# Numerical Study of Heat Transfer Enhancement for Laminar Nanofluids Flow
## 2 Mathematical Formulation
### Thermophysical Properties  
In SPH, the thermophysical properties of the fluids are assumed constant. The dissipation and pressure work are neglected. It was also assumed that the fluid phase and nanoparticles are in thermal equilibrium with zero relative velocity. The effective density (\(\rho_{eff}\)) and specific heat (\(C_{p,eff}\)) of the nanofluids were calculated based on the mixture rule. According to [PERSON] et al. [33], specific heat capacity of a particle varies with particle size. Since smaller particles have larger specific surface areas, the influence of surface energy on the effective specific heat capacity increases with a reduction in particle size. To examine the validity of \(\rho_{eff}\) and \(C_{p,eff}\) equations, other similar studies by [PERSON] et al. [34], [PERSON] et al. [35], [PERSON] et al. [36] and [PERSON] et al. [37] were reviewed. For viscosity, different models can be found in the literature for the effective viscosity of nanofluids as a function of volume fraction. [PERSON] et al. [38] presented a viscosity correlation for concentrated suspensions of nanoparticles. [PERSON] et al. [39] proposed the viscosity model in form of the Taylor series. [PERSON] et al. [40] studied the effect of Brownian motion of the rigid spherical particles on the effective viscosity. For the effective thermal conductivity, [PERSON] et al. [41] presented a definition of a two-components mixture. [PERSON] [42] proposed a model for solid-liquid mixture with relatively large particles. [PERSON] et al. [43] justified the use of the Maxwell model for calculating the thermal conductivity of water-based nanofluids. Table 2 shows the summary of the equations reviewed during this study. For computational purposes the equations from [PERSON] et al. [33], [PERSON] et al. [37], [PERSON] et al. [40] and [PERSON] et al. [28] were used to calculate the thermophysical properties of the nanofluids. The thermophysical properties equations to evaluate the SPD model were expressed as [31]:  
\[k_{eff}\ =\ k_{nf}+k_{\ disp} \tag{4}\]  
\[k_{disp}\ =\ C\left(\rho C_{p}\right)_{nf}ud_{p}R\varphi \tag{5}\]  
\[\mu_{eff}\ =\ \mu_{nf}+\mu_{\ disp} \tag{6}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Properties** & **Author** & **Equations** \\ \hline \multirow{2}{*}{Density (kg/m\({}^{3}\))} & \multirow{2}{*}{[PERSON] et al. [33]} & \(\rho_{eff}\ =\ \frac{m_{i}+m_{p}}{m_{i}+m_{p}}=\ \frac{m_{i}V_{i}+p_{i}V_{i}}{V_{i}+p_{i}}=\ (1-\phi_{i})\rho_{b}+\phi_{i}\rho_{p}\) \\  & & \(\phi_{b}\ =\ \frac{V_{p}}{V_{p}+V_{p}}\) is the volume fraction of the nanoparticles \\ \hline \multirow{2}{*}{Specific Heat (J/kg\(\cdot\)K)} & [PERSON] et al. [36] & \(c_{eff}\ =\ (1-\phi_{i})c_{f}+\phi_{i}c_{p}\) \\ \cline{2-4}  & [PERSON] et al. [37] & \(c_{eff}\ =\ \frac{(1-\phi_{i})\rho_{eff}+\phi_{i}c_{p}}{(1-\phi_{i})^{2}}\) \\ \hline \multirow{4}{*}{Viscosity (kg/m\(\cdot\)s)} & [PERSON] [38] & \(\mu_{eff}\ =\ 1/(1-\phi_{i})^{2.5}\ =\ \left(1+2.5\phi_{b}+4.375\phi_{b}^{2}+\ldots \right)\mu_{f}\) \\ \cline{2-4}  & [PERSON] [39] & \(\mu_{eff}\ =\ \frac{1}{1-2.5\phi_{b}}\mu_{f}\ =\ \left(1+2.5\phi_{b}+6.25\phi_{b}^{2}+O(\phi_{b}^{3}) \right)\mu_{f}\) \\ \cline{2-4}  & [PERSON] et al. [40] & \(\mu_{eff}\ =\ \left(1+2.5\phi_{b}+6.2\phi_{b}^{2}\right)\mu_{f}\) \\ \hline \multirow{4}{*}{Thermal Conductivity (W/m\(\cdot\)K)} & [PERSON] et al. [41] & \(k_{eff}\ =\ \frac{k_{f}(1-\phi_{i})(dT/dx_{i})+k_{f}(d_{i}/dx_{i})}{\phi_{f}(dT/dx_{i})+ (1-\phi_{i})(dT/dx_{i})}\) \\ \cline{2-4}  & [PERSON] et al. [42] & \(k_{eff}\ =\ \frac{k_{f}(1-\phi_{i})(k_{f}-k_{f})}{k_{f}+2\phi_{i}(k_{f}-k_{f})}\)\(k_{f}\ =\ k_{f}+\frac{3\phi_{b}(k_{f}-k_{f})}{k_{f}+2k_{f}-\phi_{i}(k_{f}-k_{f})}\) \\ \cline{2-4}  & [PERSON] et al. [28] & \(k_{eff}\ =\ \frac{k_{f}+2k_{f}(k_{f}/k_{f})\phi_{b}}{k_{f}+2k_{f}-(k_{f}-k_{f})\phi_{b}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Thermophysical properties equations of nanoparticles.