Core

The reactor core generates heat via nuclear fission, which is the primary energy source for nuclear power systems. The thermal-hydraulic process for the reactor core is characterized by a single-channel model [40], which describes a one-dimensional distribution along the axial direction. The governing equations of this model include the continuity, momentum, and energy equations, as follows: $\frac{\text{d}\left(\rho u\right)}{\text{d}x}=\mathrm{0,}$ (1) $\begin{array}{c}\frac{\text{d}\left(\rho uu\right)}{\text{d}x}=\frac{\text{d}}{\text{d}x}\left(\mu \frac{du}{dx}\right)-\frac{\text{d}P}{\text{d}x}\\ -\frac{\rho u^2}{2}\left(\frac{f}{D}+\frac{K_{\text{sp}}}{\text{Δ}x}\right)-\rho g\mathrm{,}\end{array}$ (2) $\frac{\text{d}\left(\rho uh+pu\right)}{\text{d}x}=\frac{\text{d}}{\text{d}x}\left(\lambda \frac{\text{d}T}{\text{d}x}\right)+q_{\text{c}}\mathrm{,}$ (3) where ρ represents the fluid density (kg/m³); x represents the axial coordinate along the core (m); u is the flow velocity (m/s); μ represents dynamic viscosity (Pa⋅s); p represents the pressure (Pa); f represents the Darcy friction coefficient; D represents the equivalent diameter of the reactor core (m) according to the coolant flow rate; K_sp represents the form-resistance pressure-drop coefficient; Δ x denotes the unit node length (m); g represents the gravitational acceleration (m/s²); T is the temperature (K); h is the specific enthalpy (J/(kg)); λ represents thermal conductivity (W/(m⋅K)); q represents the heat absorbed per unit volume of coolant (W/m³), which can be written as $q=Q/V_{\text{g}}$, where V_g represents the per-unit node volume (W/m³); and Q represents thermal power (W). For FDM discretization, the thermal power of the i^th control element is calculated as $Q_i=Q_{\text{c}}{\varphi }_i/{\displaystyle {\sum }_{i=1}^{N{X}_{\text{c}}}{\varphi }_i}$, where Q_c represents the core total power (W); NX_c denotes the discrete number of cores in the axial direction; And ϕ denotes the neutron flux (1/(m²⋅s)). The equation for the friction factor f can be written as: $f=\max \left\{\begin{array}{c}64/Re\mathrm{,}\\ 0.0055\left(1+{\left(20000\frac{\epsilon }{D}+\frac{{10}^6}{Re}\right)}^{1/3}\right)\end{array}\right\}\mathrm{,}$ (4) where ε represents the roughness (m) and Re denotes the Reynolds number.

The form resistance pressure-drop coefficient K_sp is expressed as follows [41]: $K_{\text{sp}}=C_{\text{VD}}{\xi }^2\mathrm{,}$ (5) where ξ represents the ratio of the flow area of the positioning grid to that of the bar bundle and C_VD represents the modified drag coefficient, which can be written as: $C_{\text{VD}}=3.5+\left(\frac{73.14}{{\text{Re}}^{0.264}}\right)+\left(\frac{2.79\times {10}^{10}}{{\text{Re}}^{2.79}}\right)\mathrm{.}$ (6)

Pipe

Two types of distinct pipe models are introduced in this context. The first model is the discrete pressure-drop pipe model, which does not consider heat transfer. Conversely, the second model is the discrete heat-transfer pipe model, which explicitly accounts for heat-transfer effects [42]. The governing equations for the discrete pressure-drop pipe model include the continuity and momentum equations, which are given by the following equations: $\frac{\text{d}\left(\rho u\right)}{\text{d}x}=\mathrm{0,}$ (7) $\frac{\text{d}\left(\rho uu\right)}{\text{d}x}=\frac{\text{d}}{\text{d}x}\left(\mu \frac{\text{d}u}{\text{d}x}\right)-\frac{\text{d}p}{\text{d}x}-\frac{1}{2}\frac{f_{\text{p}}}{d}\rho u^2-\rho g\sin \theta \mathrm{,}$ (8) where θ represents the pipe tilt angle, and d represents the inner diameter of the pipe (m). f_p represents the Darcy friction factor in the pipes, which can be expressed as [43]: $\{\begin{array}{ll}{f}_{\text{p}}\hfill & =\frac{64}{Re}\mathrm{,}R{e}_{\text{D}}\le R{e}_{\text{cr}}\hfill \\ \frac{1}{f_{\text{p}}{}^{1/2}}\hfill & =-2\log \left(\frac{{\epsilon }_{\text{p}}/d}{3.7}+\frac{2.51}{R{e}_{\text{D}}{f}_{\text{p}}{}^{1/2}}\right)\mathrm{,}R{e}_{\text{D}}>R{e}_{\text{cr}}\hfill \end{array}$ (9) where Re_D is the pipe Reynolds number, Re_cr is the critical Reynolds number, and ${\epsilon }_{\text{p}}$ is the pipe roughness (m).

In the discrete heat-transfer pipe model, an energy equation must be incorporated into the continuity and momentum equations. The energy equations can be expressed as follows: $\frac{\text{d}\left(\rho uh\right)}{\text{d}x}=\frac{\text{d}}{\text{d}x}\left(\lambda \frac{\text{d}T}{\text{d}x}\right)+q_{\text{p}}\mathrm{,}$ (10) where q_p denotes volumetric heat-release rate (W/m³).

Steam generator

The steam generator is the interface between the primary and second circuits, where the coolant of the primary circuit flows through the U-tube and transfers heat to the coolant of the second circuit. To simulate this equipment, the one-dimensional dynamic mathematical model proposed by Zhang et al. [44] is adopted, which introduces a distributed parameter model tailored for steam generators in PWRs. As shown in Fig.3, T_1,f and T_1,ad represent the fluid temperature of the primary side in the parallel- and counter-flow sections, respectively. T_w,in,f and T_w,out,f represent the inner and outer wall temperatures of the U-tube in the parallel-flow section, respectively. T_w,in,ad and T_w,out,ad represent the inner and outer wall temperatures of the U-tube in the counter-flow section, respectively. T_2,r and T_2,s represent the preheating and boiling section temperatures of the secondary side fluid, respectively. T_1,in represents the inlet temperature on the primary side, T_2,in is the inlet temperature on the secondary side, and T_1,out represents the outlet temperature on the primary side.

Fig. 3Full-screen View

One-dimensional dynamic model of steam generators

The steady-state heat dynamic balance equations of the parallel and counter flows of the primary side can be written as follows: $\frac{m_1}{M_1}\frac{\text{d}{T}_{\text{1,f}}}{\text{d}x}=\frac{n\pi d_{\text{in}}{K}_1}{M_1{c}_{\text{p,f}}}\left(T_{\text{w,in,f}}-T_{\text{1,f}}\right)\mathrm{,}$ (11) $\frac{m_1}{M_1}\frac{\text{d}{T}_{\text{1,ad}}}{\text{d}x}=-\frac{n\pi d_{\text{in}}{K}_1}{M_1{c}_{\text{p,ad}}}\left(T_{\text{w,in,ad}}-T_{\text{1,ad}}\right)\mathrm{,}$ (12) in which m₁ is the mass flow rate of the primary side (kg/s) defined as $m_1=\frac{n}{4}\pi {\rho }_1{u}_1{d}_{\text{in}}^2$, and ρ₁ and u₁ are the density (kg/m³) and velocity (m/s) of the primary side coolant, respectively; M₁ represents the fluid mass per unit length along the axial direction of the primary side (kg/m) and is defined as $M_1=\frac{n}{4}\pi {\rho }_1{d}_{\text{in}}^2$; n is the number of heat-transfer tubes; d_in denotes the inner diameter of the U-tubes; c_p,f represents the constant-pressure specific heat capacity of the coolant on the primary side (J/(kg⋅K)); And K₁ represents the average heat-transfer coefficient of the primary side (W/(m²⋅K)), which can be written using the Ditus–Boelter formula [45]: $K_1=0.023\frac{{\lambda }_1}{d_{\text{in}}}R{e}_1^{0.8}P{r}_1^{0.3}\mathrm{,}$ (13) where λ₁ denotes the fluid thermal conductivity on the primary side (W/(m⋅K)). Re₁ and Pr₁ are the Reynolds and Prandtl numbers on the primary side, respectively.

The heat-balance equations of the U-tube inner wall for parallel and counter flows can be written as: $\begin{array}{c}\frac{2n\pi {\lambda }_{\text{w}}}{M_{\text{w}}{c}_{\text{p,w}}\ln \left(d_{\text{out}}/d_{\text{in}}\right)}\left(T_{\text{w,out,f}}-T_{\text{w,in,f}}\right)\\ +\frac{n{K}_1\pi d_{\text{in}}}{M_{\text{w}}{c}_{\text{p,w}}}\left(T_{\text{1,f}}-T_{\text{w,in,f}}\right)=\mathrm{0,}\end{array}$ (14) $\begin{array}{c}\frac{2n\pi {\lambda }_{\text{w}}}{M_{\text{w}}{c}_{\text{p,w}}\ln \left(d_{\text{out}}/d_{\text{in}}\right)}\left(T_{\text{w,out,ad}}-T_{\text{w,in,ad}}\right)\\ +\frac{n{K}_1\pi d_{\text{in}}}{M_{\text{w}}{c}_{\text{p,w}}}\left(T_{\text{1,ad}}-T_{\text{w,in,ad}}\right)=\mathrm{0,}\end{array}$ (15) where c_p,w is the constant-pressure specific-heat capacity of the U-tube wall (J/(kg⋅K)); λ_w is the thermal conductivity of the wall (W/(m⋅K)); d_out denotes the outer diameter of the U-tube (m); and M_w denotes the tube-wall mass per unit length in the axial direction (kg/m).

The heat-balance equations of the outer wall of the U-tube for the parallel flow in the preheating and boiling sections can be written as follows: $\begin{array}{c}\frac{2n\pi {\lambda }_{\text{w}}}{M_{\text{w}}{c}_{\text{p,w}}\ln \left(d_{\text{out}}/d_{\text{in}}\right)}\left(T_{\text{w,in,f}}-T_{\text{w,out,f}}\right)\\ +\frac{n{K}_2\pi d_{\text{out}}}{M_{\text{w}}{c}_{\text{p,w}}}\left(T_{\text{2,r}}-T_{\text{w,out,f}}\right)=\mathrm{0,}\end{array}$ (16) $\begin{array}{c}\frac{2n\pi {\lambda }_{\text{w}}}{M_{\text{w}}{c}_{\text{p,w}}\ln \left(d_{\text{out}}/d_{\text{in}}\right)}\left(T_{\text{w,in,f}}-T_{\text{w,out,f}}\right)\\ +\frac{n{K}_{\text{2,s}}\pi d_{\text{out}}}{M_{\text{w}}{c}_{\text{p,w}}}\left(T_{\text{2,s}}-T_{\text{w,out,f}}\right)=0.\end{array}$ (17) Similarly, the heat-balance equation of the outer wall of the U-tube for the counter flow of the preheating and boiling sections can be written as: $\begin{array}{c}\frac{2n\pi {\lambda }_{\text{w}}}{M_{\text{w}}{c}_{\text{p,w}}\ln \left(d_{\text{out}}/d_{\text{in}}\right)}\left(T_{\text{w,in,ad}}-T_{\text{w,out,ad}}\right)\\ +\frac{n{K}_2\pi d_{\text{out}}}{M_{\text{w}}{c}_{\text{p,w}}}\left(T_{\text{2,r}}-T_{\text{w,out,ad}}\right)=\mathrm{0,}\end{array}$ (18) $\begin{array}{c}\frac{2n\pi {\lambda }_{\text{w}}}{M_{\text{w}}{c}_{\text{p,w}}\ln \left(d_{\text{out}}/d_{\text{in}}\right)}\left(T_{\text{w,in,ad}}-T_{\text{w,out,ad}}\right)\\ +\frac{n{K}_{\text{2,s}}\pi d_{\text{out}}}{M_{\text{w}}{c}_{\text{p,w}}}\left(T_{\text{2,s}}-T_{\text{w,out,ad}}\right)=\mathrm{0,}\end{array}$ (19) where K₂ and K_2,s denote the heat-transfer coefficients of the preheating and boiling sections of the secondary side, respectively.

In a natural-circulation steam generator, the heat-transfer process in the preheating section of the secondary side undergoes supercooled boiling. Therefore, the convection heat-transfer coefficient in the preheating section can be uniformly treated by following the approach used for the boiling section and the principles of boiling heat transfer. According to Rohsenow [46], the heat-transfer coefficient can be described as follows: $K_2=\frac{c_{\text{p,s}}}{C_{\text{w}}\gamma \text{P}{\text{r}}_{\text{s}}^m}{q}^{\frac{2}{3}}\cdot {\left[{\mu }_{\text{s}}\gamma \sqrt{\frac{g\left({\rho }_{\text{s}}-{\rho }_{\text{g}}\right)}{\sigma }}\right]}^{\frac{1}{3}}\mathrm{,}$ (20) where c_p,s is the constant-pressure specific heat capacity of saturated water (J/(kg⋅K)); C_w is the combination factor of a particular heating surface fluid; Pr_s represents the Prandtl number of saturated water; m is an empirical index, for water, m=1; μ_s represents the viscosity of saturated water (kg/(m⋅s)); γ denotes the latent heat of vaporization (kJ/kg); g represents gravitational acceleration (m/s²); ρ_s represents the density of saturated water (kg/m³); ρ_g is the density of saturated steam (kg/m³); σ represents the surface tension of water at the vapor-liquid interface (W/m²); and q denotes the heat-flow density on the secondary side (W/m²).

The heat-balance equations of the preheating and boiling sections of the secondary side are as follows: $\frac{m_2}{M_2}\frac{\text{d}{T}_{\text{2,r}}}{\text{d}x}=\frac{n\pi d_{\text{out}}{K}_2}{M_2{c}_{\text{p,2}}}\left(T_{\text{w,out,f}}+T_{\text{w,out,ad}}-2T_{\text{2,r}}\right)\mathrm{,}$ (21) $\frac{m_2}{M_2}\frac{\text{d}{h}_{\text{2,s}}}{\text{d}x}=\frac{n\pi d_{\text{out}}{K}_{\text{2,s}}}{M_2}\left(T_{\text{w,out,f}}+T_{\text{w,out,ad}}-2T_{\text{2,s}}\right)\mathrm{,}$ (22) where c_p,2 represents the constant-pressure specific heat capacity of the fluid-preheating section on the secondary side (J/(kg⋅K)); M₂ represents the fluid mass per unit length along the axial direction of the secondary-side preheating section (kg/m); and h_2,s is the enthalpy of the boiling section on the secondary side (J/kg).

Connection between different components

The primary circuit comprises the above components all connected. Mathematically, this connection is achieved by transferring physical quantities between different component equations, as shown in Fig. 4.

Fig. 4Full-screen View

Connection between different components

In Fig. 4, the inlet and outlet of each component are represented by dashed-line and dashed-dotted-line boxes, respectively. Along the yellow arrows, the output variables of each component, including temperature, pressure, and velocity, are transferred to the next component as input values. For example, the output temperature, pressure, and velocity of the core are transferred to heat section 1 as inlet variables.

Numerical computation methods of the full-order model

The AP1000 primary-circuit model constructed in this study comprises a core, heat section, pressurizer, steam generator, pressure-drop pipe, cold section, and pump, as shown in Fig.1. Among these components, the physical field distributions in the core, heat section, steam generator, pressure-drop pipe, and cold section are computed using the FDM. The calculated area is discretized using a uniform mesh configuration along the coolant-flow direction. The physical properties of the coolant, including thermal conductivity, density, kinematic viscosity, and specific heat capacity, are defined as functions of temperature and pressure based on the IF97 thermophysical property table [47].

In the governing equations of the pipes and cores, the convective terms are discretized using the first-order upwind scheme [48], whereas the diffusion terms are discretized using a central differencing scheme [49]. The discretization formats of the governing equations for both the pipes and reactor core can be expressed as follows, where i represents the index of the internal grids and the subscripts ‘p’ and ‘c’ represent the pipe and core, respectively. Continuity equation: $\frac{{\left(\rho u\right)}_{\text{p}\mathrm{,}i}-{\left(\rho u\right)}_{\text{p}\mathrm{,}i-1}}{\text{Δ}{x}_{\text{p}}}=0,$ (23) $\frac{{\left(\rho u\right)}_{\text{c}\mathrm{,}i}-{\left(\rho u\right)}_{\text{c}\mathrm{,}i-1}}{\text{Δ}{x}_{\text{c}}}=0.$ (24) Momentum equation: $\begin{array}{c}\frac{{\left(\rho uu\right)}_{\text{p}\mathrm{,}i}-{\left(\rho uu\right)}_{\text{p}\mathrm{,}i-1}}{\text{Δ}{x}_{\text{p}}}\\ =\frac{1}{\text{Δ}{x}_{\text{p}}}\left(\frac{{\left(\mu u\right)}_{\text{p}\mathrm{,}i+1}-2{\left(\mu u\right)}_{\text{p}\mathrm{,}i}+{\left(\mu u\right)}_{\text{p}\mathrm{,}i-1}}{\text{Δ}{x}_{\text{p}}}\right)\\ -\frac{p_{\text{p}\mathrm{,}i}-p_{\text{p}\mathrm{,}i-1}}{\text{Δ}{x}_{\text{p}}}-\frac{f_{\text{p}}}{2d}{\rho }_{\text{p}\mathrm{,}i}{u}_{{}_{\text{p}\mathrm{,}i}}^2\\ -{\rho }_{\text{p}\mathrm{,}i}g\sin \theta \mathrm{,}\end{array}$ (25) $\begin{array}{c}\frac{{\left(\rho uu\right)}_{\text{c}\mathrm{,}i}-{\left(\rho uu\right)}_{\text{c}\mathrm{,}i-1}}{\text{Δ}{x}_{\text{c}}}\\ =\frac{1}{\text{Δ}{x}_{\text{c}}}\left(\frac{{\left(\mu u\right)}_{\text{c}\mathrm{,}i+1}-2{\left(\mu u\right)}_{\text{c}\mathrm{,}i}+{\left(\mu u\right)}_{\text{c}\mathrm{,}i-1}}{\text{Δ}{x}_{\text{c}}}\right)\\ -\frac{p_{\text{c}\mathrm{,}i}-p_{\text{c}\mathrm{,}i-1}}{\text{Δ}{x}_{\text{c}}}-\frac{1}{2}\left(\frac{f}{D}+\frac{K_{\text{sp}}}{\text{Δ}{x}_{\text{c}}}\right){\rho }_{\text{c}\mathrm{,}i}{u}_{\text{c}\mathrm{,}i}^2\\ -{\rho }_{\text{c}\mathrm{,}i}g.\end{array}$ (26) Energy equation: $\begin{array}{c}\frac{{\left(\rho uh\right)}_{\text{p}\mathrm{,}i}-{\left(\rho uh\right)}_{\text{p}\mathrm{,}i-1}}{\text{Δ}{x}_{\text{p}}}\\ =\frac{1}{\text{Δ}{x}_{\text{p}}}\left(\frac{{\left(\lambda T\right)}_{\text{p}\mathrm{,}i+1}-2{\left(\lambda T\right)}_{\text{p}\mathrm{,}i}+{\left(\lambda T\right)}_{\text{p}\mathrm{,}i-1}}{\text{Δ}{x}_{\text{p}}}\right)\\ +q_{\text{p}\mathrm{,}i}\mathrm{,}\end{array}$ (27) $\begin{array}{c}\frac{{\left(\rho uh\right)}_{\text{c}\mathrm{,}i}-{\left(\rho uh\right)}_{\text{c}\mathrm{,}i-1}+{\left(pu\right)}_{\text{c}\mathrm{,}i}-{\left(pu\right)}_{\text{c}\mathrm{,}i-1}}{\text{Δ}{x}_{\text{c}}}\\ =\frac{1}{\text{Δ}{x}_{\text{c}}}\left(\frac{{\left(\lambda T\right)}_{\text{c}\mathrm{,}i+1}-2{\left(\lambda T\right)}_{\text{c}\mathrm{,}i}+{\left(\lambda T\right)}_{\text{c}\mathrm{,}i-1}}{\text{Δ}{x}_{\text{c}}}\right)\\ +q_{\text{c}\mathrm{,}i}\mathrm{.}\end{array}$ (28) The inlet boundaries for the temperature, flow velocity, and pressure are applied to the inlet of the pipes and core, and the fully developed boundaries for the outlet boundaries are applied to the pipes and core [50].

For the governing equations of the steam generator, the discretization of Eq. (11), (12), (21), and (22) employs a first-order upwind scheme, which is written as follows: $\begin{array}{l}\frac{m_1}{M_1}\frac{\left(T_{\mathrm{1,}\text{f}\mathrm{,}i}-T_{\mathrm{1,}\text{f}\mathrm{,}i-1}\right)}{\text{Δ}{x}_{\text{SG}}}=\frac{n\pi d_{\text{in}}{K}_1}{M_1{c}_{\text{p,f}}}\left(T_{\text{w,in,f}}-T_{\text{1,f}}\right)\mathrm{,}\hfill \end{array}$ (29) $\begin{array}{l}\frac{m_1}{M_1}\frac{\left(T_{\mathrm{1,}\text{ad}\mathrm{,}i+1}-T_{\mathrm{1,}\text{ad}\mathrm{,}i}\right)}{\text{Δ}{x}_{\text{SG}}}=-\frac{n\pi d_{\text{in}}{K}_1}{M_1{c}_{\text{p,ad}}}\left(T_{\text{w,in,ad}}-T_{\text{1,ad}}\right)\mathrm{,}\hfill \end{array}$ (30) $\begin{array}{c}\frac{m_2}{M_2}\frac{\left(T_{\mathrm{2,}\text{r}\mathrm{,}i}-T_{\mathrm{2,}\text{r}\mathrm{,}i-1}\right)}{\text{Δ}{x}_{\text{SG}}}\\ =\frac{n\pi d_{\text{out}}{K}_2}{M_2{c}_{\text{p,2}}}\left(T_{\text{w,out,f}}+T_{\text{w,out,ad}}-2T_{\text{2,r}}\right)\mathrm{,}\end{array}$ (31) $\begin{array}{c}\frac{m_2}{M_2}\frac{\left(h_{\mathrm{2,}\text{s}\mathrm{,}i}-h_{\mathrm{2,}\text{s}\mathrm{,}i-1}\right)}{\text{Δ}{x}_{\text{SG}}}\\ =\frac{n\pi d_{\text{out}}{K}_{\text{2,s}}}{M_2}\left(T_{\text{w,out,f}}+T_{\text{w,out,ad}}-2T_{\text{2,s}}\right)\mathrm{.}\end{array}$ (32) The inlet boundary conditions for temperature and enthalpy are applied to the primary and secondary sides of the steam generator. On the secondary side, the inlet temperature is defined as the corresponding coolant inlet temperature of the core, whereas the outlet boundary is set as a fully developed boundary.

Snapshots and POD bases

As previously mentioned, the construction of the reduced-order model involves offline and online stages, as shown in Fig. 5. In the offline stage, the full-order model is used to generate snapshots under various typical operating conditions. In the online stage, the reduced-order model is applied to predict the distributions of temperature, velocity, and pressure for unknown states. The first step in the offline stage is to calculate snapshots of different operation states, including the temperature, pressure, and velocity distributions. During this process, N operating states are selected within the operating range, and snapshots of these states are calculated using the full-order model (FOM) described in Sect. 2.1. The snapshot set generated in the offline stage based on the FOM is expressed as follows: $A=\left[{\alpha }_1\mathrm{,}{\alpha }_2\mathrm{,}\cdots \mathrm{,}{\alpha }_h\mathrm{,}\cdots \mathrm{,}{\alpha }_N\right]\mathrm{,}$ (33) where αh represents the variation set at the h^th operation condition, which can be expressed as: ${\alpha }_h={\left[{\alpha }_{{}_{h\mathrm{,1}}}\mathrm{,}\cdots \mathrm{,}{\alpha }_{{}_{h\mathrm{,}k}}\mathrm{,}\cdots \mathrm{,}{\alpha }_{{}_{h\mathrm{,}L}}\right]}^{\text{T}}\mathrm{,}$ (34) where k is the k^th variable and L is the total number of variables. Within the primary circuit, NX grids are set for the numerical calculation, and αh,k can be expressed as follows: ${\alpha }_{{}_{h\mathrm{,}k}}={\left[{\alpha }_{h\mathrm{,}k\mathrm{,1}}\mathrm{,}\cdots \mathrm{,}{\alpha }_{h\mathrm{,}k\mathrm{,}\text{g}}\mathrm{,}\cdots \mathrm{,}{\alpha }_{h\mathrm{,}k\mathrm{,}NX}\right]}^{\text{T}}\mathrm{,}$ (35) where g represents the g^th discrete grid and αh,k,g represents the value of the k^th variable at the g^th grid under the h^th operation condition.

Fig. 5Full-screen View

Reduced-order calculation steps

The main modes of the system variation, namely the POD bases, can be extracted based on a thorough analysis and processing of these snapshots, thereby realizing the objective of reducing the system intricacy to a lower-dimensional representation. POD bases consist of a set of modes or functions employed to characterize the predominant characteristics of variation within a system. These modes are extracted from snapshots and are conventionally represented as feature vectors. According to POD theory [51], the POD basis can be obtained from the eigenvalues and eigenvectors of the matrix S, which can be expressed as follows: $S=A^{\text{T}}A\mathrm{.}$ (36) The eigenvalues of S are defined as ${\lambda }_i\mathrm{,}i\in 〚\mathrm{1,}N〛$, which satisfy ${\lambda }_1>\cdots >{\lambda }_i>\cdots >{\lambda }_N>0$. The corresponding eigenvectors of these eigenvalues are $X_1\mathrm{,}\dots \mathrm{,}{X}_i\mathrm{,}\dots \mathrm{,}{X}_N$, and the POD basis Pm can be defined as follows: $P_m=\frac{1}{\sqrt{{\lambda }_m}}A{X}_m\mathrm{,}m=\mathrm{1,2,}\dots \mathrm{,}N$ (37) $P_m={\left[\begin{array}{c}{P}_{m\mathrm{,1,1}}\mathrm{,}\cdots \mathrm{,}{P}_{m\mathrm{,1,}g}\mathrm{,}\cdots \mathrm{,}{P}_{m\mathrm{,1,}NX}\mathrm{,}\cdots \\ P_{m\mathrm{,}k\mathrm{,1}}\mathrm{,}\cdots \mathrm{,}{P}_{m\mathrm{,}k\mathrm{,}g}\mathrm{,}\cdots \mathrm{,}{P}_{m\mathrm{,}k\mathrm{,}NX}\mathrm{,}\cdots \\ P_{m\mathrm{,}L\mathrm{,1}}\mathrm{,}\cdots \mathrm{,}{P}_{m\mathrm{,}L\mathrm{,}g}\mathrm{,}\cdots \mathrm{,}{P}_{m\mathrm{,}L\mathrm{,}NX}\mathrm{,}\cdots \end{array}\right]}^{\text{T}},$ (38) where $P_{m\mathrm{,}k\mathrm{,}g}$ represents the m-order POD basis corresponding to variable k^th at the g^th grid.

As the governing equations and snapshots generated by the full-order model for each component are different, special POD bases are formed for different components. In addition, for the same type of pipe, although the governing equations remain the same, different positions in the circuit result in different diameters and lengths. Consequently, their snapshots and generated POD bases are different.

According to POD theory, the significance of eigenvectors is directly associated with the magnitude of their corresponding eigenvalues. To determine the POD order Np for reduced-order modeling, the following criteria can be used: $\{\begin{array}{l}{N}_p=\arg \min \left\{I\left(N_p\right):I\left(N_p\right)\ge 1-{10}^{-8}\right\},\hfill \\ I\left(N_p\right)=\frac{{\displaystyle {\sum }_{i=1}^{N_p}{\lambda }_i}}{{\displaystyle {\sum }_{i=1}^N{\lambda }_i}},\hfill \end{array}$ (39) where I(N_p) denotes the energy fraction of the first Np POD mode. Subsequently, the snapshots ${\alpha }_{h^{\prime }\mathrm{,}k\mathrm{,}g}$ can be expanded as ${\alpha }_{h^{\prime }\mathrm{,}k\mathrm{,}g}={\displaystyle \sum _{m=1}^{N_p}{c}_m}{P}_{k\mathrm{,}g\mathrm{,}m}\mathrm{,}$ (40) where cm is the m^th-order basis function coefficient. Therefore, in the process of solving a one-dimensional thermal-circuit system, POD bases can be effectively applied to reconstruct the temperature, pressure, velocity, and relevant variables throughout the entire circuit under various operating conditions. The main task for the following processes is to establish the reduced-order model for solving the basis function coefficient cm, which, in this work, is achieved using the least-squares method (LSM) [52]. The main purpose of the ROM based on the LSM is to substitute Eq. (40) into linear equations (23) – (32). The residual functions of these equations are constructed as $E={\displaystyle \sum _i{\displaystyle \sum _m{A}_i{c}_m}}{P}_{i\mathrm{,}m}-b_i\mathrm{,}$ (41) where the subscripts i and m represent the indices of the mesh and POD order, respectively. A and b represent the stiffness matrix and source vector of the linear system, respectively.

The operating conditions to be predicted, including the inlet temperature, inlet flow rate, and core power, can be introduced by imposing the corresponding data at node i. For example, the flow rate of inlet node i is set to the given inlet value.

When using the LSM, the POD coefficient cm can be determined by minimizing the residual equation, which converts Eq. (41) to the following equations: $\frac{\partial E}{\partial c_m}=\mathrm{0,}m=\mathrm{1,2,...}$ (42) After determining cm, the distribution variables such as temperature, velocity, and pressure, can be obtained using Eq. (40). In the following section, the reduced-order model, mainly the residual functions of the primary circuit, are described in detail.

Primary circuit POD reduced-order model

To construct an entire reduced-order model for the primary circuit of the reactor, reduced-order models of the primary components within the circuit must first be established. For the AP1000 primary circuit considered in this study, the components primarily encompassed the core, pipes, and steam generator. Other components with relatively minor effects on computational efficiency, such as regulators and pumps, do not require the development of reduced-order models. The spatial arrangement of AP1000 primary-circuit components along the coolant traversing distance is shown in Fig. 6.

Fig. 6Full-screen View

Distribution of components along the flow direction

As previously mentioned, the establishment of the reduced-order model is based on a collection of snapshots. In this section, the variables used to construct the reduced-order model are systematically selected based on the governing equations associated with the various components within the circuit. In the following sections, the selection of snapshots and establishment of the ROM for different components are introduced. To simplify the subsequent construction of the residual function and compute the coefficient matrix of the POD bases, the functions differentiated within the governing equations are chosen as the variables.

Core

Table 1 lists the selected variables contained in the snapshots, POD bases, and boundary conditions for the reactor core, which are determined from the single-channel thermal-hydraulic equations (1)-(3) and their discretization. The core variables can be reconstructed using the POD bases according to $\{\begin{array}{l}\varphi ={\displaystyle {\sum }_{m=1}^{N_{p\mathrm{,}\text{c}}}{c}_{m\mathrm{,}\text{c}}{P}_{\text{c}\mathrm{,}\varphi \mathrm{,}m},}\hfill \\ \varphi =\left\{\rho u\mathrm{,}\rho uu\mathrm{,}P\mathrm{,}\mu u\mathrm{,}\rho \mathrm{,}\rho uh\mathrm{,}Pu\mathrm{,}\lambda T\right\},\hfill \end{array}$ (43) where $c_{m\mathrm{,}\text{c}}$ is the coefficient corresponding to the core POD basis and $N_{p\mathrm{,}\text{c}}$ represents the core POD order.

For each internal node, by applying Eq.(43) into Eqs.(24), (26), and (28) and calculating the summation of the residue over all internal nodes, the accumulated residual functions of the core are generated as Eqs. (45), (49), and (46), where superscripts ‘con’, ‘mom’, and ‘ene’ represent continuity, momentum, and energy, respectively, and NX_c represents the number of discretized grid points in the reactor core.

Similarly, to satisfy the inlet and outlet boundaries of the eight selected variables, the following two residual functions for the inlet and outlet are defined in Eqs. (47) and (48), respectively. Then, the total residual function of the core can be written as $E_{\text{c}}=E_{\text{c}}^{\text{con}}+E_{\text{c}}^{\text{mom}}+E_{\text{c}}^{\text{ene}}+E_{\text{c}}^{\text{b}\mathrm{,}\text{in}}+E_{\text{c}}^{\text{b}\mathrm{,}\text{out}}\mathrm{.}$ (44) Subsequently, the coefficients cm can be determined using the LSM to minimize the residual function E_c. With the determined cm, the variables in Table 1 can be reconstructed using the POD bases, according to Eq.(43). The temperature, velocity, and pressure fields at each grid point of the core can be reconstructed based on the IF97 thermophysical properties. $\begin{array}{l}{E}_{{}_{\text{c}}}^{{}_{\text{con}}}={\displaystyle \sum _{i=2}^{N{X}_{\text{c}}}{\left[{\displaystyle \sum _{m=1}^{N_{p\mathrm{,}\text{c}}}{c}_{m\mathrm{,}\text{c}}}\left(\frac{P_{\text{c}\mathrm{,}\rho u\mathrm{,}i\mathrm{,}m}-P_{\text{c}\mathrm{,}\rho u\mathrm{,}i-\mathrm{1,}m}}{\text{Δ}{x}_{\text{c}}}\right)\right]}^2}\mathrm{,}\hfill \end{array}$ (45) $\begin{array}{c}{E}_{\text{c}}^{\text{ene}}={\displaystyle \sum _{i=2}^{N{X}_{\text{c}}-1}[{\displaystyle \sum _{m=1}^{N_{p\mathrm{,}\text{c}}}}{c}_{m\mathrm{,}\text{c}}\left(\frac{P_{\text{c}\mathrm{,}\rho uh\mathrm{,}i\mathrm{,}m}-P_{\text{c}\mathrm{,}\rho uh\mathrm{,}i-\mathrm{1,}m}}{\text{Δ}{x}_{\text{c}}}\right)}\\ +{\displaystyle \sum _{m=1}^{N_{p\mathrm{,}\text{c}}}{c}_{m\mathrm{,}\text{c}}}\left(\frac{P_{\text{c}\mathrm{,}pu\mathrm{,}i\mathrm{,}m}-P_{\text{c}\mathrm{,}pu\mathrm{,}i-\mathrm{1,}m}}{\text{Δ}{x}_{\text{c}}}\right)\\ {-\frac{1}{\text{Δ}{x}_{\text{c}}}{\displaystyle \sum _{m=1}^{N_{p\mathrm{,}\text{c}}}{c}_{m\mathrm{,}\text{c}}}(\frac{P_{\text{c}\mathrm{,}\lambda T\mathrm{,}i+\mathrm{1,}m}}{\text{Δ}{x}_{\text{c}}}-2\frac{P_{\text{c}\mathrm{,}\lambda T\mathrm{,}i\mathrm{,}m}}{\text{Δ}{x}_{\text{c}}}+\frac{P_{\text{c}\mathrm{,}\lambda T\mathrm{,}i-\mathrm{1,}m}}{\text{Δ}{x}_{\text{c}}})-q_{\text{c}\mathrm{,}i}]}^2\mathrm{,}\end{array}$ (46) $\begin{array}{c}{E}_{\text{c}}^{\text{b}\mathrm{,}\text{in}}={\left({\displaystyle \sum _{m=1}^{N_{p\mathrm{,}\text{c}}}{c}_{m\mathrm{,}\text{c}}}{P}_{\text{c}\mathrm{,}\rho u\mathrm{,1,}m}-{\rho }_{\mathrm{0,}\text{c}}{u}_{\mathrm{0,}\text{c}}\right)}^2+{\left({\displaystyle \sum _{m=1}^{N_{p\mathrm{,}\text{c}}}{c}_{m\mathrm{,}\text{c}}}{P}_{\text{c}\mathrm{,}\rho uu\mathrm{,1,}m}-{\rho }_{\mathrm{0,}\text{c}}{u}_{\mathrm{0,}\text{c}}{u}_{\mathrm{0,}\text{c}}\right)}^2+{\left({\displaystyle \sum _{m=1}^{N_{p\mathrm{,}\text{c}}}{c}_{m\mathrm{,}\text{c}}}{P}_{\text{c}\mathrm{,}p\mathrm{,1,}m}-p_{\mathrm{0,}\text{c}}\right)}^2\\ +{\left({\displaystyle \sum _{m=1}^{N_{p\mathrm{,}\text{c}}}{c}_{m\mathrm{,}\text{c}}}{P}_{\text{c}\mathrm{,}\mu u\mathrm{,1,}m}-{\mu }_{\mathrm{0,}\text{c}}{u}_{\mathrm{0,}\text{c}}\right)}^2+{\left({\displaystyle \sum _{m=1}^{N_{p\mathrm{,}\text{c}}}{c}_{m\mathrm{,}\text{c}}}{P}_{\text{c}\mathrm{,}\rho \mathrm{,1,}m}-{\rho }_{\mathrm{0,}\text{c}}\right)}^2+{\left({\displaystyle \sum _{m=1}^{N_{p\mathrm{,}\text{c}}}{c}_{m\mathrm{,}\text{c}}}{P}_{\text{c}\mathrm{,}\rho uh\mathrm{,1,}m}-{\rho }_{\mathrm{0,}\text{c}}{u}_{\mathrm{0,}\text{c}}{h}_{\mathrm{0,}\text{c}}\right)}^2\\ +{\left({\displaystyle \sum _{m=1}^{N_{p\mathrm{,}\text{c}}}{c}_{m\mathrm{,}\text{c}}}{P}_{\text{c}\mathrm{,}pu\mathrm{,1,}m}-p_{\mathrm{0,}\text{c}}{u}_{\mathrm{0,}\text{c}}\right)}^2+{\left({\displaystyle \sum _{m=1}^{N_{p\mathrm{,}\text{c}}}{c}_{m\mathrm{,}\text{c}}}{P}_{\text{c}\mathrm{,}\lambda T\mathrm{,1,}m}-{\lambda }_{\mathrm{0,}\text{c}}{T}_{\mathrm{0,}\text{c}}\right)}^2\mathrm{,}\end{array}$ (47) $\begin{array}{c}{E}_{\text{c}}^{\text{b}\mathrm{,}\text{out}}=[{\displaystyle \sum _{m=1}^{N_{p\mathrm{,}\text{c}}}{c}_{m\mathrm{,}\text{c}}}\left(\frac{P_{\text{c}\mathrm{,}\rho uu\mathrm{,}N{X}_{\text{c}}\mathrm{,}m}-P_{\text{c}\mathrm{,}\rho uu\mathrm{,}N{X}_{\text{c}}-\mathrm{1,}m}}{\text{Δ}{x}_{\text{c}}}\right)+\frac{1}{\text{Δ}{x}_{\text{c}}}{\displaystyle \sum _{m=1}^{N_{p\mathrm{,}\text{c}}}{c}_{m\mathrm{,}\text{c}}}\left(\frac{P_{\text{c}\mathrm{,}p\mathrm{,}N{X}_{\text{c}}\mathrm{,}m}-P_{\text{c}\mathrm{,}p\mathrm{,}N{X}_{\text{c}}-\mathrm{1,}m}}{\text{Δ}{x}_{\text{c}}}\right)\\ +{\displaystyle \sum _{m=1}^{N_{p\mathrm{,}\text{c}}}{c}_{m\mathrm{,}\text{c}}}\left(\frac{P_{\text{c}\mathrm{,}\mu u\mathrm{,}N{X}_{\text{c}}\mathrm{,}m}-P_{\text{c}\mathrm{,}\mu u\mathrm{,}N{X}_{\text{c}}-\mathrm{1,}m}}{\text{Δ}{x}_{\text{c}}}\right)+\frac{1}{2}\left(\frac{f}{D}+\frac{K_{\text{sp}}}{\text{Δ}{x}_{\text{c}}}\right){\displaystyle \sum _{m=1}^{N_{p\mathrm{,}\text{c}}}{c}_{m\mathrm{,}\text{c}}}{P}_{\text{c}\mathrm{,}\rho uu\mathrm{,}N{X}_{\text{c}}\mathrm{,}m}+{g{\displaystyle \sum _{m=1}^{N_{p\mathrm{,}\text{c}}}{c}_{m\mathrm{,}\text{c}}}{P}_{\text{c}\mathrm{,}\rho \mathrm{,}N{X}_{\text{c}}\mathrm{,}m}]}^2\\ +[{\displaystyle \sum _{m=1}^{N_{p\mathrm{,}\text{c}}}{c}_{m\mathrm{,}\text{c}}}\left(\frac{P_{\text{c}\mathrm{,}\rho uh\mathrm{,}N{X}_{\text{c}}\mathrm{,}m}-P_{\text{c}\mathrm{,}\rho uh\mathrm{,}N{X}_{\text{c}}-\mathrm{1,}m}}{\text{Δ}{x}_{\text{c}}}\right)+{\displaystyle \sum _{m=1}^{N_{p\mathrm{,}\text{c}}}{c}_{m\mathrm{,}\text{c}}}\left(\frac{P_{\text{c}\mathrm{,}pu\mathrm{,}N{X}_{\text{c}}\mathrm{,}m}-P_{\text{c}\mathrm{,}pu\mathrm{,}N{X}_{\text{c}}-\mathrm{1,}m}}{\text{Δ}{x}_{\text{c}}}\right)\\ +{\displaystyle \sum _{m=1}^{N_{p\mathrm{,}\text{c}}}{c}_{m\mathrm{,}\text{c}}}\left(\frac{P_{\text{c}\mathrm{,}\lambda T\mathrm{,}N{X}_{\text{c}}\mathrm{,}m}-P_{\text{c}\mathrm{,}\lambda T\mathrm{,}N{X}_{\text{c}}-\mathrm{1,}m}}{\text{Δ}{x}_{\text{c}}^2}\right)-{q_{N{X}_{\text{c}}}]}^2\mathrm{,}\end{array}$ (48) $\begin{array}{c}{E}_{\text{c}}^{\text{mom}}={\displaystyle \sum _{i=2}^{N{X}_{\text{c}}-1}[{\displaystyle \sum _{m=1}^{N_{p\mathrm{,}\text{c}}}{c}_{m\mathrm{,}\text{c}}}\left(\frac{P_{\text{c}\mathrm{,}\rho uu\mathrm{,}i\mathrm{,}m}-P_{\text{c}\mathrm{,}\rho uu\mathrm{,}i-\mathrm{1,}m}}{\text{Δ}{x}_{\text{c}}}\right)+{\displaystyle \sum _{m=1}^{N_{p\mathrm{,}\text{c}}}{c}_{m\mathrm{,}\text{c}}}\left(\frac{P_{\text{c}\mathrm{,}p\mathrm{,}i\mathrm{,}m}-P_{\text{c}\mathrm{,}p\mathrm{,}i-\mathrm{1,}m}}{\text{Δ}{x}_{\text{c}}}\right)}\\ {-{\displaystyle \sum _{m=1}^{N_{p\mathrm{,}\text{c}}}{c}_{m\mathrm{,}\text{c}}}\left(\frac{P_{\text{c}\mathrm{,}\mu u\mathrm{,}i+\mathrm{1,}m}}{\text{Δ}{x}_{\text{c}}^2}-2\frac{P_{\text{c}\mathrm{,}\mu u\mathrm{,}i\mathrm{,}m}}{\text{Δ}{x}_{\text{c}}^2}+\frac{P_{\text{c}\mathrm{,}\mu u\mathrm{,}i-\mathrm{1,}m}}{\text{Δ}{x}_{\text{c}}^2}\right)+\frac{1}{2}\left(\frac{f}{D}+\frac{K_{\text{sp}}}{\text{Δ}{x}_{\text{c}}}\right){\displaystyle \sum _{m=1}^{N_{p\mathrm{,}\text{c}}}{c}_{m\mathrm{,}\text{c}}}{P}_{\text{c}\mathrm{,}\rho uu\mathrm{,}i\mathrm{,}m}+g{\displaystyle \sum _{m=1}^{N_{p\mathrm{,}\text{c}}}{c}_{m\mathrm{,}\text{c}}}{P}_{\text{c}\mathrm{,}\rho \mathrm{,}i\mathrm{,}m}]}^2.\end{array}$ (49)

Steam generator

The variable selections based on the steam generator (SG) governing equations are shown in Table 2. Because the temperature at the last grid point of the parallel-flow section is transferred to the last grid point of the counter-flow section, the inlet boundary condition for the coolant flow on the primary side is set as T_1,f,1=T_1,0. The outlet boundary condition of the coolant flow on the primary side is considered to be a fully developed boundary, indicating a temperature gradient of zero at the outlet of the counter-flow section.

Similar to Eq.(43), the SG variables can be reconstructed using the POD bases according to $\begin{array}{l}\varphi ={\displaystyle \sum _{m=1}^{N_{p\mathrm{,}\text{SG}}}{c}_{m\mathrm{,}\text{SG}}}{P}_{\text{SG}\mathrm{,}\varphi \mathrm{,}m},\\ \varphi =\{\begin{array}{l}{T}_{\mathrm{1,}\text{f}}\mathrm{,}{T}_{\mathrm{1,}\text{ad}}\mathrm{,}{T}_{\text{w}\mathrm{,}\text{in}\mathrm{,}\text{f}}\mathrm{,}{T}_{\text{w}\mathrm{,}\text{out}\mathrm{,}\text{f}}\mathrm{,}{T}_{\text{w}\mathrm{,}\text{in}\mathrm{,}\text{ad}}\mathrm{,}{T}_{\text{w}\mathrm{,}\text{out}\mathrm{,}\text{ad}}\mathrm{,}\hfill \\ T_{\mathrm{2,}\text{r}}\mathrm{,}\frac{T_{\text{w}\mathrm{,}\text{in}\mathrm{,}\text{f}}-T_{\mathrm{1,}\text{f}}}{{\rho }_1{c}_{\text{p}\mathrm{,}\text{f}}}\mathrm{,}\frac{T_{\text{w}\mathrm{,}\text{in}\mathrm{,}\text{ad}}-T_{\mathrm{1,}\text{ad}}}{{\rho }_1{c}_{\text{p}\mathrm{,}\text{ad}}}\mathrm{,}{h}_{\text{s}}\mathrm{,}\hfill \\ \frac{T_{\text{w}\mathrm{,}\text{out}\mathrm{,}\text{f}}-T_{\text{w}\mathrm{,}\text{out}\mathrm{,}\text{ad}}-2T_{\mathrm{2,}\text{r}}}{c_{\text{p}\mathrm{,2}}}\mathrm{,}\hfill \end{array}\end{array}$ (50) where $c_{m\mathrm{,}\text{SG}}$ is the coefficient corresponding to the SG POD basis and $N_{p\mathrm{,}\text{SG}}$ represents the POD order of the SG.

Let the number of discrete meshes in the parallel-flow and counter-flow sections of the SG U-tube be defined as $N{X}_{\text{SG}}$. The dividing point between the preheating and boiling sections on the secondary side is positioned at grid location $X_{\text{SG}}$. Based on the discretized equations for the flow and wall heat transfer on the primary side, Eqs.(29) and (30), and applying Eq.(50), the residual function corresponding to these two equations can be formulated as follows, where superscript 1 represents the primary side: $\{\begin{array}{l}\begin{array}{c}{E}_{\text{SG}}^{\text{primary}}={\displaystyle {\sum }_{i=2}^{N{X}_{\text{SG}}}{\left(\frac{m_1}{M_1\text{Δ}{x}_{\text{SG}}}{A}_1-\frac{4K_1}{M_1{d}_{\text{in}}}{A}_2\right)}^2}\\ +{\displaystyle {\sum }_{i=1}^{N{X}_{\text{SG}}-1}{\left(\frac{m_1}{M_1\text{Δ}{x}_{\text{SG}}}{A}_3+\frac{4K_1}{M_1{d}_{\text{in}}}{A}_4\right)}^2}\mathrm{.}\end{array}\hfill \\ A_1={\displaystyle {\sum }_{m=1}^{N_{p\mathrm{,}\text{SG}}}{c}_{m\mathrm{,}\text{SG}}}\left(P_{\text{SG}\mathrm{,}{T}_{\mathrm{1,}\text{f}}\mathrm{,}i\mathrm{,}m}-P_{\text{SG}\mathrm{,}{T}_{\mathrm{1,}\text{f}}\mathrm{,}i-\mathrm{1,}m}\right)\hfill \\ A_2={\displaystyle {\sum }_{m=1}^{N_{p\mathrm{,}\text{SG}}}{c}_{m\mathrm{,}\text{SG}}}{P}_{\text{SG}\mathrm{,}\text{ta}\mathrm{,}i\mathrm{,}m}\hfill \\ A_3={\displaystyle {\sum }_{m=1}^{N_{p\mathrm{,}\text{SG}}}{c}_{m\mathrm{,}\text{SG}}}\left(P_{\text{SG}\mathrm{,}{T}_{\mathrm{1,}\text{ad}}\mathrm{,}i+\mathrm{1,}m}-P_{\text{SG}\mathrm{,}{T}_{\mathrm{1,}\text{ad}}\mathrm{,}i\mathrm{,}m}\right)\hfill \\ A_4={\displaystyle {\sum }_{m=1}^{N_{p\mathrm{,}\text{SG}}}{c}_{m\mathrm{,}\text{SG}}}{P}_{\text{SG}\mathrm{,}\text{tb}\mathrm{,}i\mathrm{,}m}\mathrm{.}\hfill \end{array}$ (51) Similarly, for the discretized equations representing the flow in the secondary side and the heat transfer on the outer surface of the U-tube, Eqs. (31) and (32), the residual function corresponding to these two equations can be formulated as follows, where superscript 2 represents the secondary side: $\{\begin{array}{l}\begin{array}{c}{E}_{\text{SG}}^{\text{secondary}}={\displaystyle {\sum }_{i=2}^{X_{\text{SG}}}\left(\frac{m_2}{M_2\text{Δ}{x}_{\text{SG}}}{A}_1{-\frac{n{K}_2\pi d_{\text{out}}}{M_2}{A}_2)}^2\right)}\\ +{\displaystyle {\sum }_{i=X_{\text{SG}}+1}^{N{X}_{\text{SG}}}\left(\frac{m_2}{M_2\text{Δ}{x}_{\text{SG}}}{A}_3{-\frac{n{K}_{\mathrm{2,}\text{s}}\pi d_{\text{out}}}{M_2}{A}_4)}^2\right)}\mathrm{.}\end{array}\hfill \\ A_1={\displaystyle {\sum }_{m=1}^{N_{p\mathrm{,}\text{SG}}}{c}_{m\mathrm{,}\text{SG}}\left(P_{\text{SG}\mathrm{,}{T}_{\mathrm{2,}\text{r}}\mathrm{,}i\mathrm{,}m}-P_{\text{SG}\mathrm{,}{T}_{\mathrm{2,}\text{r}}\mathrm{,}i-\mathrm{1,}m}\right)}\hfill \\ A_2={\displaystyle {\sum }_{m=1}^{N_{p\mathrm{,}\text{SG}}}{c}_{m\mathrm{,}\text{SG}}{P}_{\text{SG,tc}\mathrm{,}i\mathrm{,}m}}\hfill \\ A_3={\displaystyle {\sum }_{m=1}^{N_{p\mathrm{,}\text{SG}}}{c}_{m\mathrm{,}\text{SG}}}\left(P_{\text{SG}\mathrm{,}{h}_{\text{s}}\mathrm{,}i\mathrm{,}m}-P_{\text{SG}\mathrm{,}{h}_{\text{s}}\mathrm{,}i-\mathrm{1,}m}\right)\hfill \\ A_4={\displaystyle {\sum }_{m=1}^{N_{p\mathrm{,}\text{SG}}}{c}_{m\mathrm{,}\text{SG}}}(P_{\text{SG}\mathrm{,}{T}_{\text{w,out,f}}\mathrm{,}i\mathrm{,}m}\\ +P_{\text{SG}\mathrm{,}{T}_{\text{w,out,ad}}\mathrm{,}i\mathrm{,}m}-T_{\mathrm{2,}\text{s}}).\end{array}$ (52) Based on the heat-balance equations for the U-tube inner wall, Eqs. (14) and (15), the corresponding residual function can be expressed as follows: $\{\begin{array}{l}\begin{array}{c}{E}_{\text{SG}}^{\text{w,in}}={{\displaystyle \sum _{i=1}^{N{X}_{\text{SG}}}\left(\frac{2n\pi {\lambda }_{\text{w}}}{M_{\text{w}}{c}_{\text{p,w}}\ln \left(\frac{d_{\text{out}}}{d_{\text{in}}}\right)}{A}_1+\frac{n{K}_1\pi d_{\text{in}}}{M_{\text{w}}{c}_{\text{p,w}}}{A}_2\right)}}^2\\ +{{\displaystyle \sum _{i=1}^{N{X}_{\text{SG}}}\left(\frac{2n\pi {\lambda }_{\text{w}}}{M_{\text{w}}{c}_{\text{p}\mathrm{,}\text{w}}\ln \left(\frac{d_{\text{out}}}{d_{\text{in}}}\right)}{A}_3+\frac{n{K}_1\pi d_{\text{in}}}{M_{\text{w}}{c}_{\text{p,w}}}{A}_4\right)}}^2\mathrm{.}\end{array}\hfill \\ \begin{array}{c}{A}_1={\displaystyle \sum _{m=1}^{N_{p\mathrm{,}\text{SG}}}{c}_{m\mathrm{,}\text{SG}}(P_{\text{SG}\mathrm{,}{T}_{\text{w,out,f}}\mathrm{,}i\mathrm{,}m}}\\ -P_{\text{SG}\mathrm{,}{T}_{\text{w,in,f}}\mathrm{,}i\mathrm{,}m})\end{array}\hfill \\ \begin{array}{c}{A}_2={\displaystyle \sum _{m=1}^{N_{p\mathrm{,}\text{SG}}}{c}_{m\mathrm{,}\text{SG}}(P_{\text{SG}\mathrm{,}{T}_{\text{1,f}}\mathrm{,}i\mathrm{,}m}}\\ -P_{\text{SG}\mathrm{,}{T}_{\text{w,in,f}}\mathrm{,}i\mathrm{,}m})\end{array}\hfill \\ A_3={\displaystyle \sum _{m=1}^{N_{p\mathrm{,}\text{SG}}}{c}_{m\mathrm{,}\text{SG}}}(P_{\text{SG}\mathrm{,}{T}_{\text{w,out,ad}}\mathrm{,}i\mathrm{,}m}\\ -P_{\text{SG}\mathrm{,}{T}_{\text{w,in,ad}}\mathrm{,}i\mathrm{,}m})\\ A_4={\displaystyle \sum _{m=1}^{N_{p\mathrm{,}\text{SG}}}{c}_{m\mathrm{,}\text{SG}}}(P_{\text{SG}\mathrm{,}{T}_{\text{w,out,ad}}\mathrm{,}i\mathrm{,}m}\\ -P_{\text{SG}\mathrm{,}{T}_{\text{w,in,ad}}\mathrm{,}i\mathrm{,}m})\end{array}$ (53) $\begin{array}{c}{E}_{\text{SG}}^{\text{b}}={\left[{\displaystyle \sum _{m=1}^{N_{p\mathrm{,}\text{SG}}}{c}_{m\mathrm{,}\text{SG}}}{P}_{\text{SG}\mathrm{,}{T}_{\mathrm{1,}\text{f}}\mathrm{,1,}m}-T_{\mathrm{1,0}}\right]}^2\\ +{\left[{\displaystyle \sum _{m=1}^{N_{p\mathrm{,}\text{SG}}}{c}_{m\mathrm{,}\text{SG}}}{P}_{\text{SG}\mathrm{,}{T}_{\mathrm{2,}\text{r}}\mathrm{,1,}m}-T_{\mathrm{2,0}}\right]}^2\\ +{\left[{\displaystyle \sum _{m=1}^{N_{p\mathrm{,}\text{SG}}}{c}_{m\mathrm{,}\text{SG}}}{P}_{\text{SG}\mathrm{,}{T}_{\mathrm{1,}\text{f}}\mathrm{,}N{X}_{\text{SG}}\mathrm{,}m}-{\displaystyle \sum _{m=1}^{N_{p\mathrm{,}\text{SG}}}{c}_{m\mathrm{,}\text{SG}}}{P}_{\text{SG}\mathrm{,}{T}_{\mathrm{1,}\text{ad}}\mathrm{,}N{X}_{\text{SG}}\mathrm{,}m}\right]}^2\end{array}$ (54) Based on the heat-transfer equations Eqs. (16) and (17), the corresponding residual function can be expressed as follows: $\{\begin{array}{l}\begin{array}{c}{E}_{\text{SG}}^{\text{w,out,parallel}}\\ ={{\displaystyle \sum _{i=1}^{X_{\text{SG}}}\left(\frac{2n\pi {\lambda }_{\text{w}}}{M_{\text{w}}{c}_{\text{p,w}}\ln \left(d_{\text{out}}/d_{\text{in}}\right)}{A}_1+\frac{n{K}_2\pi d_{\text{out}}}{M_{\text{w}}{c}_{\text{p,w}}}{A}_2\right)}}^2\\ +{\displaystyle \sum _{i=X_{\text{SG}}+1}^{N{X}_{\text{SG}}}(\frac{2n\pi {\lambda }_{\text{w}}}{M_{\text{w}}{c}_{\text{p,w}}\ln \left(d_{\text{out}}/d_{\text{in}}\right)}{A}_3}\\ {+\frac{n{K}_{\mathrm{2,}\text{s}}\pi d_{\text{out}}}{M_{\text{w}}{c}_{\text{p,w}}}\left(T_{\mathrm{2,}s}-A_4\right))}^2.\end{array}\hfill \\ \begin{array}{c}{A}_1={\displaystyle \sum _{m=1}^{N_{p\mathrm{,}\text{SG}}}{c}_{m\mathrm{,}\text{SG}}}(P_{\text{SG}\mathrm{,}{T}_{\text{w,in,f}}\mathrm{,}i\mathrm{,}m}\\ -P_{\text{SG}\mathrm{,}{T}_{\text{w,out,f}}\mathrm{,}i\mathrm{,}m})\end{array}\hfill \\ \begin{array}{c}{A}_2={\displaystyle \sum _{m=1}^{N_{p\mathrm{,}\text{SG}}}{c}_{m\mathrm{,}\text{SG}}}(P_{\text{SG}\mathrm{,}{T}_{\text{2,r}}\mathrm{,}i\mathrm{,}m}\\ -P_{\text{SG}\mathrm{,}{T}_{\text{w,out,f}}\mathrm{,}i\mathrm{,}m})\end{array}\hfill \\ A_3={\displaystyle \sum _{m=1}^{N_{p\mathrm{,}\text{SG}}}{c}_{m\mathrm{,}\text{SG}}}(P_{\text{SG}\mathrm{,}{T}_{\text{w,in,f}}\mathrm{,}i\mathrm{,}m}\\ -P_{\text{SG}\mathrm{,}{T}_{\text{w,out,f}}\mathrm{,}i\mathrm{,}m})\\ A_4={\displaystyle \sum _{m=1}^{N_{p\mathrm{,}\text{SG}}}{c}_{m\mathrm{,}\text{SG}}}{P}_{\text{SG}\mathrm{,}{T}_{\text{w,out,f}}\mathrm{,}i\mathrm{,}m}\mathrm{.}\end{array}$ (55) Based on the heat-transfer equations Eqs. (18) and (19), we can express the corresponding residual function using Eq. (56): $\{\begin{array}{l}\begin{array}{c}{E}_{\text{SG}}^{\text{w,out,counter}}\\ ={\displaystyle {\sum }_{i=1}^{X_{\text{SG}}}[\frac{2n\pi {\lambda }_{\text{w}}}{M_{\text{w}}{c}_{\text{p,w}}\ln \left(d_{\text{out}}/d_{\text{in}}\right)}{A}_1}\\ +\frac{n{K}_2\pi d_{\text{out}}}{M_{\text{w}}{c}_{\text{p,w}}}{A}_2{]}^2+\\ {\displaystyle {\sum }_{i=X_{\text{SG}}+1}^{N{X}_{\text{SG}}}[\frac{2n\pi {\lambda }_{\text{w}}}{M_{\text{w}}{c}_{\text{p,w}}\ln \left(d_{\text{out}}/d_{\text{in}}\right)}{A}_3}\\ {+\frac{n{K}_{\text{2,s}}\pi d_{\text{out}}}{M_{\text{w}}{c}_{\text{p,w}}}\left(T_{\text{2,s}}-A_4\right)]}^2\end{array}\hfill \\ A_1={\displaystyle {\sum }_{m=1}^{N_{p\mathrm{,}\text{SG}}}{c}_{m\mathrm{,}\text{SG}}}\left(P_{\text{SG}\mathrm{,}{T}_{\text{w,in,ad}}\mathrm{,}i\mathrm{,}m}-P_{\text{SG}\mathrm{,}{T}_{\text{w,out,ad}}\mathrm{,}i\mathrm{,}m}\right)\hfill \\ A_2={\displaystyle {\sum }_{m=1}^{N_{p\mathrm{,}\text{SG}}}{c}_{m\mathrm{,}\text{SG}}}\left(P_{\text{SG}\mathrm{,}{T}_{\text{w,in,ad}}\mathrm{,}i\mathrm{,}m}-P_{\text{SG}\mathrm{,}{T}_{\text{w,out,ad}}\mathrm{,}i\mathrm{,}m}\right)\hfill \\ A_3={\displaystyle {\sum }_{m=1}^{N_{p\mathrm{,}\text{SG}}}{c}_{m\mathrm{,}\text{SG}}}\left(P_{\text{SG}\mathrm{,}{T}_{\text{w,in,ad}}\mathrm{,}i\mathrm{,}m}-P_{\text{SG}\mathrm{,}{T}_{\text{w,out,ad}}\mathrm{,}i\mathrm{,}m}\right)\\ A_4={\displaystyle {\sum }_{m=1}^{N_{p\mathrm{,}\text{SG}}}{c}_{m\mathrm{,}\text{SG,}{T}_{\text{w,out,ad}}\mathrm{,}i\mathrm{,}m}}\end{array}.$ (56) The residual functions corresponding to the inlet boundary conditions of the coolant on both the primary and secondary sides, as well as the outlet of the parallel flow and inlet of the counter flow on the primary side, can be expressed by Eq. (54). The overall residual function E_SG is as follows: $\begin{array}{c}{E}_{\text{SG}}=E_{\text{SG}}^{\text{primary}}+E_{\text{SG}}^{\text{secondary}}+E_{\text{SG}}^{\text{w,in}}+\\ E_{\text{SG}}^{\text{w,out,parallel}}+E_{\text{SG}}^{\text{w,out,counter}}+E_{\text{SG}}^{\text{b}}\mathrm{.}\end{array}$ (57) Subsequently, the coefficients $c_{m\mathrm{,}\text{SG}}$ are determined using the LSM, and the temperature distribution inside the SG is reconstructed based on Eq. (50).