## 5 Conclusions  
Incorporating the external force method with the three-region methodology in analyzing the supercritical heated channel, this study develops a non-linear dynamic model to investigate the transient behaviors of single channel or parallel channels with supercritical water under the impacts of vertical sinusoidal and seismic accelerations. The present model is validated reasonably based on the experimental data of a single heated channel vertical vibration system. Selecting an actual vertical seismic acceleration wave and the normal operating state as a reference case, the conclusions in the present study can be summarized as:  
1. The strength of seismic-induced oscillation is relevant to the inherent stability natures of the initial state.
2. The effects of parameters on the seismic-induced oscillation conducted in single heated channel indicate that unstable influences could be caused by a longer heated length, uprating operation power and the increase of outlet loss coefficient, while  
Figure 12: Seismic-induced inlet velocity oscillations of parallel three channel system having a constant total flow rate at S1 state subject to a real vertical seismic acceleration.  
Figure 13: Seismic-induced inlet velocity oscillations of parallel three channel system having a constant total flow rate at S3 state subject to a real vertical seismic acceleration.  
stable effects could be illustrated by the increase of inlet loss coefficient, a larger channel diameter and a lower inlet fluid temperature.
3. For a three supercritical heated channel system with a heat flux ratio of 1.2:1.0:0.8, in the region with \(N_{P\text{-}SUB}\) greater than 1.67, the dimensionless natural frequency on the stability boundary is mildly increased as \(N_{P\text{-}SUB}\) is decreased, where the heavy fluid region contributes a substantial role in the channels. However, in the area with \(N_{P\text{-}SUB}\) less than 1.67, it would significantly increase as \(N_{P\text{-}SUB}\) is further decreased, which is due to the effect of light fluid region dominating over the channels.
4. The seismic-induced effects on parallel heated channels are related to the inherent stability characteristics of initial state and the interactions among channels. The effects of inlet loss coefficient and outlet loss coefficient on the vibration-induced oscillations among channels may depend on the operating conditions, which are affected by the power-to-flow difference among channels and stability nature of the system. The more uneven heat flux distribution among channels would cause a larger vibration-induced oscillation.
5. The strength of vibration-induced resonance oscillation among parallel channels is closely related to the inherent stability characteristics of initial state. It may trigger much larger resonance oscillations than in the case of non-resonance.  
Conceptualization, J.D.L.; methodology, J.D.L.; software, J.D.L.; validation, J.D.L. and S.W.C.; investigation, J.D.L.; data curation, S.W.C.; writing--original draft preparation, J.D.L.; writing--review and editing, S.W.C.; visualization, J.D.L.; supervision, J.D.L. and S.W.C. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Ministry of Science and Technology of Taiwan, grant number MOST 107-2221-E-007-096-MY2 and MOST 108-2221-E-007-046-MY3.  
Not applicable.  
Data Availability Statement: Data is contained within the article.  
The authors declare no conflict of interest.