Thermal Conductivity and Stability of Novel Aqueous Graphene Oxide–Al₂O₃ Hybrid Nanofluids for Cold Energy Storage

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The paper not only provides method and theoretical guidance for the preparation of long-term stable high-performance hybrid nanofluids, but also gives support for their efficient applications in phase change cold energy storage.

Abstract

Thermal ice storage has gained a lot of interest due to its ability as cold energy storage. However, low thermal conductivity and high supercooling degree have become major issues during thermal cycling. For reducing the cost and making full use of the advantages of the graphene oxide–Al₂O₃, this study proposes heat transfer enhancement of thermal ice storage using novel hybrid nanofluids of aqueous graphene oxide–Al₂O₃. Thermal conductivity of aqueous graphene oxide–Al₂O₃ nanofluid was measured experimentally over a range of temperatures (0–70 °C) and concentrations. Thermal conductivity of ice mixing with the hybrid nanoparticles was tested. The influences of pH, dispersant, ultrasonic power and ultrasonic time on the stability of the hybrid nanofluids were examined. A new model for the effective thermal conductivity of the hybrid nanofluids considering the structure and Brownian motion was proposed. The results showed that pH, dispersant, ultrasonic power level and ultrasonication duration are important factors affecting the stability of the hybrid nanofluids tested. The optimum conditions for stability are pH = 11, 1% SDS, 375 W ultrasonic power level and 120 min ultrasonic application time. The thermal conductivity of hybrid nanofluids increases with the increase of temperature and mass fraction of nanoparticles. A newly proposed thermal conductivity model considering the nanofluid structure and Brownian motion can predict the thermal conductivity of hybrid nanofluids reasonably well.

1. Introduction

Thermal ice storage is a very important kind of phase change cold energy storage, which can be used in the solar and wind energy system to reduce the fluctuations in the energy flow. The ice storage system not only can run fluently under a decoupled source–load condition, but can reduce maximum output power, average power and energy consumption by 20–40% in comparison with conventional systems in centralized and decentralized environments [1]. The thermal ice storage is also employed in heating, ventilation and air conditioning (HVAC) cooling, food processing, chemical reactions or pharmaceutical processing, inlet air cooling of turbine and district cooling plant [2].

As one type of phase change material (PCM), ice storage, however, has higher supercooling degree and lower thermal conductivity that reduce heat transfer performance. A high supercooling degree leads to reduced performance of the PCM thermal energy storage system because of the requirement of a large operating temperature range [3]. Fortunately, many researchers found that addition of nanoparticles not only enhance the thermal conductivity, but also reduce the supercooling degree, and improve the heat transfer performance greatly [4]. Nonetheless, the stability of the nanoparticles suspended in base fluids during the thermal cycling is becoming a key problem.

Colloidal suspensions containing dispersed metal or nonmetal nanoparticles are called nanofluids, which are widely used in several heat transfer or performance improvement areas such as solar energy collectors [5,6,7,8,9,10,11,12,13], phase change energy storage [14,15], micro- and power electronics [16], magnetic refrigeration [17,18], diesel or biodiesel fuels [19,20], ventilation and air conditioning systems [21], reactors in power generating plants, cancer treatment therapy, metallurgical operations and combustion engines [22,23,24]. Nanofluids can be divided into two categories according to the types of nanoparticles. One is a single nanofluid formed by a single nanoparticle and base liquids, and the other is a hybrid nanofluid formed by mixing multiple nanoparticles and base liquids. Hybrid nanofluid is a solid–liquid mixture formed by dispersing two or more different nanoparticles suspended in a liquid, which displays enhanced thermophysical properties and rheological characteristics compared to mono nanoparticles nanofluid due to a synergistic effect [25]. Properly selected hybrid nanofluids can not only have higher thermal conductivity than single-phase nanofluids, but also have better properties such as less mechanical resistance, better stability and lower cost. These superior characteristics make them particularly attractive for applications in many heat transfer areas [22,23]. However, there are still several challenges that have impeded the development of this field as concluded in many review articles [23,24,25,26,27,28,29,30,31,32,33,34], such as disagreement in results of various researchers on the same types of nanofluids; preparation of long-term, stable and homogeneous nanofluids; a lack of theoretical understanding of the mechanisms accountable for variety in properties and high cost of the hybrid nanoparticles. In particular, stability of nanofluids is an essential characteristic that impacts greatly the thermophysical properties and rheological characteristics as well as their field applications. Surfactants have been used to keep good dispersibility of nanofluids. However, surfactants may impact the thermal conductivity and viscosity of the hybrid nanofluid as well [35]. Among various nanomaterials used in nanofluids, graphene oxide (GO) is hydrophilic, has a superior dispersion stability and high thermal conductivity and can significantly reduce the degree of supercooling of water in phase change thermal storage [36,37,38,39,40,41], but the preparation technology is complex and expensive. Nano-Al₂O₃ is a common metal oxide material and is widely used in various nanofluids because the Al₂O₃ nanofluid has lower viscosity and better rheological behavior as compared to other particles such as copper oxide [42,43,44]. Moreover, nano-Al₂O₃ has obvious advantages such as high thermal conductivity, cheap and safe [45,46,47,48]. For example, nanofluids of aluminum oxide can enhance thermal conductivity from 0.3% to 38% with a particle size range from 5 to 80 nm [49,50,51]. Moreover, the supercooling degree of water is reduced under the combined effect of ultrasound and aluminum oxide nanoparticles. A 63.7% reduction in the supercooling degree of water can be obtained at Al₂O₃ nanoparticle concentration of 0.2 wt % with the proper ultrasonic intensity [52]. However, long-term stability of aqueous Al₂O₃ is not as good as that of graphene oxide.

To reduce the cost and make full use of the advantages of the two materials, our study prepares a novel aqueous Al₂O₃ and GO nanosheet hybrid nanofluid for thermal ice storage as we have done in our previous work about its specific heat [53]. The thermal conductivity and stability under thermal cycles of the hybrid nanofluid are investigated, the thermal conductivity of the ice mixed with the hybrid nanoparticles is tested, and a new thermal conductivity model of the hybrid nanofluids considering the influence of Brownian and construction is proposed, which presents the theoretical understanding of the mechanisms responsible for the enhancement of thermal conductivity.

2. Experimental Apparatus and Procedure

The GO–Al₂O₃ hybrid nanofluids were created by the two-step method [54] using sodium dodecyl sulfonate (SDS) as a surfactant. Different masses of the graphene oxide nanoplatelet (purity ≥99 wt %, number of layers <10, from Shenzhen Turing Evolution Technology Co., Ltd.) and nano-Al₂O₃ (content ≥ 99.99%, average particle size (30 ± 5 nm)) were fully dispersed in deionized water (DW). High-powered ultrasonication was used to disperse nanoparticles. Sodium hydroxide solution was used to adjust the pH.

The hybrid nanomaterials were analyzed and characterized by an X-ray diffractometer (Model: XRD Ultima IV). WGZ-2000 turbidity meter (indication error of ±3% F.S and repeatability ≤0.5%, from Shanghai Xinrui Instrument Co., Ltd. China) was used to test the absorbance of nanofluids with time. It can measure the degree of scattering or attenuation of light generated by insoluble particulate matter suspended in water or a transparent liquid, and quantitatively characterize the content of these suspended particulate matter with NTU (Nephelometric Turbidity Unit) as the turbidity unit.

The transient hot wire method has been regarded as the most accurate method as well as the primary method for determination of thermal conductivity of a fluid and solid. The method has become a standard method in many countries [55,56,57]. Thermal conductivity is measured by a TC3000E transient hot wire instrument (Xiaxi Electronic Technology Co., Ltd. Xi’an, China), which has been employed widely by many researchers [58,59,60]. The instrument works according to transient hot-wire method that complies with standards of ASTM C1113, ASTM D5930, GB/T 10297, and GB/T 11205. A water bath with an accuracy of ± 0.05 °C was employed to control temperature.

X-Ray Diffraction (XRD) is very useful to show the crystal structure [61], the hybrid nanomaterials are characterized by XRD as indicated in Figure 1. It can be seen that the diffraction peak at 2θ = 11.06° was the characteristic peak of graphene oxide (001) [62]; the diffraction peak at 2θ = 25.57°, 35.15°, 37.77°, 43.35°, 52.55°, 57.51°, 68.21° and 77.2° is the crystal phase of α- Al₂O₃ (012), (104), (110), (113), (024), (116), (300) and (119), respectively, which is consistent with the standard α-Al₂O₃ spectrum [63].

Figure 2 and Figure 3 are scanning electron microscope (SEM) for the hybrid nanoparticles dispersion in the base solution for 120 min and 30 min of ultrasonication, respectively. The results showed that Al₂O₃ nanoparticles of about 40 nm distribute homogeneously on the graphene oxide nanosheets when vibrated for 120 min using ultrasonic. However, a large number of nanoparticles aggregate together under 30 min of ultrasonication, which indicates poor dispersion of the nanoparticles.

3. Results and Discussion

3.1. Effect of Different Factors on the Stability of the Hybrid Nanofluids

According to the theory of electrostatic stabilization and steric stabilization [64], nanofluids stability is affected greatly by many factors such as the pH, surfactants and ultrasonic power and time. In the following parts, the aforesaid factors are investigated.

3.1.1. Influence of pH

The pH of nanofluids is usually adjusted to increase the charges on the surface of the nanoparticles, which keeps the surface potential of the nanoparticles away from isoelectric point so as to enhance the electrostatic repulsion and stability of the nanofluids. The pH adjustment is suitable for electrostatic stabilization mechanism of colloidal particles that can be described by the DLVO (Derjaguin, Landau, Verwey and Overbeek) theory, i.e., the total interaction between electrostatic stabilized particles is the combination of van der Waals attraction and electrostatic repulsion [64].

Hybrid nanofluids of 0.1% mass fraction with pH = 3, 7, and 11 were prepared and ultrasonically dispersed for 60 min. Figure 4 and Figure 5 show, respectively, the condition of the nanofluids after one and seven days. The hybrid nanofluid with pH = 3 has completely precipitated after 7 days, and the hybrid nanofluid with pH = 7 had a large amount of precipitation, while the hybrid nanofluid with pH = 11 had no obvious change and showed very good stability. The same trend is observed for bottom precipitation, as shown in Figure 5. The pH had a great influence on the stability of the hybrid nanofluid, and the hybrid nanofluid of pH = 11 was shown to have the best stability. This is mainly due to the fact that Al₂O₃ nanoparticles hydrate to produce hydroxyl groups, and the hydroxyl groups dissociate to charge the surface of the nanoparticles [65]. There are many hydrophilic acidic functional groups on the surface of the graphene oxide nanoparticles, which have strong ability to adsorb anions. Under alkaline conditions, a large amount of OH⁻ ions were continuously adsorbed on the surfaces of Al₂O₃ and GO nanoparticles to form double electric layers around the nanoparticles, which produced repulsive forces between the particles and prevent agglomeration and precipitation caused by collision between particles. Moreover, larger electrostatic repulsion also increased the distance between particles, which decreased van der Waals attraction and, thus, reduced the probability of particles agglomerating and sedimentation, and further improved the stability of the nanofluids [66].

3.1.2. Influence of Dispersant

One of the dispersing mechanisms for nanofluid stabilization is steric stabilization, which can be attained by the addition of a dispersant [64]. In order to enhance the stability of the nanofluid, polymer chains are attached to charged particle surfaces or polyelectrolyte chains are attached onto uncharged particle surfaces to form a steric barrier between nanoparticles to screen the interparticle van der Waals attraction [67,68]. Compared to electrostatic stabilization, steric stabilization has many advantages [64]:

(1) It is widely used in various dispersion systems as a thermodynamic stabilization method;

(2) It is applicable to high concentration of nanoparticles and it redisperses the temporarily aggregated particles;

(3) The dispersion is not sensitive to categories of electrolyte;

(4) It shows good compatibility for simultaneously dispersing hybrid particles within the same base fluid. SDS, a hydrophilic anionic surfactant with good penetration and dispersion, which has superior compatibility with anions and no-ions, was used in this study. The hybrid nanofluid of 0.1 wt % (mass ratio of GO and Al₂O₃ is 1:1), pH = 11, was ultrasonically dispersed for 90 min, and a different fraction of SDS was added. Figure 6 shows the state of the hybrid nanofluid.

It shows good compatibility for simultaneously dispersing hybrid particles within the same base fluid. SDS, a hydrophilic anionic surfactant with good penetration and dispersion, which has superior compatibility with anions and no-ions, was used in this study. The hybrid nanofluid of 0.1 wt % (mass ratio of GO and Al₂O₃ was 1:1), pH = 11, was ultrasonically dispersed for 90 min, and a different fraction of SDS was added. Figure 6 shows the state of the hybrid nanofluid samples with (left side) and without (right side) SDS after 3 and 30 days. It can be seen that after 30 days, the hybrid nanofluid without dispersant had more sediments and a faded color. In comparison, the hybrid nanofluid with dispersant SDS had neither obvious agglomeration nor color change. It is clearly shown that the SDS can keep the nanofluid stable for a relatively long time.

For quantitative analysis of the stability, the absorbance of the hybrid nanofluid was measured with time using the WGZ-2000 turbidity meter. At some intervals after preparation of the nanofluid, 4 mL of the upper layer of the nanofluid was taken and diluted three times with deionized water, and the absorbance was sequentially measured. The absorbance reduction ratio (ARR) was defined as:

$$ARR\%=\frac{A_0-A_n}{A_0}\times 100$$

(1)

A₀ and A_n are the initial absorbance and the nth hour absorbance of each nanofluid, respectively. Figure 7 indicates the influence of different concentration of SDS on ARR. It was found that the nanofluid’s absorbance decreased rapidly (ARR increases) in the first four days, then decreased slowly after (slower ARR increase), as can be seen in Figure 7a. The stability of the hybrid nanofluid was improved greatly with the addition of proper amount of SDS. Moreover, the absorbance increased (ARR decreased) at first, then decreased (ARR increased) with the addition of the SDS fraction of above 1% as shown in Figure 7b. This is due to SDS being attached on the surface of nanoparticles to form double electric layers that increase the distance between nanoparticles, which in turn decrease the van der Waals attraction potential. The particles can be completely shielded by anti-ions with a suitable increase of SDS, and the two electric layers are in electrostatic equilibrium by repulsive force, which keep good stability of the nanofluid [68,69]. However, if the SDS amount surpasses the critical micelles concentration (CMC), surfactant monomers can join up together to form micelles via the hydrophilic group instead of attaching on the surface of nanoparticle. These strongly charged micelles could repel the charged nanoparticles and prompt nanoparticles to agglomerate [70,71,72].

3.1.3. Influence of Ultrasonic Power

The absorbance was measured at different ultrasonic power for the hybrid nanofluid sample and graphene oxide aqueous nanofluid, at pH = 11, 0.1% nanoparticle concentration, 1% SDS and ultrasonic dispersion of 120 min. The relation between ARR and ultrasonic power is shown in Figure 8 and Figure 9.

Figure 8a shows rapid decrease in hybrid nanofluid absorbance in the first four days, indicated by the rapid increase of ARR, and relatively stable absorbance afterwards. Moreover, the absorbance firstly increased, then decreased as ultrasonic power increased, as can be inferred from Figure 8b. The results indicate that the stability of the hybrid nanofluid reached optimum at 375 W of ultrasonic power, where the ARR was 17.16% lower than that at 450 W after 7 days. This is because the ultrasonic cavitation became stronger as ultrasonic power increased, which reduced the size of the nanoparticles and increased the spacing distance between the nanoparticles by the electrostatic and steric mechanism. The van der Waals attraction between nanoparticles is thus reduced, which hinders agglomeration and enhances the stability of the hybrid nanofluid. However, when the ultrasonic power is larger than the optimum value, it leads surfactant monomers to form micelles rather than attaching on the surface of nanoparticle. The intensity and frequency of nanoparticle collision increased greatly, and colliding nanoparticles generated larger agglomerates and precipitates.

For the GO nanofluid shown in Figure 9, the influence of ultrasonic power on the stability was consistent with the trend of the GO–Al₂O₃ hybrid nanofluids. Nevertheless, the ARR of the GO nanofluid was lower than that of the hybrid nanofluid, which means higher absorbance and better stability of GO nanofluid as compared to the stability of the hybrid nanofluid, which was slightly reduced with the addition of the alumina nanoparticles.

3.1.4. Influence of Ultrasonic Time

The absorbance was studied at different ultrasonic time for the hybrid nanofluid sample and graphene oxide aqueous nanofluid, at pH = 11, 0.1% nanoparticle concentration, 1% SDS and ultrasonic power of 375 W. The relationship between ARR and ultrasonic time is shown in Figure 10 and Figure 11.

It can be seen from Figure 10 that the trend of change in ARR with holding time and ultrasonic time was similar to that of ultrasonic power; a decrease in absorbance (ARR increases) in the first 4 days, and stable afterwards. From Figure 10b, it can be inferred that the stability of the hybrid nanofluid was optimal at 120 min of ultrasonic time, where the ARR was 29.65% lower than that at 30 min after 7 days. This is due to the fact that the ultrasonic cannot completely break the agglomerates when ultrasonic time is short. As the ultrasonic time increases, there is enough time to smash agglomerates into smaller nanoparticles, and the nanoparticles can maintain suitable spacing under the coaction of dispersants and ions in the nanofluid, so that the van der Waals force keeps balance with electrostatic repulsion. However, when the ultrasonic time is longer than the optimal value, similar to the influence of ultrasonic power, it produces surfactant micelles, and simultaneously increases the frequency of nanoparticle collision greatly, leading to agglomeration and precipitation. In comparison with ARR of the GO nanofluid shown in Figure 11, the ultrasonic time had a different influence on the graphene oxide nanofluid; the longer ultrasonic time the better the stability of the nanofluid, because the GO nanosheet was hydrophilic and had superior dispersion stability in water, so even longer ultrasonic time could not form large agglomerates and sediment.

3.1.5. Influence of Thermal Cycling

The absorbance was measured, as shown in Figure 12, with different numbers of thermal cycle of solidification and melting for the hybrid nanofluid sample at pH = 11, 0.1% nanoparticle concentration, 1% SDS, ultrasonic power 300 W and ultrasonic time of 120 min.

The absorbance of hybrid nanofluids decreased slightly before the 6th cycle, and stayed stable afterwards, with an overall decrease of 6.2% at the 10th cycle as compared with the initial absorbance. This is mainly because some aggregation and deposition occurred as the balance between the van der Waals attraction and electrostatic repulsion was broken by crystallization and melting. Nevertheless, the decrease in stability was slight, owing to the screen function of dispersants surrounding the nanoparticles.

3.2. Thermal Conductivity of the Hybrid Nanofluids

3.2.1. Reliability Verification

In order to verify the accuracy of the experimental system, the thermal conductivity of deionized water was measured from 0 to 70 °C. Experimental results shown in Figure 13 are compared with reference values [73]. The maximum error was 1.12%, which was used as experimental uncertainty shown in the corresponding figures. The enhancement in thermal conductivity and the uncertainty of thermal conductivity enhancement of the hybrid nanofluid are defined as in our previous work [54] by Equations (2) and (3) respectively:

$$k_E\%=\left(\frac{k_{nf}-k_{bf}}{k_{bf}}\right)\times 100$$

(2)

where k_nf and k_bf are the thermal conductivity of the nanofluid and base fluid, respectively.

The uncertainty of the thermal conductivity enhancement of the nanofluids was calculated by

$$k_E\%\times \sqrt{{(\frac{{\sigma }_{nf}}{k_{nf}})}^2+{(\frac{{\sigma }_{bf}}{k_{bf}})}^2}$$

(3)

where ${\sigma }_{nf}$ and ${\sigma }_{bf}$ are uncertainty of k_nf and k_bf, respectively.

3.2.2. Effect of Mass Fraction of the Nanoparticle

Figure 14 shows the influence of mass fraction of nanoparticle on thermal conductivity of the hybrid nanofluid. It indicates that the thermal conductivity of the hybrid nanofluid increased with the mass fraction of the nanoparticles in the range of 0–0.2%. Moreover, the influence of the mass fraction on thermal conductivity enhancement at a lower temperature was more significant than at a high temperature. At 0 °C, the thermal conductivity of 0.2 wt % hybrid nanofluids increased by 7.2% as compared to the base fluid, and the thermal conductivity enhancement of 0.2 wt % was about 2.1 times larger than that of 0.025 wt %. At a higher temperature, 70 °C, the thermal conductivity of 0.2 wt % hybrid nanofluid increased by 1.76% as compared to the base fluid, and the thermal conductivity enhancement was 14.7 times that of the 0.025 wt % hybrid nanofluid. It shows that the nanoparticle mass fraction had a significant influence on the thermal conductivity of the hybrid nanofluid, owing to the fact that thermal conductivities of GO and Al₂O₃ nanoparticles were larger than that of deionized water. The liquid molecules on the surface of the nanoparticles form ordered structure, which acts as a thermal bridge between the fluid and the solid particles, resulted in a higher thermal conductivity and enhanced energy transfer within the system. Meanwhile, Brownian motion of nanoparticles caused microconvection in the liquid, which enhanced the energy transfer.

3.2.3. Effect of Temperature

Thermal conductivity of the hybrid nanofluid increased with temperature, as displayed in Figure 15, which was consistent with the trend of thermal conductivity of deionized water with temperature. Moreover, the thermal conductivity enhancement of the hybrid nanofluid decreased with increasing temperature. The thermal conductivity enhancement at 0 °C was 28.5 times that at 70 °C for 0.025 wt %, while the thermal conductivity enhancement at 0 °C was 4 times that at 70 °C for 0.2 wt %. This means that the influence of temperature on the thermal conductivity enhancement was more significant at a lower concentration, even though a higher concentration led to a higher thermal conductivity enhancement. This can be attributed to the fact that Brownian motion and thermal diffusion of nanoparticles increased as the temperature increased.

3.2.4. Comparison of Thermal Conductivity of GO Nanofluid with the Hybrid Nanofluid

The thermal conductivity of aqueous GO nanofluid is measured at the same mass fraction and temperature as the GO–Al₂O₃ hybrid nanofluid, as shown in Figure 16. It displays a similar trend and thermal conductivity values, suggesting the potential benefit for using less GO and reducing the cost of the hybrid nanofluid.

3.2.5. Fitting of Thermal Conductivity of the Hybrid Nanofluid

There are many correlations for the thermal conductivity of hybrid nanofluids considering the influence of temperature and volume fraction of nanoparticles [23].

Mohammad et al. [74] proposed a formula as Equation (4) for thermal conductivity of DWCNT-ZnO/water-EG hybrid nanofluids,

$$\frac{k_{nf}}{k_{bf}}=1.085e^{\left(0.001351T+0.13{\phi }^2\right)}+0.0288\ln \left(\phi \right)$$

(4)

where T is temperature in °C and $\phi$ is the volume fraction.

Arash et al. [75] used Equation (5) for thermal conductivity of multiwalled carbon nanotube (MWCNT)-Fe₃O₄/EG hybrid nanofluids:

$$\frac{k_{nf}}{k_{bf}}=1+0.0162{\phi }^{0.7038}{T}^{0.6009}$$

(5)

Equation (6) was proposed by Masoud et al. [76] for thermal conductivity of MgO-f-MWCNT/EG at 25–50 °C, volume fraction of 0.05%–0.6%.

$$\frac{k_{nf}}{k_{bf}}=0.8341+1.1{\phi }^{0.243}{T}^{-0.289}$$

(6)

Hamid et al. [77] studied thermal conductivity of SiO₂-TiO₂ (60:40)/water and EG (60:40), and proposed a formula as Equation (7) for temperature range of 30 to 70 °C, volume fraction 0–3%.

$$\frac{K_{nf}}{K_{bf}}={\left(1+\frac{\phi }{100}\right)}^{5.5}{\left(\frac{T}{80}\right)}^{0.01}$$

(7)

Esfahani et al. [78] investigated thermal conductivity of ZnO-Ag/water from 25 to 50 °C, volume fraction 0.125%–2%, and proposed Equation (8).

$$\frac{K_{nf}}{K_{bf}}=1+0.0008794{\phi }^{0.5899}{T}^{1.345}$$

(8)

Equation (9) is the Maxwell model [79], which was developed based on the thermal conductivity of solid suspensions in liquids. It is a base correlation to predict the thermal conductivity of nanofluids [5].

$$k_{nf}=\frac{k_p+2k_{bf}+2\phi \left(k_p-k_{bf}\right)}{k_p+2k_{bf}-\phi \left(k_p-k_{bf}\right)}{k}_{bf}$$

(9)

The Takebi model [80] provided correlations for the thermal conductivity of hybrid nanofluid shown in Equation (11), which should have extensive applicability because it is based on the principle of Maxwell model and mixture rule that

$$k_p=\frac{\left({\phi }_{np1}{k}_{np1}+{\phi }_{np2}{k}_{np2}\right)}{{\varnothing }_{hnf}}$$

(10)

Then

$$k_{hnf}=k_{bf}\left[\frac{\frac{\left({\phi }_{np1}{k}_{np1}+{\phi }_{np2}{k}_{np2}\right)}{{\phi }_{hnf}}+2k_{bf}+2\left({\phi }_{np1}{k}_{np1}+{\phi }_{np2}{k}_{np2}\right)-2\phi k_{bf}}{\frac{\left({\phi }_{np1}{k}_{np1}+{\phi }_{np2}{k}_{np2}\right)}{{\phi }_{hnf}}+2k_{bf}-\left({\phi }_{np1}{k}_{np1}+{\phi }_{np2}{k}_{np2}\right)+\phi k_{bf}}\right]$$

(11)

The comparison between experimental data and calculated values using the Takebi model is shown in Figure 17. The values of properties of the nanomaterials used in Equation (11) are listed in

Table 1. Nanomaterials properties.

Nanomaterials	Density, kg/m3	Thermal Conductivity, W/(m·K)	Specific Heat, J/(kg·K)
α-Al2O3	3987	40 [46]	773 [53]
GO nanosheet	1800 [82]	3000 [81]	790 [53]

. The intrinsic thermal conductivity of GO was different for various fabrication processes and measurement methods [81], but it had very little influence on the fitting results even for 1000–3000 W/(m·K) using Equation (11) due to the low concentration of nanoparticles. Figure 17 shows that the relative error was larger at a lower temperature, with a maximum relative error of 6.5%.

Regression Equation (12) was obtained considering the influence of temperature and mass fraction.

$$k_{nf}=a+b{\varnothing }_m+cT+d{\varnothing }_m{}^2+e\times {\varnothing }_m\times T+g{T}^2$$

(12)

where ${\varnothing }_m$ is the mass fraction, from 0.025% to 0.2%; T is the temperature in °C, ranging from 0 to 70 °C and a, b, c, d, e and g are the parameters fitted by experiment data, as shown in

Table 2. Parameters used in regression Equation (12).

a	b	c	d	e	g	R2
0.5705	9.041	0.001868	1738	−0.09111	−7.805 × 10−6	0.9968

. The relationship between volume fraction and mass fraction of the particles can be obtained according to the following equation:

$$\phi =\frac{1}{\left[\left(\left(1-{\varphi }_m\right)/{\varphi }_m\right)\left({\rho }_{np}/{\rho }_{bf}\right)+1\right]}$$

(13)

where ${\rho }_{bf}$ and ${\rho }_{np}$ are the density of the base fluid and density of the nanoparticle, respectively.

Figure 18 displays the fitted values calculated by the regression Equation (12) and experimental data. The maximum relative error was 0.5% and R-square was 0.996. Compared with the Takebi model [80] for thermal conductivity of hybrid nanofluids, Equation (12) can provide a more accurate prediction for the thermal conductivity of water-based GO–Al₂O₃ hybrid nanofluids, but it cannot reflect any mechanisms that explain the reason behind thermal conductivity enhancement.

3.2.6. Mechanism Model of Thermal Conductivity of the Hybrid Nanofluid

Many researchers found [54,83] that suspended nanoparticles enhance the energy transfer of nanofluids in two ways: the first is static effective thermal conductivity, which can be modeled according to the effective medium theory [84]; the second is the stochastic motions of the nanoparticles and the interfacial interactions between particles and the liquid molecules. Assuming that the two mechanisms can be separated, the total effective heat transfer in the nanofluidic suspension can be written as follows [85]:

$$k_{eff}=k_{eff,EMT}+k_{eff,Brown}$$

(14)

Al₂O₃ is a spherical particle, thus, its static thermal conductivity can be established by the H-C model [86], which introduces the influence of particle shape on thermal conductivity. The model is as follows:

$$\frac{k_{eff,EMT,1}}{k_{bf}}=\frac{k_{np}+\left(n-1\right)k_{bf}-\left(n-1\right)\phi \left(k_{bf}-k_{np}\right)}{k_{np}+\left(n-1\right)k_{bf}+\phi \left(k_{bf}-k_{np}\right)}$$

(15)

where n is an empirical shape factor, and n is 3 for spherical particles. $K_{eff,EMT,1}$ is the effective thermal conductivity of the Al₂O₃ nanofluid.

Considering the effect of thickness t, length L, interfacial thermal resistance R_k and average flatness ratio η, the effective thermal conductivity of the GO nanofluid, $K_{eff,EMT,1}$, was calculated as Equation (16) [54,87]:

$$\frac{k_{eff,EMT,2}}{k_{bf}}=\frac{3+2{\eta }^2\phi /\left[k_{bf}\left(2\frac{R_k}{L}+13.4\sqrt{t}\right)\right]}{3-\eta \phi }$$

(16)

For the enhancement of thermal conductivity by random motion of nanoparticles, especially in lower viscosity liquid, Equation (17) was employed by Xuan et al. [88] considering the effects of viscosity, temperature and properties of nanoparticles on thermal conductivity.

$$K_{eff,Brown}=\frac{{\rho }_{np}\phi C_{p,np}}{2}\sqrt{\frac{K_{B}T}{3\pi \mu r_c}}$$

(17)

ρ_np and C_p,np are the density and specific thermal capacity of nanoparticles, respectively, μ is the viscosity of the base fluid, k_B is the Boltzmann constant, T is the Kelvin temperature and r_c is the mean radius of gyration of the cluster. For spherical Al₂O₃ nanoparticles, r_c is the radius. For the GO nanosheet, as in our previous study [54], r_c is approximated to half of the equivalent diameter $d_{p,eq}$ of the nanosheet.

$$d_{p,eq}={(\frac{6v_{non-sph}}{\pi })}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}$$

(18)

Non-spherical particles volume v_non-sph can be deemed as theoretical spherical particles that have the same volumes as the non-spherical particles [89].

Combining the influence of static thermal conductivity and Brownian motion of the hybrid nanofluid, a new thermal conductivity model of the Al₂O₃-GO hybrid nanofluid, $k_{\mathrm{eff},\mathrm{hnf}},$ was developed as Equation (19):

$$k_{\mathrm{eff},\mathrm{hnf}}=\phi \left[k_{bf}\left(\frac{k_{eff,EMT,1}}{K_{bf}}+k_{eff,Brown,1}\right)\right]+\left(1-\phi \right)\left[k_{bf}\left(\frac{k_{eff,EMT,2}}{k_{bf}}+k_{eff,Brown,2}\right)\right]$$

(19)

where $\phi$ is the volume fraction of Al₂O₃ nanoparticle in the total hybrid nanoparticles. Subscript 1 and 2 are for Al₂O₃ and GO, respectively. The exact values for L, t, η and R_k are fitted based on the experimental data. According to our previous research work [54], the range of L, t, η and R_k can be determined as: L ∊ (8 × 10⁻⁶, 6 × 10⁻⁵), t ∊ (1.6 × 10⁻⁹, 6 × 10⁻⁸), η ∊ (0.2, 0.8) and R_k ∊ (1 × 10⁻¹⁰, 1 × 10⁻⁶). The fitting results are shown in

Table 3. The fitting values of η, R_k, L, t and R-square.

η	Rk/m2Kw−1	L/m	t/m	R2
0.3928	7.494 × 10−9	1.974 × 10−5	9.000 × 10−9	0.8658
0.3192	4.817 × 10−9	3.296 × 10−5	6.244 × 10−9	0.8658
0.3211	3.113 × 10−9	2.167 × 10−5	6.312 × 10−9	0.8658

when R-square was at the maximum with 95% confidence bounds. Moreover, L was approximately 20 μm according to SEM, t was about 5 nm as reported by the supplier. The values of the four parameters could be determined as L = 2.167 × 10⁻⁵ m, t = 6.312 × 10⁻⁹ m, η = 0.3211 and R_k = 3.113 × 10⁻⁹ m²Kw⁻¹.

It is demonstrated that R_k of the GO–Al₂O₃ aqueous hybrid nanofluid was less than 1.5 × 10⁻⁸ m²kw⁻¹ of graphene and water interfacial thermal resistance [54]. This is because GO had better hydrophilicity and better wettability at the solid–liquid interface, which led to a lower interfacial thermal resistance. On the other hand, the interaction between the GO nanosheet and Al₂O₃ nanoparticles might change the properties of the interface and reduce the interfacial thermal resistance. Figure 19 compares the calculated results using the new model with the experimental data; the average error was 0.18%, and the maximum deviation was 3.8% at 0 °C. The main reason is that the model underestimates the influence of Brownian motion at a lower temperature, as many researchers [83] found the temperature dependence of this model ($\propto \sqrt{T}$) is still small and presents little agreement with several experimental studies, which means that the impact of Brownian motion is greater than that obtained in this work.

3.3. Thermal Conductivity of Ice Containing the Hybrid Nanoparticles

The thermal conductivity of ice mixed with the hybrid nanoparticles may be different from that of pure ice. The thermal conductivity enhancement of the ice as functions of temperature and the mass fraction of hybrid nanoparticles is shown in Figure 20, and the standard thermal conductivity of ice refers to [90]. The thermal conductivity enhancement of ice follows a different behavior in comparison with that of the hybrid nanofluids. It seems that the low mass fraction of hybrid nanoparticles can enhance the thermal conductivity with an enhancement increase of about 8% at 0.025%, −10 °C. This is probably because the addition of hybrid nanoparticles changes the collective vibrations of atoms in the crystal lattice, and affects the transfer of phonons. However, only three plots were achieved at each temperature because of the influence of coronavirus, further research should be carried out to give more definite relations between mass fraction and the thermal conductivity enhancement.

4. Conclusions

Aqueous GO–Al₂O₃ hybrid nanofluids used in thermal ice storage were prepared and characterized. The influences of pH, dispersant, ultrasonic power and ultrasonic time on the stability of hybrid nanofluids were quantitatively analyzed. The thermal conductivities of the hybrid nanofluids were measured and correlated with temperature and concentration. Thermal conductivity of ice containing the hybrid nanoparticles was tested. A new model for the effective thermal conductivity of the hybrid nanofluids considering the structure and Brownian motion was proposed. The key results of this study were as follows:

(1) The optimal conditions for stability of the aqueous GO–Al₂O₃ hybrid nanofluids (mass fraction of GO: Al₂O₃ = 1:1) were: pH = 11, 1% SDS, ultrasonic power of 375 W and ultrasonication time of 120 min.

(2) With the increase in temperature and the mass fraction of the nanoparticles, thermal conductivity of the hybrid nanofluid increased. The influence of mass fraction on the thermal conductivity enhancement at a lower temperature was greater than that at a high temperature. At 0 °C, the thermal conductivity of 0.2 wt % hybrid nanofluids increased by 7.1% as compared to that of the base fluid, which means that this hybrid nanofluid had a potential advantage to enhance heat transfer performance.

(3) The nanoparticle mass fraction had a significant influence on the thermal conductivity of the hybrid nanofluid. The influence of temperature on thermal conductivity enhancement at a lower concentration was larger, even though at a higher concentration there was greater thermal conductivity enhancement.

(4) Comparison of thermal conductivity of the GO nanofluid with the hybrid nanofluid displays similar thermal conductivity. Using less GO in the hybrid nanofluid could reduce the cost of the hybrid nanofluid. Moreover, a low mass fraction of hybrid nanoparticles could enhance the thermal conductivity of ice.

(5) The correlation generated for thermal conductivity of the hybrid nanofluid provided an accurate prediction of the thermal conductivity of water-based GO–Al₂O₃ hybrid nanofluids. The new model for thermal conductivity of hybrid nanofluid shows the influence of thickness, length, flat rate and the interface thermal resistance of the graphene oxide, particle radius of Al₂O₃, fluid viscosity, temperature and Brownian motion. Results presented in this study gave a better understanding of the influence of various important factors on the enhancement of thermal conductivity of hybrid nanofluids.