## 2 Materials and Methods
### Methodology
#### 2.2.4 Experimental Heat Transfer Analysis  
Figure 1 illustrates the layout of the experimental facility where the nanofluids' heat transfer performance of convection was experimentally determined. A detailed description of the experimental facility is reported in previous works [25]. Tests were conducted at three cold fluid average temperatures (base fluid or nanofluid), _cf_, specifically 298, 308, and 318 K. The temperature of the hot fluid (water), _hf_, was kept 15 K higher than these values. The cold fluid flow rate was adjusted between 0.3 and 0.7 m\({}^{3}\).h\({}^{-1}\), in increments of 0.1 m\({}^{3}\).h\({}^{-1}\), while the flow rate of the hot fluid remained constant at 0.7 m\({}^{3}\).h\({}^{-1}\) for all experiments. The following hypotheses were considered for the experimental heat transfer coefficients determination: steady-state conditions; no heat losses to the surroundings; unidirectional flow through the heat exchanger with the fluids velocity perpendicular to the cross-section; constant and uniform thermophysical properties throughout the heat exchanger; and the thermal resistances associated with fouling being ignored.  
Convective heat transfer coefficients (_h_) were determined for the base fluid and three of the analysed nanofluids (0.25, 1.0, and 2.0 wt%) by means of the thermal resistances method, which accounts for the heat transferred from the hot fluid to the steel wall by convection, the heat conduction through the steel wall, and the transferred heat from the wall to the cold fluid by convection,  
\[R_{hf}=1/\left(h_{hf}\cdot\pi\cdot d_{2}\cdot L_{h}\right) \tag{6}\]  
\[R_{s}=\ln\left(\frac{d_{2}}{d_{1}}\right)/\left(2\pi\cdot k_{s}\cdot L_{h}\right) \tag{7}\]  
Figure 1: Diagram of the experimental test rig utilised for obtaining the convective heat transfer coefficients.  
\[R_{cf}=1/\left(h_{cf}\cdot\pi\cdot d_{1}\cdot L_{h}\right) \tag{8}\]  
where \(d_{1}\) and \(d_{2}\) are the internal and outer diameters of the inside pipe within the double-tube heat exchanger, respectively, \(L_{h}\) represents the heat exchange effective length, and \(k_{s}\) denotes the thermal conductivity of the steel. The overall thermal resistance, \(R_{T}\), is equal to the sum of the three abovementioned thermal resistances, which can also be obtained as:  
\[R_{T}=\frac{\Delta T_{ML}}{\dot{Q}} \tag{9}\]  
where \(\Delta T_{ML}\) is the heat exchanger logarithmic mean temperature difference and \(\dot{Q}\) is the exchanged heat flux [26]. The hot fluid coefficients of convective heat transfer through the annular section, \(h_{hf}=k_{hf}/\left(Nu_{hf}\cdot d_{h}\right)\), were calculated by means of Gnielinski correlations applicable to fully developed turbulent flow in annular ducts [27]:  
\[Nu_{hf}=\left[\frac{\frac{f_{an}}{8}\cdot Re_{hf}\cdot Pr_{hf}}{a_{1}+12\cdot 7 \cdot\sqrt{\frac{f_{an}}{8}\cdot\left(Pr_{hf}^{2/3}~{}-1\right)}}\cdot\left[1+ \left(\frac{d_{h}}{L_{h}}\right)^{\frac{2}{3}}\right]\right]\cdot\psi \tag{10}\]  
with  
\[a_{1}=1.07+\frac{900}{Re_{hf}}-\frac{0.63}{1+10\cdot Pr_{hf}} \tag{11}\]  
where \(f_{an}\) is the friction factor for annular pipes, \(Re_{hf}\) is the Reynolds number of the hot fluid, \(Pr_{hf}\) is the Prandtl number of the hot fluid, \(d_{h}\) is the annular section hydraulic diameter, \(d_{h}=d_{3}-d_{2}\), \(\psi\) is the boundary condition factor, \(\psi=0.75\cdot\left(d_{2}/d_{3}\right)^{-0.17}\), and \(d_{3}\) is the internal diameter of the outside pipe of the double-tube heat exchanger.