## 3 Results and Discussion
### Thermal Conductivity of the Hybrid Nanofluids
#### 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]:  
\[k_{eff}=k_{eff,EMT}+k_{eff,Brown} \tag{14}\]  
Al\({}_{2}\)O\({}_{3}\) 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:  
\[\frac{k_{eff,EMT,1}}{k_{bf}}=\frac{k_{up}+(n-1)k_{bf}-(n-1)\varphi\Big{(}k_ {bf}-k_{up}\Big{)}}{k_{up}+(n-1)k_{bf}+\varphi\Big{(}k_{bf}-k_{up}\Big{)}} \tag{15}\]  
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 Al\({}_{2}\)O\({}_{3}\) nanofluid. Considering the effect of thickness \(t\), length \(L\), interfacial thermal resistance \(R_{k}\) and average flatness ratio \(\eta\), the effective thermal conductivity of the GO nanofluid, \(K_{eff,EMT,1}\), was calculated as Equation (16) [54; 87]:  
\[\frac{k_{eff,EMT,2}}{k_{bf}}=\frac{3+2\eta^{2}\varphi/\Big{[}k_{bf}/\Big{(}2 \frac{R_{k}}{L}+13.4\sqrt{t}\Big{)}\Big{]}}{3-\eta\varphi} \tag{16}\]  
For the enhancement of thermal conductivity by random motion of nanoparticles, especially in lower viscosity liquid, Equation (17) was employed by [PERSON] et al. [88] considering the effects of viscosity, temperature and properties of nanoparticles on thermal conductivity. \[K_{eff,Brown}=\frac{\rho_{up}\varphi C_{p,up}}{2}\sqrt{\frac{K_{B}T}{3\pi\mu r _{c}}} \tag{17}\]  
\(\rho_{\rm np}\) and \(C_{p,np}\) are the density and specific thermal capacity of nanoparticles, respectively, \(\mu\) is the viscosity of the base fluid, \(k_{\rm B}\) is the Boltzmann constant, \(T\) is the Kelvin temperature and \(r_{c}\) is the mean radius of gyration of the cluster. For spherical Al\({}_{2}\)O\({}_{3}\) 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 \(d_{p,q}\) of the nanosheet. \[d_{p,eq}=(\frac{6v_{non-sph}}{\pi})^{\frac{1}{3}} \tag{18}\]  
Figure 18: Comparison of fitted values and experimental data. Non-spherical particles volume \(v_{\text{non-sph}}\) 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 Al\({}_{2}\)O\({}_{3}\)-GO hybrid nanofluid, \(k_{\text{eff,hInf}}\), was developed as Equation (19):  
\[k_{\text{eff,hInf}}=\varphi\Bigg{[}k_{b}\Bigg{(}\frac{k_{eff,EMT,1}}{K_{bf}}+ k_{eff,Brown,1}\Bigg{)}\Bigg{]}+(1-\varphi)\Bigg{[}k_{bf}\Bigg{(}\frac{k_{eff,EMT,2}}{k_{bf}}+k_{eff,Brown,2} \Bigg{)}\Bigg{]} \tag{19}\]  
where \(\varphi\) is the volume fraction of Al\({}_{2}\)O\({}_{3}\) nanoparticle in the total hybrid nanoparticles. Subscript 1 and 2 are for Al\({}_{2}\)O\({}_{3}\) and GO, respectively. The exact values for \(L\), \(t\), \(\eta\) and \(R_{k}\) are fitted based on the experimental data. According to our previous research work [54], the range of \(L\), \(t\), \(\eta\) and \(R_{k}\) can be determined as: \(L\in(8\times 10^{-6}\), \(6\times 10^{-5})\), \(t\in(1.6\times 10^{-9}\), \(6\times 10^{-8})\), \(\eta\in(0.2\), \(0.8)\) and \(R_{k}\in(1\times 10^{-10}\), \(1\times 10^{-6})\). The fitting results are shown in Table 3 when R-square was at the maximum with 95% confidence bounds. Moreover, L was approximately 20 \(\upmu\)m according to SEM, t was about 5 nm as reported by the supplier.