5 %) [34], and n-heptane (Fluka, ≥99.5 %) [35, 55]. Overall uncertainty in thermal conductivity is about 1.5 % [33].

In order to measure the specific heat capacity of nanoparticles and nanofluids, a Setaram Micro DSC II microcalorimeter provided with “batch” vessels designed in our laboratory, was used. The method was also described in a previous work [38], and measurements have a relative uncertainty of 0.3 % [56].

Results and discussion

Thermal conductivity measurements

Experimental values of the thermal conductivity as a function of temperature and nanoparticle volume fraction (φ) of CuO/water and CuO/EG nanofluids are presented in Tables 1 and 2, respectively. Figures 2 and 3 show the relative thermal conductivity k r , (ratio of the thermal conductivity of the nanofluid with respect to that of the base fluid) as a function of the temperature and volume fraction at 298.15 and 338.15 K for the CuO/water nanofluid. Figures 3 and 4 depict the same information for the CuO/EG nanofluid. The thermal conductivity data of water [35, 54] and ethylene glycol [34] were taken from the literature.

Table 1 Experimental thermal conductivity of CuO/water nanofluid as a function of the temperature and nanoparticle volume fraction
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Table 2 Experimental thermal conductivity of CuO/EG nanofluid as a function of the temperature and nanoparticle volume fraction
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Fig. 2
figure 2

Relative thermal conductivity of CuO/water nanofluid versus temperature, at several volume fractions (circles 0.4 %, inverted triangles 0.7 %, squares 1 %, diamonds 1.5 %). Full lines (linear correlations) are intended to guide the eyes

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Fig. 3
figure 3

Relative thermal conductivity of CuO/water (filled circles 298.15, open circles 338.15) and CuO/EG (filled squares 298.15, open squares 338.15) nanofluids versus volume fraction at 298.15 and 338.15 K. Full lines (correlations) are intended to guide the eyes

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Fig. 4
figure 4

Relative thermal conductivity of CuO/EG nanofluid versus temperature, at several volume fractions (circles 0.4 %, inverted triangles 0.7 %, squares 1.5 %, diamonds 3 %). Full lines (linear correlations) are intended to guide the eyes

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Figures 5 and 6 compare our results of k r as a function of the particle volume concentration at 298.15 K for the CuO/water and CuO/EG nanofluids, with recent published data for the same nanofluids in similar conditions (nanoparticle size and temperature), along with the reported uncertainties. A large degree of randomness and scatter were observed in the literature data; in any case, the results of the present paper are within the general range of values obtained from various sources. The scatter of the experimental data could possibly be attributed to several factors including nanofluid manufacturing process, particle size, and clustering, as well as the thermal conductivity experimental technique. The two-step method with sonication used in this work has been shown to be suitable to obtain a stable nanofluid over long periods and with an appropriate size distribution. Furthermore, the microcalorimetric technique ensures that our thermal conductivity measurements are performed at a constant temperature and with practical absence of natural convection, as has been justified elsewhere [33]. These facts make us confident that our measurements are correct. They represent a relevant contribution to the issue of scatter of data as they have been obtained with a technique that eliminates possible sources of error (high temperature gradients and convection). The relative thermal conductivity increases almost linearly with concentration. For the CuO/water and CuO/EG nanofluids, the thermal conductivity increments observed in this paper are in good agreement with the literature values within the reported uncertainties, except with data from [62].

Fig. 5
figure 5

Relative thermal conductivity of CuO/water nanofluid versus volume fraction: Present work (filled circles, 23–37 nm, 298 K), Lee et al. [57] (open circles, 24 nm, 298 K), Eastman et al. [58] (open triangles, 24 nm, 298 K), Das et al. [59] (open squares, 29 nm, 294 K), Patel et al. [60] (open inverted triangles, 31 nm, 293 K), Zhang et al. [61] (open stars, 32 nm, 296 K), Mintsa et al. [62] (filled triangles, 29 nm, 298 K), Hwang et al. [63] (filled squares, 22 nm, 298 K)

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Fig. 6
figure 6

Relative thermal conductivity of CuO/EG nanofluid versus volume fraction: present work (filled circles, 23–37 nm, 298 K), Lee et al. [57] (open circles, 24 nm, 298), Wang et al. [64] (open triangles, 23 nm, 298 K), Patel et al. [60] (open squares, 31 nm, 293 K), Hwang et al. [63] (open inverted triangles, 33 nm, 298 K), Gowda et al. [6] (open stars, 30 nm, 297 K)

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Figures 7 and 8 compare our experimental results of k r as a function of the temperature for the CuO/water (1.5 vol%) and CuO/EG (3.0 vol%) nanofluids, with recently published data for the same nanofluids in similar conditions (nanoparticle size and concentration), along with the reported uncertainties. In the scarce literature data available, it is shown that the relative thermal conductivity increases with temperature; nevertheless, our measurements indicate that it is essentially temperature independent. The same behavior is found in Al2O3 nanofluids made with the same base fluid [38, 65] and in CuO with diathermic oil nanofluids [29]. We conclude that the observed growth of the thermal conductivity of our nanofluids with increasing temperature is due mainly to the base fluids (water and ethylene glycol) rather than to the nanoparticles and, therefore, that the thermal conductivity enhancement is temperature independent.

Fig. 7
figure 7

Relative thermal conductivity of CuO/water nanofluid versus temperature: present work (filled circles, 23–37 nm, 1.5 %), Das et al. [59] (open circles, 29 nm, 2 %), Patel et al. [60] (open triangles, 31 nm, 1.2 %), Zhang et al. [61] (open diamonds, 33 nm, 2.6 %), Mintsa et al. [62] (open inverted triangles, 29 nm, 3.3 %)

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Fig. 8
figure 8

Relative thermal conductivity of CuO/EG nanofluid versus temperature: Present work (filled circles, 23–37 nm, 3 %), Patel et al. [60] (open circles, 31 nm, 3 %)

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Figure 9 compares the measurements of this work with those calculated with the Hamilton and Crosser [66] model, an extension of Maxwell model [67] (low-concentration diffusion of identical solid spheres in a liquid) to apply it to non-spherical particles. They show good agreement. We have also analyzed the predictions of other models such as that of Nan et al. [68], which take into account the Kapitza thermal contact resistance or the more complex model of Jang et al. [69], which also includes the effect of Brownian motion of nanoparticles at the molecular and nanoscale level. The improvements obtained with these models are in all cases within the experimental error range. It can be concluded that the experimental thermal conductivities of the nanofluids studied in this work can be properly justified using a simple classical model and that no anomalous enhancement of thermal conductivity was observed [37, 61, 70].

Fig. 9
figure 9

Thermal conductivity of CuO/water nanofluids at different nanoparticle volume fractions (open circles 0.4 %, open inverted triangles 0.7 %, open squares 1.0 %, open diamonds 1.5 %); and CuO/EG nanofluid at different nanoparticle volume fractions (filled circles 0.4 %, filled inverted triangles 0.7 %, filled squares 1.5 %, filled diamonds 3 %) in function of temperature. Full lines represent the predictions of the Hamilton–Crosser model

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Specific heat capacity measurements

Experimental values of the specific heat capacities of CuO nanoparticles and nanofluids obtained with water and EG as base fluids as a function of the particle volume concentration and at temperatures between 298 and 338 K are presented in Tables 3 and 4. All these data fit nicely by a linear curve-fit:

Table 3 Experimental specific heat capacities of CuO nanoparticles and CuO/water nanofluid as a function of the temperature and nanoparticle volume fraction
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Table 4 Experimental specific heat capacities of CuO/EG nanofluid as a function of the temperature and nanoparticle volume fraction
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$$ c_{\text{p}} = a + b \times T $$
(2)

Table 5 shows the coefficients of linear regressions with their standard errors and the root-mean-square deviations of the fits.

Table 5 Coefficients of Eq. (2) and root-mean-square deviations
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In the literature, there are very few reliable data on specific heat capacities of water- or EG-based CuO nanofluids to compare with. Zhou et al. [44] have studied the specific heat capacities of EG-based CuO nanofluid at room temperature as a function of the nanoparticle volume fraction. Pantzani et al. [46] have measured the specific heat capacities of water-based CuO nanofluids at 298 K and three-particle volume fraction with a microcalorimeter. Recently, O’Hanley et al. [40] have measured the specific heat capacities of water-based CuO nanofluids at 308, 318, and 328 K and various nanoparticle concentrations using a heat-flux type differential scanning calorimeter. It is worth noting that our results are in excellent agreement with those measurements within the experimental error range.

Assuming that the nanoparticles and the base fluid are in thermal equilibrium, the specific heat capacity of nanofluid can easily be deduced on the basis of the First Law of Thermodynamics. As has been explained previously [38], the specific heat capacity of nanofluid c p,nf is related to the specific heat capacities of nanoparticles c p,np and base fluid c p,bf by:

$$ {\text{c}}_{\text{p,nf}} = \frac{{\varphi \cdot \rho_{\text{np}} \cdot {\text{c}}_{\text{p,np}} + \left( {1 - \varphi } \right) \cdot \rho_{\text{bf}} \cdot {\text{c}}_{\text{p,bf}} }}{{\varphi \cdot \rho_{\text{np}} + \left( {1 - \varphi } \right) \cdot \rho_{\text{bf}} }} $$
(3)

where \( \rho_{\text{nf}} \), \( \rho_{\text{np}}, \) and \( \rho_{\text{bf}} \) are the densities of the nanofluid, nanoparticles, and base fluid, respectively, and φ is the nanoparticle volume fraction.

Figure 10 shows a comparison of our experimental results of the specific heat capacities at 303 and 333 K as a function of the volume fraction concentration for the CuO/water and CuO/EG nanofluids, with the predictions of Eq. (3). An excellent agreement can be seen.

Fig. 10
figure 10

Specific heat capacity of CuO/water nanofluid versus nanoparticle volume fraction at two temperatures (open circle 303 K, filled circle 333 K), and of CuO/EG nanofluid (open square 303 K, filled square 333 K). Full lines represent Eq. (3)

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Conclusions

At present, nanofluid thermal conductivity data and measurement methods lack of consistency. In this paper, the authors have measured thermal conductivities of nanoparticles of CuO dispersed in water and EG, as a function of the particle volume fraction. The temperatures varied between 298 and 338 K. A microcalorimetric technique has been used. It is particularly suitable for studying nanofluids because the measurements are made with very small temperature gradients and with practical absence of natural convection.

Hence, these measurements represent a relevant contribution to the issue of scatter of the nanofluids relative thermal conductivity data found in the literature. In the scarce literature data available, it is shown that the relative thermal conductivity increases. Nevertheless, our results indicate that it is essentially temperature independent. We conclude that the observed growth of the thermal conductivity of our nanofluids with increasing temperature is mainly due to the base fluids—water and EG—rather than to the nanoparticles.

There are very few reliable data on specific heat capacities of water or EG-based CuO nanofluids. Thus, this thermal study has been completed with very precise specific heat capacity measurements of the same nanofluids.

The classical Hamilton–Crosser model properly accounts for the thermal conductivity of the studied nanofluids. Moreover, assuming that the nanoparticles and the base fluid are in thermal equilibrium, the experimental specific heat capacities of nanofluids are correctly justified by the First Law of Thermodynamics.