## 1 Introduction
Nevertheless, due to the coupling of steeply varying thermo-physical properties of the fluid and the formation of special flow patterns in heating tubes [23; 24; 25], the flow and heat transfer in heating or cooling channels has become more complex. At present, there is still no consensus on the effects of various parameters on the convective heat transfer, especially in cases with high heat flux (\(q\))/ mass flux (\(G\)) ratios. Figure 2 shows three kinds of completely different temperature distributions for various bulk enthalpy values at different mass fluxes, which were derived from [PERSON] [26] (used electric heater, experimental range: \(G\) = 285-1200 kg/m\({}^{2}\)s, \(q\) = 30-170 kW/m\({}^{2}\)), [PERSON] [27], and [PERSON] [28]. At moderate (\(G\) = 700 kg/m\({}^{2}\)s; \(q\)/\(G\) = 0.14 kJ/kg, as seen in Figure 2b, data from [PERSON] [27] (used electrical heater, inlet temperature is 7.1-13.8 \({}^{\circ}\)C)) and high (\(G\) = 1200 kg/m\({}^{2}\)s, \(q\)/\(G\) = 0.216 kJ/kg, as seen in Figure 2c, data from [PERSON] [28] (used electric heater, \(Re_{\rm in}\) = 2.3 \(\times\) 10\({}^{5}\)-1.18 \(\times\) 10\({}^{6}\))) mass fluxes, deteriorated heat transfer occurred, and a temperature peak appeared in the region of 240-280 kJ/kg. At low mass flux (\(G\) = 232 kg/m\({}^{2}\)s, \(q\)/\(G\) = 0.259 kJ/kg, as seen in Figure 2a, data from [PERSON] [23]), the temperature increased monotonously, and no temperature peak was observed, even for high heat fluxes. At low mass flux (\(G\) = 100 kg/m\({}^{2}\)s), the temperature changed smoothly without any distinct temperature peak, as seen in Figure 3 This behavior was different from that observed for the moderate (\(G\) = 500 kg/m\({}^{2}\)s) and high (\(G\) = 1000 kg/m\({}^{2}\)s) mass fluxes, where an apparent temperature peak  
Figure 2: Profiles of the inner-wall temperatures for different mass fluxes. (**a**) Mass flux (\(G\)) = 232 kg/m\({}^{2}\)s (data from [PERSON] [26]); (**b**) \(G\) = 700 kg/m\({}^{2}\)s (data from [PERSON] [27]); (**c**) \(G\) = 1200 kg/m\({}^{2}\)s (data from [PERSON] [28]). occurred. The uneven temperature profiles indicated the presence of convective instability. For the conditions of heat transfer in supercritical CO\({}_{2}\) at low mass fluxes, no obvious temperature peaks were observed, even if the \(q/G\) was identical (\(q/G\) = 0.2) or much higher (\(q/G\) = 0.3). It is worth mentioning that this phenomenon has not been studied in detail in the literature. Until now, satisfactory analytical methods were not available, due to the difficulty of dealing with temperature-dependent nonlinear property variations. The corresponding and precise mechanisms of heat transfer regimes that cause deteriorated heat transfer at moderate and high mass fluxes are still not well understood. Instead, the viewpoint that supercritical fluid was analogous to subcritical fluid has prevailed in the past few decades. [PERSON] [29] briefly surveyed the existing experimental data of supercritical fluids, and suggested that heat transfer at supercritical pressure was accompanied by a \"boiling-like\" phenomenon. [PERSON] [30] proposed that an unusual heat transfer mode could be explained by a boiling-like phenomenon resembling nucleate boiling. The author proposed a model with liquid-like clusters of molecules below the pseudocritical temperature, and gas-like aggregates above the pseudocritical temperature. Later, [PERSON] [31] argued that this interesting pseudo-boiling was a characteristic of heat transfer under the conditions of \(P>P_{\mathrm{cr}}\) and \(T_{\mathrm{b}}<T_{\mathrm{cr}}<T_{\mathrm{w}}\). [PERSON] [32] also observed the unusual mode of heat transfer, and attributed it to a crisis-like process analogous to the process by which film boiling occurs at subcritical pressures. [PERSON] [33] claimed that more evidence had been collected, and that one of causes for unusual temperature peaking at higher heat fluxes is ascribed to the pseudo film-boiling process.