Corresponding author: Artavazd M. Sujyan (

Academic editor: Georgy Tikhomirov

The paper presents a review of modern studies on the potential types of coolant flow instabilities in the supercritical water reactor core. These instabilities have a negative impact on the operational safety of nuclear power plants. Despite the impressive number of computational works devoted to this topic, there still remain unresolved problems. The main disadvantages of the models are associated with the use of one simulated channel instead of a system of two or more parallel channels, the lack consideration for neutronic feedbacks, and the problem of choosing the design ratios for the heat transfer coefficient and hydraulic resistance coefficient under conditions of supercritical water flow. For this reason, it was decided to conduct an analysis that will make it possible to highlight the indicated problems and, on their basis, to formulate general requirements for a model of a nuclear reactor with a light-water supercritical pressure coolant. Consideration is also given to the features of the coolant flow stability in the supercritical water reactor core. In conclusion, the authors note the importance of further computational work using complex models of neutronic thermal-hydraulic stability built on the basis of modern achievements in the field of neutron physics and thermal physics.

An important task in the research of nuclear power plants is to determine the boundaries of their stable operation modes, which is necessary to ensure safety. In this context, ‘

A typical example of static instability is the Ledinegg instability (

The characteristic types of dynamic instability are thermoacoustic oscillations and oscillations of the ‘density wave’ type (

To analyze the dynamic instability of the coolant flow in the core of nuclear reactors, it is necessary to consider complex oscillatory flow regimes in parallel channels. In this case, instabilities of the coolant flow appear when there is a noticeable difference in the specific flow volumes in the parallel channels or in modes with natural circulation with noticeably different coolant heating in the parallel channels. Therefore, in designing reactor plants, it is practically important to establish the boundaries of the region of existence of oscillatory modes as well as their main dynamic characteristics. For this reason, the purpose of the work is to analyze the existing results in the field of computational research of thermal-hydraulic and neutronic thermal-hydraulic stability of supercritical water reactors as well as to specify problems for further research in this area.

A significant number of works have been devoted to the problem of establishing the boundaries of the region of existence of oscillatory modes. Most of them were performed by computational methods using various types of computational models built for heated channels of a simple geometric shape (round pipes) or individual cells that include fuel elements and a coolant. Computational models are basically built by two methods, i.e, frequency and time analyses, which differ in how the initial equations and uniqueness conditions are written.

In the frequency analysis method, the nonlinear differential equations of the model are linearized near the operating point, and a stability map is constructed for the system under study. However, due to this linearization, information about the system nonlinear properties is lost.

To simulate nonlinear processes in the system under study, the time analysis method is used, which involves nonlinear models with partial differential equations in space and time. As a rule, this method is used to study transient processes. At the initial stage, a stationary solution to the problem is obtained, after which a small disturbance is imposed on the stable state to obtain a dynamic response of the system. If the disturbance grows over time and leads to oscillations in the coolant flow rate, the system is considered unstable. If, after the disturbance, the oscillations damp, and the initial stationary state is established, then the system is stable.

As the analysis of the research results shows (_{cr}, the flow of a supercritical medium should be more stable than a two-phase one, while the stability region should expand as the pressure increases.

The main parameters influencing the stability of the flow of a supercritical medium in channels of a simple geometric shape include pressure _{in} = _{heat}_{in}; hydraulic diameter _{hyd} and heated length _{heat} of the channel; coefficients of local resistances at the inlet ζ_{in} and outlet ζ_{out} of the channel. The flow stability also depends on the orientation of the channel in the gravity field and on the direction of the flow with respect to the vector of this force.

One of the modern approaches to determining the areas of design and operating parameters at which it is possible to guarantee the absence of an unstable coolant flow in relation to the operating conditions of supercritical water reactors is to select generalized dimensionless parameters that determine the boundaries of the stable flow regions of supercritical media in channels (

_{1} = (β_{heat}_{exp}_{heat}_{heat}_{in}), (1)

_{2} = (β_{heat}_{exp}_{heat}

where _{heat}_{exp}_{heat}_{1}) can be considered as a value characterizing the fluid subcooling at the channel inlet to pseudocritical temperature, while the second parameter (_{2}) determines the change in the supercritical medium flow state as a result of heat supply to it. An example of a map with coordinates _{1}, _{2} for fluid flow regimes at supercritical pressures is shown in Fig.

Map of flow regimes of a medium of supercritical parameters in a round pipe (coefficient of local resistance at the inlet ζ_{in} = 20; coefficient of local resistance at the outlet ζ_{out}: 1–20; 2–10; 3–5; 4–2).

The threshold values of the defining parameters, shown in the figure by lines 1–4, were obtained as a result of calculations according to the linearized model carried out for water (

The suitability of the parameters _{1} and _{2} was checked for a general description of the stability of the system. Using the RELAP5 program code in (_{1} and _{2} were recorded. As a result, the considered conditions in the channel were determined as stable and located far enough from the boundaries of the unstable region.

The paper (

_{1}′ = [(_{out} – _{in})/_{in}](_{0}/_{h}), (3)

_{2}′ = (_{out} – _{in})/_{in}, (4)

the physical meaning of which is similar to that considered earlier for _{1} and _{2}. In this case, the change in the state of the supercritical medium due to heating in a channel of length _{h} (dimensionless number _{2}′, formula (4)) is expressed through the difference in specific volumes _{1}′, formula (3)) is calculated as the fraction _{0}/_{h} of number _{2}′, where _{0} is the distance from the inlet at which the temperature value _{0}, is reached, which determines the beginning of the ‘pseudophase transition’. In calculations at a water pressure of 25 MPa, _{0} = 350 °С. is conventionally assumed in this work. In the case of uniform water heating along the length of the channel, the ratio _{0}/_{h} can be replaced by (_{0} – _{in})/(_{out} – _{in}), where _{0} is the single-valued function of temperature _{0} at a given pressure.

Work (

The paper (

Influence of water temperature at the inlet to the core on the thermal-hydraulic stability (

The studies (

The results of the frequency analysis method showed that the coolant flow rate in the thermal region is stable in a wide range of operating parameters. An increase in the mass flow rate of the coolant has a positive effect on the stability of the system. The importance of the second natural frequencies of the system was identified. This led to the need for a nonlinear stability analysis. The time analysis method showed that the oscillations in the mass flow rate of water at the inlet and outlet of each channel do not coincide in phase. Apparently, this was due to the use of a fixed boundary value for the flow rate at the inlet to the channels.

The stability of the SCWR-M fast-spectrum core zone was determined in a wide range of operating conditions. The stability of the parallel channel system is determined by the hottest channel, the parameters of which are most susceptible to oscillations in unstable modes. The higher the specific power of the hottest channel, the more unstable the system is. An increase in the mass flow rate of the coolant has a positive effect on stability. Systems with a uniform axial power distribution are less stable than systems with a cosine or forked distribution.

The results of the analysis by two methods in both works are in good agreement in assessing the limiting stability of the system.

The authors of (

In (

It was found that the range of parameters in which the thermal-hydraulic instability is observed is directly determined by the point at which the coolant reaches the pseudocritical temperature. In addition, the boundary of the stability region of the CSR1000 strongly depends on the value of the hydraulic resistance coefficient adopted in the calculations. It is noted that, depending on the operating parameters in the reactor, both the Ledinegg instability and the density wave instability can occur.

In (

The work (

In (

It is shown that stability is greatly influenced by the pressure drop, the change in the value and distribution of the mass flow rate of the coolant between the heated channels. The stability of the parallel channels is mainly determined by the pressure drop. With an increase in the mass flow rate and pressure in the system, as well as with a decrease in the heat flow, the stability of the flow rate in the parallel channels increases. A dimensionless analysis of the boundaries of the thermal-hydraulic stability indicated an analogy between the key criteria in systems with water of sub- and supercritical pressure. The influence of the temperature at the inlet is ambiguous at low and high values of water ‘pseudo-subcooling’. Small changes in the coolant density and in the pressure drop cause instability in the parallel channels. The authors note the importance of taking into account the lag effects and feedbacks between the mass flow rate, the density of the medium and the pressure drop.

In (

It is argued that predicting the flow stability boundaries using a single-channel model may be inaccurate due to the fact that this model does not take into account the radial change in the power of heat release in the reactor and the possibility of instability in the form of oscillations of the coolant flow in the parallel channels in antiphase.

The analysis of neutronic thermal-hydraulic stability takes into account that one or another type of instability of the coolant flow in a particular reactor facility should always be considered as a complex phenomenon, in which various processes (hydrodynamic, thermal or neutronic) are simultaneously involved, occurring under certain boundary conditions and closely related with the features of the scheme and design of this facility.

In (

It is noted that the neutronic effects have a significant impact on the reactor stability. This illustrates the comparison of the parameters of the oscillations of the

Influence of water temperature at the inlet to the core on the neutronic thermal-hydraulic stability (

It was found that, on the whole, the SCLWR-H design meets the requirements for the neutronic thermal-hydraulic stability when the reactor is operating at its rated power. Instability can occur at low loads. At the same time, a negative influence is exerted by the effect of feedback with the moderator density. An increase in the density coefficient of reactivity decreases stability. Due to the long delay of heat transfer to water in the moderator rods, the joint consideration of the interacting neutronic and thermal-hydraulic characteristics acquires particular importance in the analysis of the stability of this reactor.

The results of the study of the thermal-hydraulic and neutronic thermal-hydraulic stability of the US SCWR, obtained using the frequency analysis method to a one-dimensional numerical model, are presented in (

The calculations of instabilities, when oscillations in operating parameters coincide in phase within the entire core, showed that, under normal conditions, such oscillations quickly damp. The rate of this process is characterized by the obtained

In (

Below is a summary table, where, in addition to the purpose and main characteristics of the computational models, additional information is provided regarding the use of ratios to determine the coefficients of friction and heat transfer on the channel walls in the analysis of the thermal-hydraulic stability.

Based on the data in the table, we can note the following: of all the works, only two (

To determine the boundaries of the supercritical coolant flow stability, computational schemes of varying degrees of complexity were used, including three-dimensional thermal-hydraulic and complex neutronic thermal-hydraulic models. It is shown that the preliminary generalized stability analysis can be performed by the frequency or time analysis methods using one-dimensional one-channel models.

In the case of one-dimensional one-channel models, the heat flux density on the channel walls is usually a given value, constant or a function that varies along the channel length. Thus, the hydrodynamic connection between adjacent channels, combined by the inlet and outlet collectors in the core, is not taken into account, and the thermal interaction of the channel under consideration with its environment is also disregarded. In this respect, models that include two or more parallel channels have an undoubted advantage.

Particular consideration should be given to the development of one-dimensional models, in which the heat transfer between the coolant moving in the cells between the heat-generating and other elements of the core structure in non-stationary (transient) modes, when the heat capacity of materials can play a significant role.

In thermal SCWRs, where the so-called ‘water rods’ serve as the moderator of neutrons, taking into account the heat transfer between these rods and the coolant is especially important, since a change in the water rod temperature, and hence the moderator density, can lead to the neutronic thermal-hydraulic instability.

In any case, the frictional resistance and heat transfer in channels with supercritical water parameters should be taken into account using relations specially developed for the case of a significant change in the thermal-physical properties of the coolant with temperature and pressure near the critical point.

It is important to keep in mind that at supercritical pressure in the vicinity of the ‘pseudophase transition’, the regularities of friction resistance can have peculiarities, and the heat transfer by its nature can vary greatly, i.e., depending on conditions it can be normal, improved or deteriorated. The ratios for the coefficients of friction and heat transfer should be tested for use in the calculations of such complex structures as fuel assemblies for nuclear reactors. Unfortunately, the above considerations, as can be seen from Tab.

As for three-dimensional models, in which a detailed calculation of flow characteristics is carried out using modern CFD codes, it should be noted that the methods for determining the coefficients of turbulent transfer of momentum and heat in supercritical media have not yet been sufficiently developed. The results obtained in this way are not entirely reliable and, therefore, do not have any special advantages over the data found by one-dimensional models, in which the parameters averaged over the channel cross-section and empirical coefficients of resistance and heat transfer are used.

Purpose and main characteristics of computational models in works (

Work | Simulated reactor _{in/out}, °С |
Thermal- hydraulic model^{1)} |
Ratios for the friction coeff.^{2)} |
Ratios for the heat-trans. coeff.^{3)} |
Neutronic model |
---|---|---|---|---|---|

( |
SCWR 25; 280/500 | S | Ha | – | – |

( |
US SCWR 25; 280/500 | S | – | D-B | – |

( |
SCLWR-H 25; 280/500 | S | Bl | K-O | – |

( |
SCWR-M 25; 280/407.7 | PCh | – | – | – |

( |
SCWR-M 25; 407.7/510 | PCh | Bl-McA | – | – |

( |
SCWR 25; 220/450 | PCh | Ha | – | – |

( |
CSR1000 25; 280/500 | S | Ha, Bl, F | K-O | – |

( |
VVER-SCP 24.5; 290/540 | S | – | – | – |

( |
SCPS-600 24.5; 390/500 | S | – | – | – |

( |
SCWR 25; 280/500 | PCh | Bl-McA | – | – |

( |
CANDU SCWR 25; 350/625 | S | Ch-Ch, C-W, F | – | – |

( |
CANDU SCWR 25; 350/625 | PCh | Ch-Ch | – | – |

( |
SCLWR-H 25; 280/500 | S | Bl | K-O | Point, 6 groups |

( |
US SCWR 25; 280/500 | S | – | – | Point |

( |
US SCWR 25; 280/500 | S | – | – | Point |

( |
SCWR 25; 280/500 | S | F, P | M | Point, 6 groups |

^{1)} S = single-channel; PCh – parallel channels; ^{2)} Ha – Haaland; Bl – Blasius; F – Filonenko; McA – McAdams; Ch-Ch – Churchill-Chu; C-W – Colebrook–White; P – Popov; ^{3)} D-B – Dittus-Boelter; K-O – Koshizuka-Oka; M – Mokry.

Undoubtedly, to obtain the recommendations necessary to substantiate the stable operation and safety of any nuclear power plant, one should use complex neutronic thermal-hydraulic computational models that fully take into account the feedbacks between the reactor reactivity, pressure drop, flow rate, and coolant temperature.

The main feature of the stability of the coolant flow in the core of supercritical water reactors is associated with the strong dependence of the physical properties of water on temperature and pressure in the region of the ‘pseudo-phase transition’ near the critical point.

By tradition and by analogy with a boiling coolant, in computational studies of the thermal-hydraulic stability of systems with supercritical water, frequency and time analysis methods are used with process models of different levels of complexity. One-dimensional non-stationary models with one or two channels combined by inlet and outlet collectors have received the greatest development. Based on the results of calculations using these models, a number of useful parametric and generalized dependencies have been obtained that determine the boundaries of the existence of stable flow and heat removal regimes.

Further improvement of the computational models involves the inclusion of the hydrodynamic and thermal interaction of the channel (cell) under consideration with the coolant with the surrounding elements of the core (adjacent channels, fuel rods, moderator water rods, etc.) into the non-stationary process under study. In this case, as the closing relations for the frictional resistance and heat transfer, universal dependences should be mainly used that take into account both the variability of the properties of water and the possibility of changing the flow and heat transfer regimes. These dependencies should be tested for use in calculations of structures that are as close as possible to real fuel assemblies of nuclear reactors.

The final conclusions about the reliability and safety of a nuclear power plant in nominal and transient modes should be based on the results of calculations performed using complex models of neutronic thermal-hydraulic stability, built on the basis of modern advances in neutron physics and thermal physics.

^{th}International Symposium on Supercritical Water-Cooled Reactors (ISSCWR-5). March 13–16, Vancouver, Canada: 115.

* Russian text published: Izvestiya vuzov. Yadernaya Energetika (ISSN 0204-3327), 2021, n. 3, pp. 29–43.