## 4 Results and Discussion  
### Stability Map of the Static System  
In SCWR, the reactor core involves parallel supercritical heated channels. Table 1 lists the reference condition for three representative supercritical heating channels, which are extracted from the design data for U.S. reference SCWR [25]. In the present model, the fluid properties of supercritical water are obtained from the database of Nation Institute of Standards and Technology (NIST) [26], which have been widely used in the literature [5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16]. Based on the parameter values given in Table 1, this non-linear model can explore the stability maps of the static system by setting \(a_{scs}^{+}=0\) through time domain evolutions. With a perturbation in the inlet velocity of the highest heating channel (Ch.1) at a constant pseudo-subcooling number of \(N_{P\text{-}SUB}\) = 1.55, Figure 3a illustrates the examples of transient responses corresponding to a stable state, a boundary state and an unstable state of a three supercritical water-heated channel system with a radial power distribution of 1.2:1.0:0.8. Under the constraint by constant total flow rate, all the results in Figure 3a show that the highest amplitude oscillation in the highest heating channel (Ch.1) is out-of-phase with those of the other less-heated channels (Ch.2 and Ch.3). The stability boundary could be judged by the transient responses among three asymmetric heating channels. For a stable state of \(N_{P\text{-}SUB}\) = 1.55 and \(N_{TTC}\) = 5.645, the perturbation would decay to its original steady state, while for an unstable state of \(N_{P\text{-}SUB}=1.55\) and \(N_{TPC}=5.704\), the time domain oscillations among channels are growing and eventually trigger the reversed flow. In addition, the results in Figure 3a display the fact that transient responses among channels finally evolve to stable limit cycle oscillations at a boundary state of \(N_{P\text{-}SUB}=1.55\) and \(N_{TPC}=5.669\). Through non-linear analyses as the examples illustrated in Figure 3a, the results in Figure 3b reveal the comparison of stability boundary between parallel three heated channels and single heated channel with supercritical water under the system pressure of 25 MPa. These stability maps are presented based on the pseudo-subcooling number (\(N_{P\text{-}SUB}\)) and the average true trans-pseudo-critical number (\(N_{TPC}\)), which were originally introduced by [PERSON] and [PERSON] [8]. \[N_{TPC}=\frac{Q_{tot}}{W_{tot}}\frac{\beta_{PC}}{C_{P,PC}},\ N_{P-SUB}=\frac{ \beta_{PC}}{C_{P,PC}}(h_{PC}-h_{I}) \tag{18}\]  
The three-region methodology for the supercritical water-heated system at 25 MPa can be applied when the inlet flow temperature is between 0 \({}^{\circ}\)C and 349 \({}^{\circ}\)C while two-region model, excluding region 1, is employed instead when inlet flow temperature is beyond \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Channel 1** & **Channel 2** & **Channel 3** \\ \hline Heat flux ratio & above 1.2 & 0.9 to 1.19 & below 0.9 \\ Length (\(L_{H}\)) & 4.2672 m & 4.2672 m & 4.2672 m \\ Diameter (\(D_{H}\)) & 3.4 mm & 3.4 mm & 3.4 mm \\ Outlet loss coefficient (\(k_{c}\)) & 1.0 & 1.0 & 1.0 \\ Inlet loss coefficient (\(k_{i}\)) & 22.7 & 93.0 & 241.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The reference case of three supercritical heated channels [25]. Figure 3: Investigations of supercritical water heated systems through non-linear analysis: (**a**) examples of stable state, boundary state and unstable state; (**b**) stability map of the static system with parallel three heated channels or single heated channel. 349 \({}^{\circ}\)C (N\({}_{P\text{-}SUB}\) < 0.904). Moreover, the case with the inlet fluid temperature below 0 \({}^{\circ}\)C (N\({}_{P\text{-}SUB}\) > 3.59) implies a restricted area that prohibits the supercritical water flow. The results in Figure 3b suggest that the system of parallel three channels with a heat flux ratio of 1.2:1.0:0.8 is more stable than the single channel system in this study.