## 2 The Model and Solution Method  
Based on our preliminary work [22] employing the external force method to evaluate seismic impact, the non-linear dynamic model of parallel supercritical water uniformly-heated channels in the static system developed previously by the authors [14] is extended to the vertical vibration system. The modelling structure involving M parallel uniformly-heated channels subject to vertical seismic acceleration at a supercritical pressure is displayed in Figure 1. Adopting the three-region methodology suggested by [PERSON] et al. [7], each supercritical water-heated channel is separated into region 1 of heavy fluid, region 2 of heavy and light fluid mixture and region 3 of light fluid, as shown in Figure 1. At a supercritical pressure of 25 MPa, the dividing point \"A\" between region 1 and 2 is about 349 \({}^{\circ}\)C and \"B\" between region 2 and 3 is nearly 449 \({}^{\circ}\)C, respectively. Considering the j-th channel in the parallel supercritical heated channel system in Figure 1, the analytical problem can be simplified on the basis of following assumptions:  
* The whole system is assumed as a rigid body;  
Figure 1: Modelling structure of parallel supercritical water uniformly-heated channels analyzed by three-region methodology [7] under vertical seismic acceleration. (“A” and “B” indicate the separation points between regions). * The system is retained at a supercritical pressure of 25 MPa under both static and external vibration conditions, while the pressure drop through each channel is usually far less than the system pressure;
* Each channel is supposed to have a uniform heat flux distribution along the axial direction;
* The same constant inlet enthalpy is distributed over all channels under both static and external vibration conditions;
* The parallel channels possess the same pressure drop under both static and external vibration conditions, as they are connected by the common upper plenum and lower plenum. For future applications to different supercritical fluids besides supercritical water, the state variables for each channel are non-dimensionalized based on the fluid properties corresponding to the pseudocritical point at a supercritical pressure, as defined in the nomenclature. When considering the effect of seismic motion on the present system, the entire system is supposed as a rigid body, which infers no deformation occurring in system components during seismic motions. This demonstrates that all the parts are subject to the same but variant vibration acceleration, inferring they face the external force induced by the seismic motions. Thus, based on the assumptions and the external force method, with the non-dimensional methodology [8], the one-dimensional dimensionless conservation equations for the j-th supercritical uniformly-heated channel subject to vertical seismic acceleration can be written as:  
\[\frac{\partial\rho_{j}^{+}}{\partial t^{+}}+\frac{\partial\rho_{j}^{+}u_{j}^{ +}}{\partial z^{+}}=0 \tag{1}\]  
\[\frac{\partial\rho_{j}^{+}h_{j}^{+}}{\partial t^{+}}+\frac{\partial\rho_{j}^{+ }h_{j}^{+}u_{j}^{+}}{\partial z^{+}}=N_{P-PCHj} \tag{2}\]  
\[\frac{\partial\rho_{j}^{+}u_{j}^{+}}{\partial t^{+}}+\frac{\partial\rho_{j}^{ +}u_{j}^{+2}}{\partial z^{+}}=-\Lambda_{j}\rho_{j}^{+}u_{j}^{+}\rho_{j}^{-} \sum_{m=1}^{N}k_{m,j}\delta\left(z_{j}^{+}-z_{m,j}^{+}\right)\frac{\rho_{j}^{+ }u_{j}^{+}}{2}-\frac{\rho_{j}^{+}u_{V}^{+}}{Fr}-\frac{\partial P_{j}^{+}}{ \partial z^{+}} \tag{3}\]  
where \(N_{P-PCHj}\) is pseudo phase-change number for the j-th channel that was originally suggested by [PERSON] and [PERSON] [8], as defined in the nomenclature. The parameter \(a_{V}^{+}\) represents the dimensionless vertical acceleration including both the gravity and vertical seismic (\(a_{ses}^{+}\)) accelerations. Thus,  
\[a_{V}^{+}=\left\{ \begin{array}{rl}1+a_{ss}^{+},&\text{\it under vertical vibrations}\\ 1,&\text{\it under static conditions}\end{array} \right.