A Modified JFNK for Solving the HTR Steady State Secondary Circuit Problem

Abstract

A nuclear power plant is a complex coupling system, which features multi-physics coupling between reactor physics and thermal-hydraulics in the reactor core, as well as the multi-circuit coupling between the primary circuit and the secondary circuit by the shared steam generator (SG). Especially in the pebble-bed modular HTR nuclear power plant, different nuclear steam supply modules are further coupled together through the shared main steam pipes and the related equipment in the secondary circuit, since the special configuration of multiple reactor modules connects to a steam turbine. The JFNK (Jacobian-Free Newton–Krylov) method provides a promising coupling framework to solve the whole HTR nuclear power plant problem, due to its excellent convergence rate and strong robustness. In this work, the JFNK method was modified and applied to the steady-state calculation of the HTR secondary circuit, which plays an important role in simultaneous solutions for the whole HTR nuclear power plant. The main components in the secondary circuit included SG, steam turbine, condenser, feed pump, high/low-pressure heat exchanger, deaerator, as well as the extraction steam from the steam turbine. The results showed that the JFNK method can effectively solve the steady state issue of the HTR secondary circuit. Moreover, the JFNK method could converge well within a wide range of initial values, indicating its strong robustness.

1. Introduction

The pebble-bed HTR nuclear power plant is a complex coupling system [1]. In the reactor core, there is a coupling of multi-physics composed of reactor physics, thermal-hydraulics, and structural mechanics. The reactor’s primary and secondary circuits are coupled through a steam generator (SG) based on a shared heat transfer wall. Because the once-through SG is utilized in the HTR, the primary and secondary circuits are coupled tightly. At the same time, it adopts the operation mode of multi-NSSSs (Nuclear Steam Supply Systems) with one steam turbine, and different NSSS modules are also coupled together through the shared main steam pipes and the secondary circuit equipment. Therefore, the whole HTR nuclear power plant is a large-scale nonlinear and complex strong coupling system featured with multi-physics, multi-circuits, and multi-modules.

The secondary circuit is a carrier of multi-circuits and multi-modules and plays an important role in the whole coupling system [1]. The multi-component coupling system in the secondary circuit is completely different from the multi-physical field and multi-scale coupling system in the reactor core, which has three of its own features. The first one is that it is a multi-component coupled system with complex network topology. In the secondary circuit, the steam turbine, condenser, deaerator, condensate pump/feed pump, and other components are tightly coupled through the flow of steam and water, forming a set of flow and heat transfer networks. At the same time, due to the existence of steam extraction, the high-pressure turbine is closely coupled with a high-pressure heater and deaerator, and the low-pressure turbine and low-pressure heater are closely coupled. The low-pressure extraction is finally fed back to the condenser, and the high-pressure extraction is finally returned to the main circuit through the deaerator, which constitutes another flow and heat transfer network. Moreover, taking into account the operating requirements, the flow and heat transfer network are dynamically changed by using the control system, which can change the mass flow through the connecting/isolating part of the valve components or components such as pumps. Therefore, the entire secondary circuit system is a multi-component coupling system with complex network topology characteristics. The second feature is that each component of the secondary circuit itself is also a subsystem with complex phenomena. For example, the steam turbine has staged extraction, and the inlet and outlet pressure changes greatly. Also, valve switches can dramatically change the distribution of physical fields. Especially for the steam generator, there is a complex two-phase heat transfer process, which has the characteristics of a large variety of water properties and large variations of pressure. The last one is that the distribution of a single physical field in the entire secondary circuit system varies dramatically. For example, the maximum pressure of the system is on the order of mega-Pascal, but the minimum pressure is close to the vacuum at the condenser. In addition, the mass flow rate at the extraction steam is only a few percent of the system flow rate, and the enthalpy change from the steam generator outlet to the condenser is also prominent.

As a result, a stable and efficient solution is needed for the whole HTR nuclear power plant. Traditional methods for solving coupled systems are the operator splitting method [2] and the Picard iteration method [3,4], but these two methods cannot be competent for such large-scale and complex problems, due to their weak stability and linear convergence rate. The JFNK (Jacobian-free Newton–Krylov) method is a promising method to solve this challenging issue, which could realize all the physical field synchronization convergence and has a higher convergence rate and stronger stability than the traditional methods [5]. Therefore, many scholars study the JFNK method to solve complex coupling problems, such as H. Park [6], H. Zhang [7,8,9], and M. Fratoni [10]. However, most existing JFNK research mainly focuses on the reactor core and primary circuit problem. While for the secondary circuit coupling issue, the traditional coupling methods [11] are still mainly used, and only a few attempts have been made to use the JFNK method, such as Relap7 [12]. The main contribution of this work is to fully realize the coupling calculation of the secondary circuit within the JFNK framework, which is an important part of the whole HTR nuclear power plant coupling calculation.

2. Physical Model

For the high-temperature gas-cooled reactor, the helium temperature at the outlet of the primary circuit is as high as 750 °C and the heat of helium is taken away by the secondary circuit water after passing through the steam generator. The schematic diagram of the secondary circuit is shown in Figure 1. The main function of the secondary circuit is to take away the heat generated by the primary circuit, converting the heat into electricity by the steam turbine. The physical models established in this paper include SG, steam turbine, condenser, feed pump, high/low-pressure heater, and deaerator.

As a heat exchanger, the main function of the SG is to transfer the heat generated by the reactor core from hot helium to feed water, which flows through the entire HTR Secondary circuit. High temperature and high-pressure steam are used in the steam turbine, where the pressure changes greatly from mega-Pascal to kilo-Pascal. There are two steam turbines in the HTR secondary circuit; namely, a high-pressure turbine and a low-pressure turbine, and each of them has steam extraction. The high-pressure turbine has two stages of extracting steam which are to the high-pressure heater and deaerator, respectively. The extracted steam to the high-pressure heater is finally condensed and returned to the deaerator. Unlike the high-pressure turbine, the low-pressure turbine has three stages of steam extraction to three low-pressure heaters, respectively, where the steam extraction is finally returned to the condenser. The extraction heat transfer processes further enhance the coupling relationship among the steam turbine, the high-pressure heater, and the low-pressure heater, as well as the deaerator. The working pressure of the condenser is almost constant, with a complex condensation heat transfer process. The function of the condensate pump and feed pump is to increase the pressure of condensate water, from kilo-Pascal to the system pressure mega-Pascal. The details of these models are described below and all models are considered in the steady state.

2.1. Steam Generator

A helical coiled once-through steam generator (H-OTSG) is used in the HTR secondary circuit, where the feed water is heated by the hot helium from liquid water into saturated steam, and further into superheated steam, accompanied by phase change heat transfer. There are two main modeling methods for H-OTS: the fixed grid model [13] and the dynamic grid model [14]. For the fixed grid model, the grid size of the heat transfer tube in H-OTSG is pre-defined and fixed. Usually, a relatively fine grid size is utilized to resolve the temperature distribution with phase change. While for the dynamic grids model, only three computational grids are required to resolve the undersaturated water region, two-phase flow region, and steam region, respectively. Therefore, grid size is also a variable, and usually, a relatively large grid size is employed. Only resolving the average pressure, enthalpy, and temperature of the liquid phase stage, two-phase flow stage, and gases phase stage were solved and presented, but with acceptable accuracy. Due to its simplicity and practicality, the dynamic grid model was used in this work. The model divided the heat exchange tube of the steam generator into three sections: preheating section, evaporation section, and overheating section, as shown in Figure 2.

Momentum equation [14]:

$$l\left(\rho g\mathit{sin}\theta +W_{SG}{m}_{SG}/r/\rho \right)-\Delta p=0$$

(1)

Energy equation [14]:

$$\Delta h\cdot A_{SG}{m}_{SG}+\left(T_{}-T_p\right)K=0$$

(2)

$W_{SG}$ is the damping coefficient in the secondary side. $m_{SG}$ is the mass flow in the secondary side. $r$ is the radius of the heat exchange tube.$A_{SG}$ is the cross-sectional area. $K$ is the equivalent heat transfer coefficient. T is the temperature of the water. $T_p$ is the temperature of helium. $G_p$ is the mass flow of helium.$C_p$ is the specific heat capacity of helium. More information is available in reference [14].

When the steam generator was designed, throttling apertures [15] were utilized at the entrance to stabilize the two-phase flow. In order to extend the data under experimental conditions to the practical engineering design, the dimensionless Π-criterion [15] was adopted in the design to select the size of the throttling apertures, which described the relationship among the pressure drop of the throttling aperture, the subcooled region, the boiling region, and the superheated region. According to the suggestion in reference [15], the Π-criterion (throttling degree) was defined as:

$$\Pi =\frac{\mathsf{\Delta }{p}_{th}+ \mathsf{\Delta }{p}_{pre}}{\mathsf{\Delta }{p}_{eva}+\mathsf{\Delta }{p}_{over}}$$

(3)

where $\Delta p_{th}$ is the pressure drop of the throttling aperture; $\Delta p_{pre}$ is the pressure drop of the subcooled region; $\Delta p_{eva}$ is the pressure drop of the boiling region; and $\Delta p_{over}$ is the pressure drop of the superheated region. In this article, we ignore the loss of energy in the throttling apertures.

Energy equation:

$$m_{out}{h}_{out}-h_{in}{m}_{in}=0$$

(4)

Momentum equation:

$$\Delta p_{th}A+Wm+\rho gV=0$$

(5)

Mass equation:

$$m_{out}-m_{in}=0$$

(6)

$W$ is the damping coefficient with unit $\mathrm{m}/\mathrm{s}$, $A$ is the cross-section of the pipe with unit ${\mathrm{m}}^2$,$m$ is mass flow rate whose unit is $\mathrm{kg}/\mathrm{s}$, and $V$ is volume whose unit is ${\mathrm{m}}^3$.

2.2. Steam Turbine

For the purpose of simplification, the complex flow and heat transfer process was not considered inside the steam turbine, and a component-level lumped parameter model was utilized. In this work, the steam turbine is divided into two parts: a high-pressure turbine and a low-pressure turbine. The high-pressure turbine had two stages of extracting steam to the high-pressure heater and deaerator. While, the low-pressure turbine had three stages of extraction to three low-pressure heaters. Finally, the cooled hydrophobic returned to the main circuit through the condenser. The high-pressure turbine was divided into two control volumes, and the low-pressure turbine was divided into four control volumes. This was because its last stage of extraction was not at the exit compared with the high-pressure turbine. The control equations are as follows, where the classical Flugel formula [16] is used for the momentum equation.

Momentum equation:

$$m_{in}=\frac{K}{\sqrt{T_{in}}}\sqrt{p_{in}^2-p_{out}^2}$$

(7)

Energy equation:

$$\alpha m_{in}\left(h_{in}-h_{out}\right)-W_t=0$$

(8)

Mass equation:

$$m_{out}+m_{ext}-m_{in}=0$$

(9)

The classical Flugel formula describes the relationship between the pressure and the mass flow. $T_{in}$ is used to modify this equation, because the temperature difference between the inlet and outlet of the steam turbine is so large. $K$ is determined by the design of steam turbines. $\alpha$ is the efficiency of work at each stage. $W_t$ is the external work at each stage. $m_{ext}$ is the mass flow of extraction steam. Without considering the energy loss of extraction steam in the pipeline, the enthalpy of extraction steam remained unchanged before reaching the heat exchanger. However, there was a pressure drop after the extraction steam reached the heat exchanger, which was caused by the pipeline.

Steam extraction process:

$$(p_{heatexchanger}-p_{turbine})A+Wm+\rho gV=0$$

(10)

$$h_{heatexxchanger}=h_{turbine}$$

(11)

$m$ is the mass flow rate, $W$ is the pipe damping coefficient, $A$ is the cross-section of the pipe, and $V$ is the volume.

2.3. Heat Exchanger

In the secondary circuit, there are usually steam extractions from the steam turbine to heat the feedwater by the heat exchanger, raising thermal efficiency. In this work, four heat exchangers were considered: one high-pressure heater and three low-pressure heaters. Each heat exchanger had two parts: tube side and shell side. The steam extraction was on the shell side and the feedwater was on the tube side. The conservation equations are as follows.

Shell side:

$$steam\left|\begin{array}{c}{m}_{out}{h}_{out}-m_{in}{h}_{in}+Q=0\hfill \\ m_{out}-m_{in}=0\hfill \\ \left(p_{out}-p_{in}\right)A+Wm+\rho gV=0\hfill \end{array}\right.$$

(12)

Tube side:

$$feedwater:\left|\begin{array}{c}{m}_{out}{h}_{out}-m_{in}{h}_{in}-Q=0\hfill \\ m_{out}-m_{in}=0\hfill \\ \left(p_{out}-p_{in}\right)A+Wm+\rho gV=0\hfill \end{array}\right.$$

(13)

$$Q=C\left(T_g-T_w\right)$$

(14)

$Q$ is the heat exchange of heat exchanger and $C$ is the comprehensive heat transfer coefficient, which considers the equivalent effect of convection in steam and feedwater, and conduction through the tubes.

$$C=f\left(C_{convection, steam}^{}, C_{conduction}^{},C_{convection, feedwater}^{}\right)$$

(15)

$T_g$ is the temperature of extraction steam, which is calculated by function $f\left(p,h\right)$.$T_w$ is the temperature of feedwater, which is also calculated by function $f\left(p,h\right)$.

2.4. Condenser

There is a very complex heat transfer process inside the condenser, whose working pressure is close to vacuum. A relatively simple condenser model was considered in this work, in which the working pressure of the condenser was considered as the boundary condition. The outlet water temperature of the condenser was the saturated temperature under corresponding pressure. Because the main purpose of this paper was to construct the overall framework of the secondary circuit under the JFNK method, the model of the condenser needs to be refined in the future. The main equations are as follows.

Momentum equation:

$$p_{condenser }=constant$$

(16)

Energy equation:

$$m_{in}{h}_{in}-m_{out}{h}_{out}-Q_{hs}=0$$

(17)

Mass equation:

$$m_{out}+m_{lowpressure}-m_{in}=0$$

(18)

$m_{lowpressure}$ is the mass flow of condensed water from three low-pressure heaters, and finally through the condenser into the secondary circuit. $Q_{hs}$ is approximately equal to the latent heat of vaporization under the corresponding pressure.

2.5. Pump

The main function of the pump was to provide enough pressure heads for the system to compensate for the pressure loss in the pipeline and components. There was a certain relationship between pressure head and flow [17].

$$H_0=K_1+K_2{Q}_0+K_3{Q}_0^2$$

(19)

The coefficient K can be fitted by the flow characteristics of the pump. The pump model in this paper needed to be modified according to the flow characteristic curve. For the condensate pump, since the enthalpy of the inlet and outlet were changed slightly, we could ignore the heat exchange. However, for the feed water pump, the energy equation needed to be corrected, because its import and export enthalpy change greatly.

Momentum equation:

$$P_{out}-P_{in}+{\rho }_{in}gH=0 H=f\left(m\right)$$

(20)

Energy equation correction:

$$h_{in}-h_{out}+Q_{pump}=0$$

(21)

Mass equation:

$$m_{out}-m_{in}=0$$

(22)

$H$ is the pressure head generated by the pump, which can be determined according to the flow characteristic curve of the pump itself. $Q_{pump}$ is the difference between the inlet and outlet enthalpy of the pump, which is determined by using the function $h=f\left(T,p\right)$ according to the inlet and outlet pressure and temperature.

3. Solution Method

The JFNK method is a promising method to solve the nonlinear coupling system, where the Newton method is used as the external nonlinear iteration and the GMRES subspace method is used as the internal iteration to solve the linearization equation system, as shown in Algorithm 1. Instead of constructing a Jacobian matrix explicitly, the matrix-vector production $J\cdot v$ is approximated by the finite difference in the JFNK method [5]. For a nonlinear equation system with two unknowns, the matrix-vector production $J\cdot v$ approximation could be expressed as Equation (23). More details about the JFNK method can be found in Ref. [5].

Algorithm 1 Traditional JFNK method

1. For nonlinear equations F(x) = 0, set $x_0$

2. For i = 0, 1, …, until convergence (Newton iteration), do:

2.1 Solve $J\left(x_i\right)\delta x_i=-F\left(x_i\right)$ by GMRES (a kind of Krylov subspace method)

Where $J\cdot v\approx \left(F\left(x+\epsilon v\right)-F\left(x\right)\right)/\epsilon$,

$J$ is Jacobian matrix, $v$ is Krylov basis vector and $\epsilon$ is a perturbation

2.2 Set $x_{i+1}=x_i+\delta x_i$.

$$Jv=\left[\begin{array}{cc}\frac{\partial f_1}{\partial x_1^{\left(k\right)}}& \frac{\partial f_1}{\partial x_2^{\left(k\right)}}\\ \frac{\partial f_2}{\partial x_1^{\left(k\right)}}& \frac{\partial f_2}{\partial x_1^{\left(k\right)}}\end{array}\right]\cdot \left[\begin{array}{c}{v}_1\\ v_2\end{array}\right]\approx \left[\begin{array}{c}\frac{f_1\left(x_1+\epsilon v_1,x_2+\epsilon v_2\right)-f_1\left(x_1,x_2\right)}{\epsilon }\\ \frac{f_2\left(x_1+\epsilon v_1,x_2+\epsilon v_2\right)-f_2\left(x_1,x_2\right)}{\epsilon }\end{array}\right]$$

(23)

The main purpose of this work was to realize the simultaneous solution of the whole HTR secondary circuit under the framework of the JFNK method. Here, the HTR secondary circuit was simplified into a one-dimensional model, as shown in Figure 1. Moreover, each component was divided into one node or several nodes. In detail, the high-pressure turbine was divided into two nodes to consider the two stages of extraction. The low-pressure turbine had three stages of extraction, so four nodes were utilized. While, the other components, such as condenser, condensate pump, feedwater pump, deaerator, and high-pressure heater, were modeled as one node. The pipes that connected the components and the extraction bypasses were also considered as a node, which contained pressure, enthalpy, and mass flow information. The dynamic grid steam generator model in this work considered four nodes and mass flow as boundary conditions. Therefore, the entire HTR secondary circuit model was simplified into a one-dimensional circuit and contained five extraction bypasses.

Finally, there were 27 variable components of steam/water mass flow rate m, 24 variable components of steam/water pressure p, and 24 variable components of steam/water enthalpy h, as shown in

Table 1. Summarization of unknowns.

	m	p	h	Tp	l
Main circuit	15	15	15	0	0
Extraction bypass	10	5	5	0	0
SG inlet and outlet	2	0	0	0	0
Inside SG	0	4	4	3	2
Total	27	24	24	3	2

. In detail, the number of variable components of mass flow rate, pressure, and enthalpy in the secondary circuit was 15. For the extraction bypass, there were additionally 10 unknowns of mass flow, 5 unknowns of pressure, and 5 unknowns of enthalpy. Because the steam extraction finally returns to the main circuit, the missing node was the components of the main circuit corresponding to the steam extraction. The remaining two mass flow rates were the mass flow rates at the inlet and outlet of the SG, and the remaining four pressure and enthalpy values were the variables to be solved in the secondary side of SG, two of which were the inlet and outlet of the SG. For the primary side of the SG, there were additionally 3 variable components of helium temperature, which is the average temperature of different phase states (preheating section, evaporation section, and overheating section), and 2 variable components of grid size for the heat exchange tube (preheating section and evaporation section). Therefore, the solution variables were determined as follows.

$$x=\left[m,p,h,Tp,l\right]$$

(24)

Based on determining the solution variable, we needed to consider the residual function of the whole model.

$$f\left(m_i\right)=m_{i+1}-m_i+m_{ext}, i=1,2,3,\dots ,27$$

(25)

$$\begin{gathered} f\left(p_i\right)=p_{i+1}-p_i+W_{i\left(m_i\right)}{m}_i+{\rho }_{i}g{A}_i, i=1,2,3,\dots ,21,24 \\ f\left(p_{22}\right)=h_{22}-h\left(p_{22},q\right), i=22 \\ f\left(p_{23}\right)=\left(lt-l_1-l_2\right)\left({\overline{\rho }}_{over}gsin\theta +{\overline{\lambda }}_{over}{m}_{SG}/r/{\overline{\rho }}_{over}\right)-\left(p_{23}-p_{24}\right), i=23 \end{gathered}$$

(26)

Here, $q$ is the steam content of water, $lt$ is the total length of the heat exchange tube of the SG, $\overline{\rho }$ is the average density of water in the overheated section, and $\overline{\lambda }$ is the average thermal conductivity of water in the overheated section.

$$\begin{gathered} f\left(h_i\right)=m_{i+1}{h}_{i+1}-m_i{h}_i-Q, i=1,2,3,\dots ,20,24 \\ f\left(h_i\right)=\left(h_{i+1}-h_i\right)A_{SG}{m}_{SG}+\left(T-T_p\right)K, i=21,22,23 \end{gathered}$$

(27)

$$f\left(T{p}_i\right)=A_{SG}{m}_{Sg}\Delta h+\left(T_i-T_{i+1}\right)G_p{C}_p, i=1,2,3$$

(28)

$$f\left(l_i\right)=l_i\left(\overline{\rho }g\mathit{sin}\theta +500\overline{\lambda }{m}_{SG}/r/\overline{\rho }\right)-\Delta p_{l_i}, i=1,2$$

(29)

$\Delta h$ is the enthalpy difference of the corresponding heat transfer section. $\Delta p_{l_i}$ is the pressure difference of the corresponding heat transfer section.

In practice, there is a challenge for the traditional JFNK method. Since the physical quantities change dramatically, such as the pressure varying from the level of mega-Pascal to that of kilo-Pascal, as well as the difficulty of initial solution choice, the intermediate iterative solution may drop into the non-physical region, resulting in a bad break for JFNK method. According to the physical characteristics of the secondary circuit, a modified JFNK algorithm is proposed in this work, as displayed in Algorithm 2. There are two corrections (Step 2.3 and Step 2.4).

The first one is to correct the negative value of the solution vector after each newton iteration step, as shown in Step 2.3 of Algorithm 2, mainly to solve the issue caused by water’s physical properties. If the non-physical negative enthalpy or pressure occurs during the intermediate iteration, these physical quantities are forced to a positive value, so that the water’s physical properties can be calculated. The second one is to determine whether the physical variable exceeds the maximum value, as shown in Step 2.4 of Algorithm 2. If so, the local variable component is modified. The basic idea of the second one is to make the intermediate iteration solution within a reasonable range.

Algorithm 2 Modified JFNK method for HTR secondary circuit

1. For nonlinear equations $F\left(x\right)=0$, Equations (25)–(29), set $x_0=\left[m_0,p_0,h_0,T{p}_0,l_0\right]$

2. For i = 0, 1, …, until convergence (Newton iteration), do:

2.1 Solve $J\left(x_i\right)\delta x_i=-F\left(x_i\right)$ by GMRES (a kind of Krylov subspace method)

Details about the GMRES method could be found in Ref. [5]

2.2 Set $x_{i+1}=x_i+\delta x_i$

2.3 Set $x=abs\left(x\right)$

2.4 For $m_0,p_0,h_0,T{p}_0$

if $x_i>{xi{\left(\eta \ge 1\right)}_{systom\_max}}_{systom\_max}$

For $l_0$

if $x_i>l_{total},\hspace{0.33em}\hspace{0.33em}{x}_i=l_{total}/3$

4. Numerical Results and Discussion

4.1. Secondary Circuit Results

This paper focused on the steady-state calculation of the HTR secondary circuit. The model included SG, steam turbine, condenser, condensate pump, high and low-pressure heater, deaerator, feed water pump, etc. Here, the total electrical power of the HTR nuclear power plant with two nuclear steam supply systems was 211.9 MW. The SG inlet helium temperature and mass flow rate in the primary circuit were set as 750 °C and 96 kg/s, respectively. The mass flow rate of the secondary circuit was 186 kg/s. The working pressure of the condenser was 4.5 KPa. The numerical results of the secondary circuit are shown in Figure 3, where the mass flow rate, pressure, and enthalpy value are normalized.

Because of the existence of steam extraction, the mass flow rate became smaller after passing through the high-pressure turbine and low-pressure turbine, with normalized mass flow rate decreasing from 1 to 0.70. Afterward, the mass flow of the condensate pump outlet increased because the low-pressure turbine extraction finally returned to the main circuit through the condenser, as a result of which, the normalized mass flow rate increased from 0.70 to 0.88. Moreover, the high-pressure turbine steam extraction finally returned to the main circuit through the deaerator, so the mass flow rate increased after passing through the deaerator. Meanwhile, the mass flow at the outlet of the high-pressure heater was equal to that at the outlet of the steam generator, which proved that the mass of the whole flow process in the secondary circuit was conserved. The normalized pressure decreased from 0.8 to 0 after passing through the turbine, outputting power in this process. After the condensate pump and feed pump, the pressure rose back to the system pressure.

After passing through the steam turbine, the normalized enthalpy decreased from 1 to 0.65, and the steam turbine did a lot of work outside. Next, passing through the condenser, the normalized enthalpy decreased from 0.65 to 0. Because there were two phases in the condenser, water vapor directly became saturated water after passing through the condenser. The heat transfer in this process was close to the vaporization latent heat of water under the corresponding pressure. Subsequently, the enthalpy rose due to the presence of a high and low-pressure heater. Finally, after the high-pressure heater, the normalized enthalpy grew to 0.25.

The nonlinear Newton iteration residuals in Norm 2 are shown in Figure 4. Moreover, it can be seen that the asymptotical convergence rate of the JFNK method was 1.4179, indicating a super-linear convergence rate was achieved and satisfies the theorical results. In this work, there were 11 Newton steps to solve the steady-state problem of the HTR secondary circuit efficiently.

The comparisons between the numerical results and the typical values are summarised in

Table 2. Comparison between numerical results and typical values.

Mass Flow	Simulated Value	Typical Value	Relative Error/%
Steam turbine outlet	0.6810	0.6714	1.336
Condenser outlet	0.8748	0.8706	0.4551
Feedwater pump outlet	0.9999	1	−2.38 × 10−5
Pressure	Simulated value	Typical value	Relative error/%
Steam turbine outlet	0	0	0
Condenser outlet	0.1109	0.1109	5.94 × 10−11
Feedwater pump outlet	1.004	1	0.4621
Enthalpy	Simulated value	Typical value	Relative error/%
Steam turbine outlet	0.6401	0.6398	0.0486
Condenser outlet	−5.754 × 10−6	0	−0.0150
Feedwater pump outlet	0.1781	0.1775	0.2446

. The typical values of the HTR nuclear power plant can be found in reference [18]. The maximum relative errors of pressure and enthalpy are within 1%. The maximum relative error of mass flow is 1.336% at the outlet of the steam generator. The calculated errors are all within 1.5%, indicating that it has good consistency.

4.2. Sensitivity Analysis

4.2.1. Sensitivity Analysis of Initial Value

The components of the secondary circuit are closely coupled in a strong nonlinear manner; therefore, the choice of the initial value was a key issue for the nonlinear Newton iteration, which may have had a great impact on the convergence. In this work, the influence of different iterative initial values on the modified JFNK algorithm was discussed and analyzed.

In this work, the steam mass flow, pressure, and enthalpy were normalized by the steam mass flow, pressure, and enthalpy at the SG outlet, respectively. The helium temperatures at the SG inlet and outlet were 750 °C and 250 °C. Therefore, the average helium temperature $Tp$ could be approximated as 500 °C, and was used as the reference. The length of the heat exchange tube was half of the total length, and was used as a reference.

With the reference values as the standard, the value of each physical field increases by 10% or decreased by 10% as a new iterative initial value. To evaluate the robustness of the modified JFNK, 60 different cases were used. All cases were based on the above reference values. Cases 1–10 only changed mass flow; Cased 11–20 changed pressure only; Cases 21–30 only changed enthalpy; Cased 31–40 only changed the temperature of helium; Cased 41–50 only changed the length of the heat transfer section; all the physical fields changed simultaneously in Cases 51–60. The selection of different iterative initial values is shown in

Table 3. Initial value choice.

	Case 1–10	Case 11–20	Case 21–30	Case 31–40	Case 41–50	Case 51–60
Normalized steam/water mass flow	0.5, 0.6, 0.7, …, 1.4	1	1	1	1	0.5, 0.6, 0.7, …, 1.4
Normalized steam/water pressure	1	0.5, 0.6, 0.7, …, 1.4	1	1	1	0.5, 0.6, 0.7, …, 1.4
Normalized steam/water enthalpy	1	1	0.5, 0.6, 0.7, …, 1.4	1	1	0.5, 0.6, 0.7, …, 1.4
Normalized helium temperature	1	1	1	0.5, 0.6, 0.7, …, 1.4	1	0.5, 0.6, 0.7, …, 1.4
Normalized heat exchange tube length	1	1	1	1	0.5, 0.6, 0.7, …, 1.4	0.5, 0.6, 0.7, …, 1.4

. The correction factor $\eta$ was set to 1.5 for mass flow pressure and enthalpy because the maximum value of the iterative initial value exceeded 40% of the reference value.

The numerical results are presented in Figure 5 and

Table 4. Average Newton iterations.

	Change m	Change p	Change h	Change Tp	Change l	Change x	Total
Average newton steps	13.6	14.8	15.2	13.4	15.2	14.1	14.38

. The results showed that the modified JFNK can solve the HTR secondary circuit problem well with different iterative initial values. As shown in Figure 5, all cases converge within 20 Newton steps. For all these cases, the average Newton iteration is minimum when only helium temperature is perturbed, because the coupling between the primary side and the secondary side was relatively weaker than that of the flow field and temperature field of the secondary side itself. The maximum average Newton iteration is achieved for the cases which only changed the length of the heat transfer section and the enthalpy, because there was a complex two-phase heat transfer process inside the SG. The change of heat transfer section length and enthalpy in the SG not only affected the pressure field inside the SG but also had a great influence on the distribution of enthalpy. The average Newton iteration steps under the cases that changed the entire solution variable x were 14.1, which was between the minimum and the maximum. Moreover, all cases could converge and the average number of Newton iterations was 14.38, which indicated that the modified JFNK for HTR secondary circuit was robust.

4.2.2. Sensitivity Analysis of Correction Factors

The correction factor $\eta$ was a key parameter for the modified JFNK method in Algorithm 2. The influence of different correction factors on the convergence behavior was analyzed. Here, the range of correction factor $\eta$ was set to 1–1.8. Under different correction factors, the average Newton iteration steps of 60 cases in Section 4.2.1 were calculated.

The numerical results are shown in

Table 5. Average Newton iterative steps with different correction factors.

Correction Factor $\mathit{\eta }$	1	1.1	1.2	1.3	1.4	1.5	1.6	1.7	1.8
Average Newton steps	14.68	13.38	13.88	13.70	14.60	14.38	14.81	15.00	15.32

and Figure 6. The results showed that the average number of Newton iterations had an increasing trend with the increase of the correction factor. The basic idea of the modified JFNK was to limit the solution to a reasonable range during the iteration. If the limit range increases, that is, the correction factor increases, then in the iterative process, the probability of the solution variable deviating from the real solution will increase, which led to an increase in the number of iterations. However, the correction factor $\eta =1$ did not meet this trend, which was related to the upper limit we set as the maximum value of the system. This was because certain intermediate iterative solution components may have been slightly larger than the maximum value of the system. In this case, the convergence of these components was delayed, and undoubtedly the number of Newton iterations increased.

Overall, the selection of correction factors will also have a certain impact on the algorithm. The correction factor cannot be set too close to 1, the reasonable selection should be around 1.1. In this situation, the average Newton iteration steps for 60 cases was minimal.

5. Conclusions

In this paper, the steady-state problem of the HTR secondary circuit, including steam generator, steam turbine, condenser, feed pump, high/low-pressure heat exchanger, deaerator, etc, was solved by the JFNK method. However, this is still a challenge for the traditional JFNK method, since the physical quantities change dramatically and the intermediate iterative solution may drop into the non-physical region. Therefore, the modified JFNK was proposed and developed based on the physical characteristics of the system. Several conclusions can be made from this work:

(1)

The relative errors of the numerical results were within 1.5%, agreeing well with the typical values. The asymptotical convergence rate p was 1.41, achieving the super-linear convergence rate and satisfying the theorical prediction.

(2)

For the 60 different initial values selected in this paper, the modified JFNK could effectively converge, which proved that the algorithm had strong robustness.

(3)

The selection of the correction factor $\eta$ also had a certain impact on the algorithm, which could not be set too close to 1. The results showed that the average Newton iteration steps of 60 cases were the smallest with $\eta =1.1$

In conclusion, the modified JFNK method not only solved the problem whereby the traditional JFNK cannot solve the HTR secondary circuit well, but it also had strong robustness. Moreover, the modified JFNK had a good convergence accuracy and convergence rate. However, the physical models considered in this paper were relatively simple, such as the condenser and deaerator, which could be further developed and compared with the validated model in the future. Moreover, the neutronics and thermal-hydraulics coupling model based on JFNK had been developed in our existing works [1,7,8,9]. In the future, we plan to further couple the primary circuit and secondary circuit based on the JFNK method.