## 2 The Model and Solution Method
\tag{4}\]  
The parallel heated channels shown in Figure 1 exhibit the same dynamic pressure drop under seismic conditions as they are linked with the common upper plenum and lower plenum. \[\Delta P_{ch,1}^{+}=\Delta P_{ch,j}^{+}=\cdots=\Delta P_{ch,M}^{+} \tag{5}\]  
The dynamic pressure drop for each channel subject to vertical seismic acceleration could be estimated by the summation of the dynamic pressure drop in the static condition plus the additional pressure drop induced by the seismic vibration:  
\[\Delta P_{ch,j}^{+}=\Delta P_{A,j}^{+}+\Delta P_{B,j}^{+}+\Delta P_{C,j}^{+} +\Delta P_{ses,j}^{+}=Coef_{j}\frac{du_{ij}^{+}}{dt^{+}}+\Delta P_{ch0,j}^{+} +\Delta P_{ses,j}^{+},\,j=1,2,\cdots,M \tag{6}\]  
\[Coef_{j}=C_{A,j}\lambda_{A,j}^{+}+C_{B,j}(\lambda_{B,j}^{+}-\lambda_{A,j}^{+}) +C_{C,j}(1-\lambda_{B,j}^{+}) \tag{7}\]  
where \(\Delta P_{A,j}^{+}\), \(\Delta P_{B,j}^{+}\) and \(\Delta P_{C,j}^{+}\) indicate the dynamic pressure drop for region 1, region 2 and region 3 in the static condition, respectively. All these terms, including \(\Delta P_{ch0,j}^{+}\), \(C_{A,j}\), \(C_{B,j}\)and \(C_{\text{C},j}\), are defined in [PERSON] et al. [14]. Moreover, \(\Delta P_{\text{se},j}^{+}\) is seismic-induced pressure drop under vertical vibration condition, as given below:  
\[\Delta P_{\text{se},j}^{+}=\Big{[}C_{A,j}\lambda_{A,j}^{+}+C_{B,j}(\lambda_{B,j }^{+}-\lambda_{A,j}^{+})+C_{\text{C},j}(1-\lambda_{B,j}^{+})\Big{]}a_{\text{se} }^{+}/Fr \tag{8}\]  
Through Equations (5) and (6), it can be deduced that the dynamic equations of inlet flow velocity, \(u_{i,j}^{+}\), among parallel supercritical heated channels under vertical seismic acceleration are:  
\[\frac{du_{i,j}^{+}}{dt^{+}}=Coe1j\frac{du_{i,1}^{+}}{dt^{+}}+Coe2j,\,j=2,3, \cdots,M \tag{9}\]  
\[Coe1j=Coef_{1}/Coef_{j} \tag{10}\]  
\[Coe2j=(\Delta P_{ch0,1}^{+}+\Delta P_{\text{se},1}^{+}-\Delta P_{ch0,j}^{+}- \Delta P_{\text{se},j}^{+})/Coef_{j} \tag{11}\]  
In addition, for this parallel channel system, the total flow rate is the summation of every channel flow rate. The variation in total flow rate (\(W_{tot}^{+}\)) subject to seismic motion can be considered by the time-derivative term:  
\[\frac{dW_{tot}^{+}}{dt^{+}}=\sum_{j}\rho_{ij}^{+}\frac{du_{i,j}^{+}}{dt^{+}} \tag{12}\]  
Substituting Equation (9) into Equation (12) can lead to the dynamic equation of inlet flow velocity,\(u_{i,1}^{+}\), for the first channel under vertical seismic acceleration:  
\[\frac{du_{i,1}^{+}}{dt^{+}}=\Bigg{(}\frac{dW_{tot}^{+}}{dt^{+}}-\sum_{j}\rho_{ ij}^{+}Coe2j\Bigg{)}/\sum_{j}\rho_{ij}^{+}Coe1j \tag{13}\]  
Through Equation (13) together with Equation (9) and adopting all the other dimensionless dynamic equations given in [PERSON] et al. [14], the present non-linear dynamic model can be employed to explore the influences of vertical seismic accelerations on the system with parallel supercritical water uniformly-heated channels. If the analytical system has a constant total flow rate during seismic motion, the term of \(dW_{tot}^{+}/dt^{+}\) is set to zero. On the other hand, if the system only comprises single supercritical heated channel, the boundary condition can be simplified as: the dynamic pressure drop of the supercritical uniformly-heated channel under vertical seismic accelerations (\(\Delta P_{ch}^{+}\)) is equal to the summation of pressure drop for the static system and that additionally contributed by the external vertical acceleration (\(\Delta P_{\text{se}}^{+}\)). Thus,  
\[\Delta P_{ch}^{+}=\Delta P_{A}^{+}+\Delta P_{B}^{+}+\Delta P_{\text{C}}^{+}+ \Delta P_{\text{se}}^{+}=Coef\frac{du_{i}^{+}}{dt^{+}}+\Delta P_{\text{ch},0}^ {+}+\Delta P_{\text{se}}^{+} \tag{14}\]  
where \(\Delta P_{A}^{+}\), \(\Delta P_{B}^{+}\) and \(\Delta P_{\text{C}}^{+}\) represent the dynamic pressure drop for region 1, region 2 and region 3, respectively, inside single supercritical heated channel in the static condition. The corresponding formulas in a parallel channel system [14] can be applied to a single channel case.