Corresponding author: Artavazd M. Sujyan ( artavazd1994@gmail.com ) Academic editor: Georgy Tikhomirov
© 2021 Artavazd M. Sujyan, Viktor I. Deev, Vladimir S. Kharitonov.
This is an open access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Citation:
Sujyan AM, Deev VI, Kharitonov VS (2021) Analysis of numerical studies on the thermalhydraulic and neutronic thermalhydraulic stability of supercritical water reactors. Nuclear Energy and Technology 7(4): 311318. https://doi.org/10.3897/nucet.7.78368

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 lightwater 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 thermalhydraulic stability built on the basis of modern achievements in the field of neutron physics and thermal physics.
Nuclear reactor, supercritical water, coolant flow instabilities, computational models
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, ‘instability’ means the occurrence of sustained oscillations in the coolant flow rate when small shortterm disturbances are introduced into the system. There are two main types of stability related to supercritical pressure water reactors (SCWR): (1) thermalhydraulic, which is characterized by the retention of operating parameters (density, temperature, pressure and flow rate) of the coolant flow, as well as heat transfer parameters during operation, and (2) neutronic thermalhydraulic, which consists in maintaining the stability of the processes of energy release and heat removal in the core. The whole set of different types of instability can be divided into two main groups: static and dynamic (oscillatory) instabilities (
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 thermalhydraulic and neutronic thermalhydraulic 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 (
The main parameters influencing the stability of the flow of a supercritical medium in channels of a simple geometric shape include pressure p; mass flow rate in the channel ρw; heat flux density on the heattransfer surface q; fluid subcooling at the inlet to the temperature of the pseudophase transition expressed through the difference in enthalpies Δh_{in} = h_{heat} – h_{in}; hydraulic diameter d_{hyd} and heated length l_{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 (
N _{1} = (β_{heat}/c_{exp},_{heat})×(h_{heat} – h_{in}), (1)
N _{2} = (β_{heat}/c_{exp},_{heat})×(Q/G), (2)
where Q is the power supplied to the coolant in the channel; G is the fluid flow rate in the channel; β_{heat} and c_{exp},_{heat} are the coefficient of volumetric expansion and specific heat at the pseudocritical point. These parameters, as such, are analogs of dimensionless complexes that have already been used to describe the boundaries of stability on the map of flow regimes of twophase flows (
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 (p = 25 MPa), as well as for other media, i.e., carbon dioxide (8.0 MPa), freon23 (5.7 MPa) and ammonia (15.0 MPa). The calculations (
The suitability of the parameters N_{1} and N_{2} was checked for a general description of the stability of the system. Using the RELAP5 program code in (
The paper (
N _{1}′ = [(v_{out} – v_{in})/v_{in}](L_{0}/L_{h}), (3)
N _{2}′ = (v_{out} – v_{in})/v_{in}, (4)
the physical meaning of which is similar to that considered earlier for N_{1} and N_{2}. In this case, the change in the state of the supercritical medium due to heating in a channel of length L_{h} (dimensionless number N_{2}′, formula (4)) is expressed through the difference in specific volumes v of the fluid at the inlet and outlet with respect to the value of the same parameter at the inlet to the channel, and the dimensionless supercritical water ‘subcooling’ (number N_{1}′, formula (3)) is calculated as the fraction L_{0}/L_{h} of number N_{2}′, where L_{0} is the distance from the inlet at which the temperature value T_{0}, is reached, which determines the beginning of the ‘pseudophase transition’. In calculations at a water pressure of 25 MPa, T_{0} = 350 °С. is conventionally assumed in this work. In the case of uniform water heating along the length of the channel, the ratio L_{0}/L_{h} can be replaced by (h_{0} – h_{in})/(h_{out} – h_{in}), where h_{0} is the singlevalued function of temperature T_{0} at a given pressure.
Work (
The paper (
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 SCWRM fastspectrum 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 thermalhydraulic 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 thermalhydraulic 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 ‘pseudosubcooling’. 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 singlechannel 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 thermalhydraulic 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 DR system, calculated using the thermalhydraulic and neutronic thermalhydraulic models (see Figs
It was found that, on the whole, the SCLWRH design meets the requirements for the neutronic thermalhydraulic 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 thermalhydraulic characteristics acquires particular importance in the analysis of the stability of this reactor.
The results of the study of the thermalhydraulic and neutronic thermalhydraulic stability of the US SCWR, obtained using the frequency analysis method to a onedimensional 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 DR values, which in cases of the thermalhydraulic and neutronic thermalhydraulic instability are equal to 0.20 and 0.007, respectively, which is significantly lower than the limit values usually taken when the BWR stability is assessed (0.5 and 0.25). The sensitivity analysis showed that an increase in the coefficient of local resistances at the inlet sections increases the system stability, and an increase in the feedback coefficient for the coolant density decreases it. Compared with the axial profile of the cosineshaped heat release power, the uniform profile increases stability, while the asymmetric profile sloping downwards decreases it. It is shown that this reactor is characterized by a weak connection between neutron kinetics and thermal hydraulics. In this case, the main moderators of neutrons are water rods, in which oscillations in the water density are insignificant; in addition, the high subcriticality of the first subcritical regime, compared to that of the BWR, provides fast damping of neutron oscillations.
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 thermalhydraulic 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 threedimensional thermalhydraulic and complex neutronic thermalhydraulic models. It is shown that the preliminary generalized stability analysis can be performed by the frequency or time analysis methods using onedimensional onechannel models.
In the case of onedimensional onechannel 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 onedimensional models, in which the heat transfer between the coolant moving in the cells between the heatgenerating and other elements of the core structure in nonstationary (transient) modes, when the heat capacity of materials can play a significant role.
In thermal SCWRs, where the socalled ‘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 thermalhydraulic 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 thermalphysical 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 threedimensional 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 onedimensional models, in which the parameters averaged over the channel crosssection and empirical coefficients of resistance and heat transfer are used.
Purpose and main characteristics of computational models in works (
Work  Simulated reactor р, MPa; Т_{in/out}, °С  Thermal hydraulic model^{1)}  Ratios for the friction coeff.^{2)}  Ratios for the heattrans. coeff.^{3)}  Neutronic model 

( 
SCWR 25; 280/500  S  Ha  –  – 
( 
US SCWR 25; 280/500  S  –  DB  – 
( 
SCLWRH 25; 280/500  S  Bl  KO  – 
( 
SCWRM 25; 280/407.7  PCh  –  –  – 
( 
SCWRM 25; 407.7/510  PCh  BlMcA  –  – 
( 
SCWR 25; 220/450  PCh  Ha  –  – 
( 
CSR1000 25; 280/500  S  Ha, Bl, F  KO  – 
( 
VVERSCP 24.5; 290/540  S  –  –  – 
( 
SCPS600 24.5; 390/500  S  –  –  – 
( 
SCWR 25; 280/500  PCh  BlMcA  –  – 
( 
CANDU SCWR 25; 350/625  S  ChCh, CW, F  –  – 
( 
CANDU SCWR 25; 350/625  PCh  ChCh  –  – 
( 
SCLWRH 25; 280/500  S  Bl  KO  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 
Undoubtedly, to obtain the recommendations necessary to substantiate the stable operation and safety of any nuclear power plant, one should use complex neutronic thermalhydraulic 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 ‘pseudophase transition’ near the critical point.
By tradition and by analogy with a boiling coolant, in computational studies of the thermalhydraulic stability of systems with supercritical water, frequency and time analysis methods are used with process models of different levels of complexity. Onedimensional nonstationary 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 nonstationary 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 thermalhydraulic stability, built on the basis of modern advances in neutron physics and thermal physics.