## 3 Results and Discussion
### Thermal Conductivity of the Hybrid Nanofluids
#### 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]. [PERSON] et al. [74] proposed a formula as Equation (4) for thermal conductivity of DWCNT-ZnO/water-EG hybrid nanofluids,  
\[\frac{k_{nf}}{k_{bf}}=1.085e^{(0.001351T+0.13\varphi^{2})}+0.0288\ln(\varphi) \tag{4}\]  
where \(T\) is temperature in \({}^{\circ}\)C and \(\varphi\) is the volume fraction. [PERSON] et al. [75] used Equation (5) for thermal conductivity of multiwalled carbon nanotube (MWCNT)-Fe\({}_{3}\)O\({}_{4}\)/EG hybrid nanofluids:  
\[\frac{k_{nf}}{k_{bf}}=1+0.0162\varphi^{0.7038}T^{0.6009} \tag{5}\]  
Equation (6) was proposed by [PERSON] et al. [76] for thermal conductivity of MgO-f-MWCNT/EG at 25-50 \({}^{\circ}\)C, volume fraction of 0.05%-0.6%. \[\frac{k_{nf}}{k_{bf}}=0.8341+1.1\varphi^{0.243}T^{-0.289} \tag{6}\]  
[PERSON] et al. [77] studied thermal conductivity of SiO\({}_{2}\)-TiO\({}_{2}\) (60:40)/water and EG (60:40), and proposed a formula as Equation (7) for temperature range of 30 to 70 \({}^{\circ}\)C, volume fraction 0-3%. \[\frac{K_{nf}}{K_{bf}}=\Big{(}1+\frac{\varphi}{100}\Big{)}^{5.5}\Big{(}\frac{T }{80}\Big{)}^{0.01} \tag{7}\]  
[PERSON] et al. [78] investigated thermal conductivity of ZnO-Ag/water from 25 to 50 \({}^{\circ}\)C, volume fraction 0.125%-2%, and proposed Equation (8). \[\frac{K_{nf}}{K_{bf}}=1+0.0008794\varphi^{0.5899}T^{1.345} \tag{8}\]  
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]. \[k_{nf}=\frac{k_{p}+2k_{bf}+2\varphi\Big{(}k_{p}-k_{bf}\Big{)}}{k_{p}+2k_{bf}- \varphi\Big{(}k_{p}-k_{bf}\Big{)}}k_{bf} \tag{9}\]  
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  
\[k_{p}=\frac{\Big{(}\varphi_{np1}k_{np1}+\varphi_{np2}k_{np2}\Big{)}}{\varnothing _{lmf}} \tag{10}\]  
Then  
\[k_{lmf}=k_{bf}\Bigg{[}\frac{\Big{(}\varphi_{np1}k_{np1}+\varphi_{np2}k_{np2} \Big{)}}{\frac{\Big{(}\varphi_{np1}k_{np1}+\varphi_{np2}k_{np2}\Big{)}}{\Big{(} \varphi_{np1}k_{np1}+\varphi_{np2}k_{np2}\Big{)}}+2k_{bf}-\Big{(}\varphi_{np1} k_{np1}+\varphi_{np2}k_{np2}\Big{)}+\varphi k_{bf}}\Bigg{]} \tag{11}\]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. 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. \[\frac{k_{nf}}{k_{bf}}=1+0.0162\varphi^{0.7038}T^{0.6009} \tag{5}\]0  
where \(\varnothing_{m}\) is the mass fraction, from 0.025% to 0.2%; \(T\) is the temperature in \({}^{\circ}\)C, ranging from 0 to 70 \({}^{\circ}\)C and \(a\), \(b\), \(c\), \(d\), \(e\) and \(g\) are the parameters fitted by experiment data, as shown in \[\frac{k_{nf}}{k_{bf}}=1+0.0162\varphi^{0.7038}T^{0.6009} \tag{5}\]1 The relationship between volume fraction and mass fraction of the particles can be obtained according to the following equation:  
\[\frac{k_{nf}}{k_{bf}}=1+0.0162\varphi^{0.7038}T^{0.6009} \tag{5}\]2  
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.