## 2 Experimental Setup and Data Reduction
All of the independent parameters, such as the test section's inlet temperature, mass flow rate, pressure, and heat flux, were monitored and controlled by an NI (National instruments) Labview data acquisition system. During the experimental process, when the system achieved steady state, the data was recorded for 40 s at the rate of 3 Hz, and the average of these data points was used for the analysis. For each operating condition, the test system was assumed to be at steady state, while the energy balance was obtained using Equation (1). \[\eta=\frac{G\frac{\pi}{4}d_{\text{in}}^{2}\left(H_{\text{out}}-H_{\text{in}} \right)}{U\times I} \tag{1}\]  
where \(\eta\) is the thermal efficiency, \(G\) is the mass flux (kg/m\({}^{2}\)s), \(U\) is the voltage (V), \(I\) is the current (A), \(d_{\text{in}}\) is the inside diameter (equal to 5 mm), and \(H_{\text{out}}\) and \(H_{\text{in}}\) are the inlet and outlet bulk fluid enthalpies (kJ/kg), respectively. The energy balance of the whole system was found to have a deviation of \(\pm\)4%. The total averaged heat generation of the test section was determined using Equation (2). \[\overline{q}=\frac{U\times I\times\eta}{\pi\text{d}_{\text{in}}L} \tag{2}\]  
Figure 5: Installation of the thermocouples at the outer wall of the test section. where \(\overline{q}\) is the averaged heat flux, and \(L\) is the heating length, whereas all of the other symbols carry the same meaning and units of measurement as described for Equation (1). However, the ohmic resistance varies with temperature for 304 stainless steel, and does not remain constant. The electric resistivity almost varies linearly with temperature. According to the variation in slope, it is estimated that when the temperature lies within the range of 0-200 \({}^{\circ}\)C, the uncertainty in heat flux is \(\pm\)3.8%. However, the uncertainty in heat flux becomes \(\pm\)8.2% when the temperature of the test section lies within the range of 0-400 \({}^{\circ}\)C due to the variation of electric resistivity. Based on the measured outer-wall temperature data, inner wall temperatures were estimated by assuming a simple one-dimensional, steady-state conduction equation, which is given by Equation (3). \[t_{\text{in}}=t_{\text{out}}+\frac{q(\text{x})d_{\text{in}}}{2K_{\text{ss}}} \ln(\frac{d_{\text{out}}}{d_{\text{in}}}) \tag{3}\]  
where \(d_{\text{in}}\) and \(d_{\text{out}}\) are the inner and outer diameters of the test sections, respectively, \(q(\text{x})\) is the local heat flux, and \(K_{\text{ss}}\) is the thermal conductivity of the stainless steel. Furthermore, \(K_{\text{ss}}\) is calculated using Equation (4). \[K_{\text{ss}}=14.408\ (1.0+0.0011332\times T) \tag{4}\]  
where \(h\) is the heat transfer coefficient, which is given by Equation (5). \[h=\frac{q}{t_{w}-t_{f}} \tag{5}\]  
where \(t_{w}\) is the inner-wall temperature, and \(t_{f}\) is the bulk fluid temperature. To describe the large buoyancy effect in supercritical flow, [PERSON] [39] summarized a criterion defined using the buoyancy parameter for predicting the onset of deteriorated heat transfer. \[{B_{o}}^{*}={G_{r}}^{*}/{\text{Re}}_{b}^{3.425}{\text{Pr}}_{b}^{0.8} \tag{6}\]  
where, \({G_{r}}^{*}=\frac{\rho_{b}(\rho_{b}-\rho_{m})q\bar{q}d^{3}}{\mu_{b}^{2}(\mu_{ m}-t_{b})k_{b}}\ Re_{b}=\frac{Gd}{\mu_{b}}\), \(\rho\) is the density (kg/m\({}^{3}\)), \(\mu\) is the viscosity (Pa-s), \(g\) is the gravity (kg/m\({}^{2}\)), \(d\) is the inner diameter (mm), and subscript \(b\) indicates the bulk fluid. The thermal conductivity of the bulk fluid is k\({}_{b}\). [PERSON] et al. [39] proposed another criterion to determine the onset of heat transfer deterioration. \[\overline{Gr_{b}}/Re_{b}^{2.7}<10^{-5} \tag{7}\]  
where \(\overline{Gr_{b}}=\frac{\rho b(\rho b-\overline{p})gd^{3}}{\mu_{b}^{2}}\), \(Re_{b}=\frac{Gd}{\mu_{b}}\), \(\overline{p}=\frac{1}{t_{w}-t_{b}}\int_{t_{b}}^{t_{w}}\rho dT\).