On the Specific Heat Capacity of CuO Nanofluid

Abstract

This paper reviews briefly the definition of heat capacity and clarifies the defined specific heat capacity and volumetric heat capacity. The specific heat capacity and volumetric heat capacity, with our measured experimental data for CuO nanofluids, are discussed as an illustrating example. The result indicates that the specific heat capacity of CuO nanofluid decreases gradually with increasing volume concentration of nanoparticles. The measurement and the prediction from the thermal equilibrium model exhibit good agreement. The other simple mixing model fails to predict the specific heat capacity of CuO nanofluid. The nanoparticle size effect and solid-liquid interface effect on the specific heat capacity of nanofluid are discussed.

1. Introduction

Nanofluids, that is, liquids with nanometer-sized particles suspensions, have attracted a substantial amount of attention since Masuda et al. reported firstly, in 1993, their precursory observations on the thermal conductivity enhancement in liquid dispersions of nanoparticles [1]. To meet potentially the increasing demand for high thermal conductive working fluids, many researches are focused on the basic thermophysical properties of nanofluids, for example, effective thermal conductivity, effective viscosity, thermal diffusivity, specific heat capacity, Prandtl number, and so forth, with which researchers investigate the convective heat transfer and flow characteristics of nanofluids. However, only effective thermal conductivity and effective viscosity are extensively covered in relevant studies [2]. As a thermodynamic property, the specific heat capacity of a nanofluid is important to dictate the nanoparticle and fluid temperature changes, which affect the temperature field of the nanofluid and hence the heat transfer and flow status. Other thermophysical properties, such as thermal diffusivity and Prandtl number need the knowledge of specific heat capacity, too.

Only several researches involve in experiments of the specific heat capacities of nanofluids. For example, Sinha et al. [3] measured the specific heat capacity of a nanotube solution (nanofluid) by an AC calorimeter. He et al. [4] measured the specific heat capacity of BaCl₂-water-based TiO₂ nanofluid with a heat-flux differential scanning calorimeter. Peng et al. [5] measured the specific heat capacity of water-based Cu, Al, ${Al}_{2} O_{3}$ , and CuO nanofluids, and propylene glycol- (PG-) based ${Al}_{2} O_{3}$ nanofluid, with a special designed comparison calorimeter. Recently, Zhou and Ni [2] measured the specific heat capacity of water-based ${Al}_{2} O_{3}$ nanofluid with a power-compensated differential scanning calorimeter. These experiments show that the specific heat capacities of nanofluids are different from that of base fluid and vary with the size and volume concentration of nanoparticles. In most of the cases, however, there are no experimental data available for specific heat capacity, such that two kinds of models have been generally adopted to deduce it from the value of base fluid and nanoparticles. One is macroscopic, that the specific heat capacity of a nanofluid is equal to the volume average of the specific heat capacities of base fluid and nanoparticles. The other is mesoscopic, which assumes the base fluid and the nanoparticles are in their thermal equilibrium, respectively. Using the volumetric heat capacity instead, the volumetric heat capacity of a nanofluid can be expressed as the sum of the volumetric heat capacities for base fluid and nanoparticles, using their respective volume concentrations. These models are equivalent for small concentrations, but for large concentrations the models are obviously divergent. Zhou and Ni [2] also summarized the models mentioned earlier and compared them with their experimental results, concluding that their experiments are in good agreement with the second model, while the first one fails to predict the specific heat capacity of nanofluid. The divergence arises from the misunderstanding for the definition of specific heat capacity and volumetric heat capacity. However, the misunderstanding remains in many articles, so that it is necessary to further explain the controversy among models and experiments.

In this study, we will first review briefly the definition of heat capacity and clarify the relation between specific heat capacity and volumetric heat capacity. We will measure the specific heat capacity of nanofluid made by EG (ethylene glycol) with inclusion of CuO nanoparticles at room temperature using the quasisteady-state principle [6], to compare with the mentioned models. We will also consider the particle size effect and particle-liquid interface effect on the specific heat capacity of nanofluid, and then compare them with our analytical results, to attempt to find the rule of specific heat capacity variation when increasing the volume concentration of nanoparticles and to apply it in the experimental and numerical investigations of nanofluid.

2. On the Definition of Heat Capacity

To avoid misunderstanding of the specific heat capacity of nanofluid and misemploying in experimental and numerical researches, we review here briefly the definition of heat capacity, together with the relation between specific heat capacity and volumetric heat capacity.

The heat capacity C of a substance, sometimes also called total heat capacity, is the amount of heat required to change its temperature by one Kelvin, and has units of joule per Kelvin (J/K) in the SI system [7, 8]. The equation relating thermal energy to heat capacity is $Δ Q = C Δ T$ , where $Δ Q$ is the thermal energy put into or taken out of the substance, and $Δ T$ is the temperature differential. The heat capacity is therefore an extensive variable and depends simply on the amount of substance. The heat capacity for a mixture of different substances is the sum of the individual heat capacities:

C = C_{1} + C_{2} + \dots .

(1)

The specific heat capacity c of a substance, also named mass-specific heat capacity in science and engineering, is the amount of the heat required to change its temperature of unit mass (one kilogram) of the substance by one Kelvin. The unit of specific heat capacity c in the SI system is the Joule per kilogram-Kelvin, J/(kg·K) [7, 8]. The equation relating heat energy to specific heat capacity is $Δ Q = c m Δ T$ . The specific heat capacity corresponds to the quotient of heat capacity and mass, or $c = C / m$ , where m is the total mass. The specific heat capacity of a mixture of substances is equal to the sum of the individual heat capacities divided by the total mass:

c = \frac{C}{m} = \frac{(c_{1} m_{1} + c_{2} m_{2} + \dots)}{(m_{1} + m_{2} + \dots)} = ω_{1} c_{1} + ω_{2} c_{2} + \dots,

(2)

where $ω_{i} = m_{i} / m$ is the mass concentration of the ith substance.

In the measurement of physical properties, the term “specific” means the measure is an intensive property, wherein the quantity of substance must be specified. For specific heat capacity, mass is the specified quantity (unit quantity). In some books on thermodynamics, the noted specific heat capacity is used for the molar heat capacity. Furthermore, the specific heat capacity is sometimes simply denoted as specific heat. These may cause confusion. In chemistry, the term molar heat capacity c_mol of a substance may be used to more explicitly describe the measure of the amount of the heat required to change its temperature of unit quantity of substance (one mole) by one Kelvin. The unit of molar heat capacity c_mol in the SI system is the joule per mole-Kelvin, J/(mol·K) [7, 8]. The equation relating heat energy to molar heat capacity is Δ $Q = c_{mol} n Δ T$ , where n is the number of moles. The molar heat capacity is related to the heat capacity by $c_{mol} = C / n$ , and is related to the specific heat capacity by $c_{mol} = c M$ , where M is the molar mass. The molar heat capacity of a mixture of substances is equal to the sum of the individual heat capacities divided by the total number of moles:

\begin{matrix} c & = \frac{C}{n} = \frac{(c_{mol 1} n_{1} + c_{mol 2} n_{2} + \dots)}{(n_{1} + n_{2} + \dots)} \\ = x_{1} c_{mol 1} + x_{2} c_{mol 2} + \dots, \end{matrix}

(3)

where $x_{i} = n_{i} / n$ is the molar concentration of the ith substance. While the “specific heat, γ”, of a substance is the ratio of the amount of heat required to raise the temperature of a given mass of the substance through a given range of temperature to the heat required to raise the temperature of an equal mass of water through the same range: $γ = c / c_{0}$ , where c₀ is the specific heat capacity of water.

The specific volumetric heat capacity, ρc, of a substance is the amount of the heat required to change its temperature of unit volume of the substance by one Kelvin [7, 8], and ρ being the density or mass per unit volume. The volumetric heat capacity describes the ability of a given volume of a substance to store internal energy while undergoing a given temperature change, but without undergoing a phase change. The unit of volumetric heat capacity ρc in SI system is the Joule per square meter-Kelvin, J/( $m^{3} \cdot$ K). The equation relating thermal energy to volumetric heat capacity is $Δ Q = ρ c V Δ T$ , where V is the total volume. The specific heat capacity corresponds to the quotient of heat capacity and volume, or C/V. The volumetric heat capacity of a mixture of substances is equal to the sum of the individual heat capacities divided by the total volume:

ρ c = ϕ_{1} ρ_{1} c_{1} + ϕ_{2} ρ_{2} c_{2} + \dots,

(4)

where $ϕ_{i} = V_{i} / V$ is the volume concentration of the ith substance, and $ρ = (ϕ_{1} ρ_{1} + ϕ_{2} ρ_{2})$ , is the density of the mixture.

In the case of nanofluid, the specific heat capacity at constant pressure c_p can be derived from (6), which becomes

c_{p, n f} = \frac{[(1 - ϕ) ρ_{f} c_{p, f} + ϕ ρ_{n p} c_{p, n p}]}{[ϕ_{f} ρ_{f} + (1 - ϕ) ρ_{n p}]},

(5)

where ϕ is the volume concentration of nanoparticle, and the subscripts nf, f, and np represent for nanofluid, base fluid, and nanoparticle, respectively. The following equation is proposed for determining specific heat capacity of nanofluid and assessing heat transfer performance of nanofluids [9–11]:

c_{p, n f} = (1 - ϕ) c_{p, f} + ϕ c_{p, n p} .

(6)

However, it is approximately correct only for dilute suspensions when small density difference exists between base fluid and nanoparticle. For water-based ${Al}_{2} O_{3}$ nanofluid, for example, the deviation cannot be ignored, as the density ratio between nanoparticle and base fluid is large ( $ρ_{n p} = 3600 kg / m^{3}$ for ${Al}_{2} O_{3}$ , while $ρ_{f} = 994.7 kg / m^{3}$ for water), so that large discrepancy occurs when increasing the volume concentration of nanoparticle [11].

3. Data Correlated with Discussions

Now, we will discuss, as an example, on the specific heat capacity and volumetric heat capacity with our measured experimental data for CuO nanofluids [12]. We use CuO nanoparticles (produced by chemical vapor-synthesis method) and EG as base fluid to prepare nanofluid for experiments. The nanoparticles are in form of loose agglomerates under atmospheric condition and can be dispersed in EG successfully through ultrasonic vibration for about three hours. The nanoparticles show a lognormal size distribution with nominal diameter of 50 nm. The quasisteady-state method is adopted to measure the specific heat capacity of nanofluid [6].

Figure 1 shows the specific heat capacity of CuO/EG nanofluid, of which the volume concentration of nanoparticle varying from 0.1% to 0.6% by interval of 0.1%. An obvious increase of specific heat capacity can be observed for these nanofluids, comparing with the value calculated from (5) and (6). The discrepancy between them indicates that neither of the two equations can be used to predict the tendency for fluid with nanoparticle inclusions. The experiments also present a slight decrease of specific heat capacity for CuO nanofluid when increasing the volume concentration of CuO nanoparticle, while it coincides with (5) for nearly the same linear tendency.

Figure 1

Specific heat capacity of CuO/EG nanofluid.

The volumetric heat capacity of nanofluid is important to dictate the temperature change of nanofluid. This is clear for analyzing the heat transfer between nanoparticle and base fluid, with energy conservation either in thermal equilibrium form or in nonequilibrium one. Figure 2 shows the corresponding volumetric heat capacity of CuO/EG nanofluid. The variation of volumetric heat capacity is small for dilute CuO/EG nanofluid. One can conclude that the volumetric heat capacity ratio of CuO/EG nanofluid versus deionized water will be approximately a constant, which may be helpful to analyze the energy conservation in a dimensionless form.

Figure 2

Volumetric heat capacity of CuO/EG nanofluid.

As shown in Figure 1, the discrepancy between the experimental results and (5) and (6) may arouse from the surface and size effects on the specific heat capacity of nanoparticle [12], and hence, decreased with increasing ϕ due to agglomeration of nanofluid (clustering). We adopt model I (6) and II (5), respectively, to show the nanoparticle size effect on the specific heat capacity of nanofluid, as shown in Figure 3. The specific heat capacities of nanoparticles correspond to the bulk, 50 nm and 25 nm CuO, respectively [12]. As shown, the specific heat capacity of nanofluid is underestimated using the specific heat capacity of bulk CuO. While the prediction can be improved with the specific heat capacity of CuO nanoparticle obtained from either theoretical analysis or experiments. The results from our previous calculation show that the discrepancy between nanoparticles with different sizes is small when increasing the nanoparticle volume concentration, due to the large specific heat capacity of base fluid.

Figure 3

Specific heat capacity of CuO/EG nanofluid compared with model I and II.

Qualitatively, the solid-liquid interface may change the phonon vibration mode near the surface area of a nanoparticle and thus change the specific heat capacity of nanofluid. The high specific interfacial area of nanoparticle can adsorb liquid molecules to its surface and form liquid layers, which will reversely constrain nanoparticle and turns its free-boundary surface atoms to be nonfree interior atoms [12]. The varied Gibbs free energy of nanoparticle and liquid layers will further change the specific heat capacity of nanofluid.

4. Conclusion

This paper reviews briefly the definition of heat capacity and clarifies the defined specific heat capacity and volumetric heat capacity. For illustration, the specific heat capacity of nanofluid made by ethylene glycol and copper dioxide nanoparticle inclusions, measured at room temperature, were compared with two kinds of models for determination of the specific heat capacity of nanofluid, which are frequently used in researches on convective heat transfer experiments and simulations. The particle size effect and particle-liquid interface effect on the specific heat capacity of nanofluid are also discussed briefly. The effect of liquid adsorption on suspended nanoparticles’ surface will also increase the specific heat capacity of nanofluid to some extent with increasing nanoparticles’ volume concentration, which may be worthy to be investigated further for nanofluids.

Footnotes

Acknowledgment

The work was financially supported by the National Natural Science Foundation of China with Grant Number 50906024.

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