## 2 Numerical Approach
### Fluid Properties and Two-Phase Modeling
#### 2.3.2 Single-Phase Model  
The single-phase model assumes that the phases are in thermal equilibrium and the relative velocity between the base fluid and the nanoparticles is null. It treats then the nanofluid as a homogeneous fluid with effective thermophysical properties evaluated by theoretical models or empirical correlations.  
All the nanofluid properties are function of the base fluid (\(bf\)) and nanoparticles (\(np\)) properties as well as the volume fraction \(\varphi\) of the nanoparticles. It is recalled that all properties of the base fluid are temperature-dependent and evaluated using Equations (1)-(4). Plenty of correlations are available in the literature [8,10] and it appears crucial to use the most appropriate ones for the effective nanofluid properties to produce accurate results with the single-phase model. The present correlations are chosen to enable direct comparisons with [PERSON] et al. [19]. Two correlations for the thermal conductivity \(k_{nf}\) and for the dynamic viscosity \(\upmu_{nf}\) are considered here. The equations used to evaluate the nanofluid properties (density [11], heat capacity [32,33], viscosity [34,35]) are as follows:  
\[\rho_{nf}=(1-\varphi)\rho_{bf}+\varphi\rho_{np} \tag{5}\]  
\[Cp_{nf}=\frac{(1-\varphi)(\rho Cp)_{bf}+\varphi(\rho Cp)_{np}}{\rho_{nf}} \tag{6}\]  
\[\upmu_{nf}=(1+0.025\varphi+0.015\varphi^{2})\upmu_{bf} \tag{7}\]  
\[\upmu_{nf}=(1+7.3\varphi+123\varphi^{2})\upmu_{bf} \tag{8}\]  
Note that the relations for the dynamic viscosity do not take into account the hysteresis cycle observed by [PERSON] et al. [36] for commercial and highly concentrated solutions of \(Al_{2}O_{3}\)/water-based nanofluids. The thermal conductivity \(k_{nf}\) is evaluated using two different correlations suggested by [5,37] respectively:  
\[k_{nf}=\left[\frac{k_{np}(1+2\alpha)+2k_{bf}-2\varphi(k_{bf}-k_{np}(1-\alpha) )}{k_{np}(1+2\alpha)+2k_{bf}+\varphi(k_{bf}-k_{np}(1-\alpha))}\right]k_{bf} \tag{9}\]  
\[k_{nf}=k_{bf}(1-\varphi)+\gamma k_{np}\varphi+C_{d}\frac{d_{bf}}{d_{np}}k_{bf }{Re_{np}}^{2}Pr\varphi \tag{10}\]  
where \(\alpha=2R_{b}k_{bf}/d_{np}\) is the particle Biot number, \(R_{b}=0.77\times 10^{-8}\) K\(\cdot\)m\({}^{2}\)/W is the interfacial thermal resistance, \(\gamma=0.01\) is a constant taking into account the Kapitza resistance per unit area, \(C_{d}=18\times 10^{-6}\) and \(Re_{np}\) the particle Reynolds number defined as:  
\[Re_{np}=\left(\frac{\vec{C}_{RM}d_{np}}{\
u_{bf}}\right) \tag{11}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & \(\rho\) (kg\(\cdot\)m\({}^{-3}\)) & \(C_{p}\) (J\(\cdot\)kg\({}^{-1}\)\(\cdot\)K\({}^{-1}\)) & \(k\) (W\(\cdot\)m\({}^{-2}\)\(\cdot\)K\({}^{-1}\)) \\ \hline \(C\) & 220 & 710 & 129 \\ \(Cu\) & 8933 & 385 & 401 \\ \(CuO\) & 6510 & 540 & 18 \\ \(Al_{2}O_{3}\) & 3880 & 729 & 42.3 \\ \(TiO_{2}\) & 4175 & 692 & 8.4 \\ \(SiO_{2}\) & 2220 & 745 & 1.4 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Thermophysical properties of different types of nanoparticles.  
In the present case, the random motion velocity \(\bar{C}_{RM}\) is fixed to 0.1 m/s as recommended by [5].  
The general forms of the governing differential equations (conservation of mass, momentum and energy) for the single-phase model are:  
\[\
abla\cdot(\rho\vec{V})=0 \tag{12}\] \[\rho\vec{V}\cdot\
abla\vec{V}=-\
abla P+\
abla\cdot(\mu\
abla\vec{V})+p \vec{g}\] (13) \[\rho\
abla\cdot(\vec{V}H)=-\
abla\cdot q-\tau:\
abla\vec{V} \tag{14}\]