The working fluid inside the tube experiences a phase change caused by heating. In order to accurately consider the physical properties along the tube and the corresponding changes in resistance and heat transfer characteristics, the present model divides the flow region into a series of grids based on the discrete method and uses the finite difference method to solve the equations. Fig. 3 shows a schematic diagram of the mesh division along the flow region in any helical tube.

In Fig. 3, for any grid J along the tube i at time layer k, the inlet parameters are denoted by X(i, j, k). Based on the above grid mesh method, the N-S partial differential Eqs. (1)–(3) are numerically discretized. For grid J, the discretized equations are expressed as Eqs. (5)–(7). (1)

Discretized mass conservation equation (5)AΔt(ρ(i,j,k+1)+ρ(i,j−1,k+1)2−ρ(i,j,k)+ρ(i,j−1,k)2)+M(i,j,k+1)−M(i,j−1,k+1)Δz=0

The fluid inside the helical tube undergoes three stages: subcooled water region, two-phase mixture region, and superheated steam region. The present model determines the state of the fluid in each grid by comparing the inlet fluid enthalpy with the corresponding enthalpy values of saturated water and steam.

For single-phase region, the frictional pressure drop gradient is calculated using the Ito formula [22]. (8)dPfdz=fG22ρdwhere G denotes mass flow rate, d denotes hydraulic diameter, and f is coefficient of frictional resistance. Under laminar flow conditions, the formula for calculating f is below [22]. (9)f=fs⋅21.5⋅De/(1.56+log10⁡De)5.73where f_s is the corresponding straight tube resistance coefficient, De is the Dean number, which is calculated by the following equation. (10)De=RedDcwhere Re denotes Reynolds number, and D_c denotes helical diameter.

Under turbulent conditions, the formula for calculating f is below [22]: (11)f=0.304Re−0.25+0.029(d/Dc)0.5

For two-phase flow, frictional pressure drop gradient is calculated by the formula proposed by Santini et al. [23].(12)dPfdz=(−0.0373x3+0.0387x2−0.00479x+0.0108)G1.91ρ⋅d1.2where x is the steam quality.

Boundary conditions are set as below: (1)

The total inlet mass flow rate of the system is known.

In the above boundary condition (5), the heat flux density distribution is obtained from the coupled heat transfer process between two sides of the steam generator, and it can be calculated by the steady thermal hydraulic model in Section 2 of this paper.

(1) Verification for density wave instability

Wang et al. [24] conducted an experiment to study two-phase flow instability in two parallel helical tubes, and obtained the threshold heat power values for density wave instabilities that occurred under different inlet fluid quality (x_in). The related structural parameters and operation parameters are shown in Table 1.

Fig. 4 shows the calculated results of the model in this work, as well as the corresponding experimental values.

Fig. 4 shows that the calculated values show good agreement with the experimental values, and the relative deviation of the predicted results from the experimental data shall not exceed 4% within the operating range of the examples.

(2) Verification for flow excursion instability

Papini et al. [25] conducted an experimental study on two-phase instabilities in the parallel helically coiled channels of a steam generator, and the experimental data were used to verify the reliability of the prediction for flow excursion by the present model. In their experiment system, the test section consists of two channels, each with a length of 32 m. The first 24 m of the channel is heated, while the remaining 8 m is unheated. The flow rates in the two channels were recorded in the presence of a flow excursion instability (system parameters: P = 24 bar, Tin = 134°C, Q = 46.5 kW). Fig. 5 shows the comparison of the predicted results by the present model with the experimental data.

In Fig. 5, it can be found that the predicted results show a good agreement with the experimental data, and the flow rates of the two channels gradually diverge from each other, exhibiting an obvious flow excursion phenomenon under the current operating conditions.

In this section, the heat load distribution along the tube under the coupled heat transfer conditions is first calculated by the steady thermal-hydraulic model, as shown in Fig. 7. In the calculation, the inlet temperature of lead bismuth on primary side (T_lb,in) is 400°C, and inlet temperature of water on secondary side (T_f,in) is 200°C. In addition, existing related research often simplifies the heat load distribution into a uniform distribution or a linear distribution. In order to study the influence of heat load distribution, Fig. 7 also presents four different linear heat flux distributions with different slopes (K = 0, 0.5, 1.0, 2.0), and the uniform heat flux distribution can be considered as a linear distribution with slope K = 0. The average heat flux (Q_avg) remains constant under all heat flux distribution patterns.

In Fig. 7, it can be seen that under the coupled heat transfer conditions, the heat load distribution is extremely non-uniform along the tube. Firstly, the fluid at the front of the tube section is in a single-phase subcooled water state, and heat transfer coefficients on the inner wall surface are generally low, resulting in a relatively low heat flux density. As the fluid continues to be heated along the flow direction, the temperature difference between water and lead bismuth gradually decreases, resulting in a gradual decrease in heat flux density. At the position of z = 10.9 m, the fluid inside the tube reaches saturation temperature and begins to enter the nucleate boiling region. The inner wall heat transfer coefficient sharply increases, resulting in a sudden increase in heat flux density between two sides. At the outlet of the tube section, the heat flux density reaches its peak. The maximum and minimum values of heat flux density are about 1087.4 and 25.1 kW/m², and there exists a difference of 43 times between them. The relative standard deviation (RSD) is adopted to characterize the uniformity of heat flux density distribution along the tube, and the calculation method is as follows.(13)RSD=∑j=1N(Q(j)−Q¯)2N−1Q¯where Q¯ is the average value of the heat flux corresponding to all grids along the tube.

According to the calculation, the RSD of the heat flux distribution under coupled heat transfer conditions is 1.47, which is much higher than the corresponding values under uniform heat flux distribution and linear heat flux distribution conditions.

Fig. 8 indicates the oscillation curve of inlet mass flow rate in parallel helical tubes calculated under different heat flux boundary conditions, as defined in Fig. 7, calculated by the dynamic instability model.

Fig. 8 declares that the oscillation curve of mass flow rate tends to converge, and the system quickly returns to stability under the coupled heat transfer conditions. Under the condition of linear heat flux distribution, the oscillation curve of mass flow rate in the tube shows a significant divergence trend. As the distribution slope K increases, the divergence trend of the oscillation curve slows down significantly, and the system stability is enhanced. Under the condition of uniform heat flux distribution (K = 0), the divergence trend of the oscillation curve is the most severe, the oscillation amplitude rapidly expands, and the system is the most unstable. The main reason can be explained below. The heat flux density in a long tube section from the tube inlet (0 m < z < 15 m) under the coupled heat transfer conditions is further lower than that under the condition of linear heat flux distribution. This directly leads to a significantly longer length of the single-phase subcooled water section and a relatively shorter length of the two-phase evaporation section inside the tube. Fig. 9 shows the fluid quality distribution (x) along the helical tube under different heat boundary conditions. The length of subcooled water section (L_s) and the length of two-phase evaporation section (L_t) inside the tube under different heat boundary conditions are also shown in Fig. 9.

It can be clearly seen from Fig. 9 that under the coupled heat transfer conditions, the length of subcooled water region (L_s) is 10.9 m, and the length of the two-phase evaporation section (L_t) is only 9.1 m. Under the condition of uniform heat flux distribution, the corresponding length of the single-phase subcooled water region is 3.1 m, which is only one-third of L_s under the coupled heat transfer conditions. Meanwhile, as the slope K of the linear distribution increases, L_s also increases and L_t gradually decreases. The main reason for this is that as the slope K increases, the wall heat flux density of the front section of the tube near the inlet gradually decreases, which means the inlet working fluid needs to flow through a longer distance to be heated to saturation temperature. In general, the compressibility of the liquid in the subcooled water region is very weak, which can hinder the oscillation of the flow rate inside the tube. In the two-phase region, the working fluid undergoes a phase transition, resulting in significant density changes and promoting the occurrence of density wave oscillation. The larger the ratio of L_s to L_t, the more stable the system becomes [26]. Therefore, as the slope K of the linear distribution increases, the density wave oscillation trend weakens and the system stability enhances.

Fig. 10 declares the heat load distribution of the inner tube wall at different inlet lead bismuth temperatures (T_lb,in = 320°C~480°C) calculated by the steady thermal-hydraulic model. Among them, the inlet water temperature in secondary side is 160°C.

Fig. 10 shows that the heat load between two sides gradually increases with the increase of T_lb,in, especially in the section near the outlet of the helical tube, where the heat flux density increases significantly. As T_lb,in gradually increases from 320°C to 480°C, the average heat flux density (Q_avg) along the steam generator gradually increased from 105.85 to 257.38 kW/m², with an increase of approximately 2.5 times. The peak heat flux density increased from 427.82 to 1515.86 kW/m², increasing by nearly 3.5 times. The main reason for this can be explained below. As the inlet lead bismuth temperature increases, the temperature difference between the two fluids gradually increases. Moreover, the increase in heat transfer temperature difference is more significant in the section near the outlet of the helical tube, due to the counter flow arrangement of the steam generator. This leads to an increase in heat transfer between the two sides. Fig. 10 also indicates that as T_lb,in gradually increases from 320°C to 480°C, the relative standard deviation (RSD) of the heat flux distribution along the steam generator shows a gradually increasing trend from 0.85 to 1.27, increasing by 49%. This indicates that the heat flux density distribution gradually becomes more uneven. At T_lb,in = 480°C, the maximum value and minimum value of heat flux density are 1515.86 and 39.07 kW/m², and the difference between them is approximately 40 times. Fig. 11 represents the fluid quality distribution results along the helical tube under different inlet lead bismuth temperatures ranging from 320°C to 480°C.

In Fig. 11, it can be seen that as T_lb,in increases, the heat load between two sides gradually increases, and the fluid quality at the outlet of the secondary side gradually increases. Especially in the two-phase region, the variation in fluid quality along the tube becomes more pronounced as T_lb,in increases. When T_lb,in increases to 480°C, the fluid at the outlet has reached a superheated steam state, with a thermal equilibrium quality of nearly 1.1. Fig. 12 further shows the oscillation curves of the inlet mass flow rate in two parallel helical tubes under different inlet lead bismuth temperatures of 320°C~480°C calculated by the dynamic instability model.

Fig. 12 indicates that the inlet mass flow rate in parallel helical tubes exhibits two typical flow instability phenomena at different inlet lead bismuth temperatures. Firstly, under the condition of relatively low inlet lead bismuth temperature (T_lb,in = 320°C~400°C), the flow excursion instability occurs in the inlet mass flow rate of two tubes. The flow rate of the working fluid in the tube gradually deviates from the initial steady-state value, accompanied by the appearance of out-of-phase density wave oscillation. As T_lb,in increases, the phenomenon of flow excursion intensifies, and the mass flow rate of the two tubes rapidly deviates from the steady-state value, indicating that the system becomes more unstable. When T_lb,in increases to 440°C, the flow excursion phenomenon disappears, and the inlet mass flow rate of the two tubes shows single out-phase density wave oscillation and tends to diverge. As T_lb,in continues to increase, the oscillation amplitude of mass flow rate also increases, the divergence trend becomes more obvious, and the system stability is significantly weakened. The above law of two typical flow instability phenomena is related to the variation curve of pressure drop with flow rate of the parallel helical tube system in the secondary side (“∆P–G curve”). The flow excursion instability generally occurs in the negative slope region of ∆P–G curve [27]. Fig. 13 shows the corresponding ∆P–G curve under different inlet lead bismuth temperatures (T_lb,in = 320°C~480°C). In the calculation of the above examples in Fig. 12, the mass flow rate of the parallel helical tube system is 1000 kg/(m²·s).

Fig. 13 indicates that the working conditions in the tube are all located in the negative slope region of ∆P–G curve under the conditions of low inlet lead bismuth temperature (T_lb,in = 320°C~400°C), resulting in the occurrence of the flow excursion phenomenon. Under these conditions of T_lb,in = 320°C~400°C, the heat load values along the tube are relatively small, and the steam quality at the outlet is low. Therefore, the fluid density variation along the helical tube is relatively weak, the amplitude of density wave oscillation is also relatively weak. As T_lb,in increases (T_lb,in = 440°C~480°C), the wall heat flux density gradually increases, and the steam quality at the outlet of the secondary side gradually increases, even reaching the superheated steam state. At this time, the working condition in the tube is located in the positive slope region of ∆P–G curve, and the flow excursion phenomenon disappears. However, the density changes experienced by the fluid are significantly intensified under higher T_lb,in of 440°C~480°C, so stronger density wave oscillation between the tubes is induced.

Fig. 14 indicates the heat load distribution at different inlet water temperatures in the secondary side (T_f,in = 160°C~240°C) calculated by the steady thermal-hydraulic model. Among them, T_lb,in is 400°C.

Fig. 14 indicates that as T_f,in increases, the temperature difference between water and lead bismuth is bound to decrease, resulting in a weakened heat exchange between two sides. The average heat flux density (Q_avg) gradually decreases from 183.41 to 140.04 kW/m², decreasing by about 24%. Fig. 14 also declares that as T_f,in gradually increases from 160°C to 240°C, the relative standard deviation of heat load distribution along the steam generator (RSD) increases from 1.19 to 1.60, increasing by 34%. This indicates that heat load distribution gradually becomes more uneven. Besides, it should be noted that there is almost no difference in the outlet fluid temperature under different inlet water temperatures of 160°C~240°C. Fig. 15 further illustrates the fluid quality distribution (x) of the secondary side along the helical tube under different inlet water temperatures (T_f,in = 160°C~240°C).

Fig. 15 indicates that there is almost no obvious change in the fluid quality at the outlet, due to the constant T_lb,in. When the inlet water temperature is low, the wall heat flux density near the inlet of the tube is also high, caused by a higher heat transfer temperature difference, and the fluid quality also changes more dramatically. Fig. 16 further shows the oscillation curves of the inlet mass flow rate in two parallel helical tubes under different inlet water temperatures of the secondary side (T_f,in = 160°C~240°C) calculated by the dynamic instability model.

Fig. 16 declares that when T_f,in is 160°C, there is a significant flow excursion phenomenon, accompanied by divergent density wave oscillation characteristics, resulting in a rapid deviation of the mass flow rate from the steady-state value. When the inlet water temperature increases to 180°C, the phenomenon of flow excursion disappears, and the flow rate between two tubes exhibits divergent out-of-phase density wave oscillation. As the inlet water temperature continues to increase, the oscillation trend of the mass flow rate between two tubes tends to converge, the amplitude gradually weakens, and the system stability is enhanced. When the inlet water temperature increases to 240°C, the oscillation trend of the flow rate is already very weak, and quickly recovers to the initial steady-state value.