The method of express analysis of nuclear and ecological safety during the modernization of nuclear fuel

. The original method for the nuclear safety analysis is given. The method is based on an adequate interpretation of the safety conditions changes based on the temperature of fuel elements and nuclear fuel claddings in the process of the nuclear power plants modernization. The temperature limitations for nuclear fuel and cladding made of structural material are an - alyzed. A method based on the nuclear safety analysis conservative criteria is proposed. The safety conditions according to the fuel element and its cladding temperature are obtained. The proposed method is based on conservative criteria of nuclear safety analysis and does not require modeling of all possible sequenc - es using special codes. Therefore, the computational research amount is significantly reduced. In addition, the method ensures a rapid adaptation of the criteria method for the nuclear safety changes express assessment for various initial events and conditions, as well as modifications and/or changes in the nuclear fuel design. In accordance with the international experience of nuclear energy, the mo - nopolization of the types of operated nuclear power plants and their systems has a negative impact both on ensuring the conditions for nuclear and radiation safety, and on their competitiveness and operational efficiency. Taking into account the successful experience of the Czech Republic and Finland, Ukraine has launched and continues a program to diversify fuel assemblies of Westinghouse nuclear fuel for reactor plants with VVER-type reactors. In accordance with the nuclear legislation of Ukraine, any modernization of reactor facilities, including the diversification of nuclear fuel assemblies, requires an additional analysis of ensuring nuclear and radiation safety conditions. The use of traditional methods of safety analysis by deterministic codes for modeling the entire range of possible emergency situations in such a situation is limited due to the possible negative influence of the effects of differences in codes and users of codes, which indicates the importance and relevance of the work.


Introduction
The main purpose of the analysis of nuclear reactor systems for spent nuclear fuel is to assess the safety criteria feasibility in the accident transient process for various initial events.
The fuel barriers in nuclear power plants are directly the protective fuel matrix and the zirconium shell. According to Pantak, Melnik, 2020) it is possible to list the characteristic damage stages at the reactor core with UO 2 fuel depending on temperature conditions (Table 1).  Safety criteria and their critical interpretation. In the information on the nuclear power plant safety analysis with the accident modeling only one criterion of reaching the rapid oxidation temperature for fuel rod cladding, namely Т 0 С = 1473 о K, is considered without any sufficient justification. It is assumed that a severe accident begins at the temperature of Т 0 С and causes the nuclear fuel damages. The admissibility of such an approach to the nuclear safety assessment needs some further comprehension.
As is known, during the reactor nominal operation, the maximum heat flow is approximately 109 W/m 3 , and the fuel rod surface cooling by the cool-ant flow ensures a heat transfer coefficient of 3.104 W/(m 2 о K) (Solodov, 1986). The analysis of these data shows that the maximum fuel temperature in the central part reaches 2573 о K at coolant operating temperatures and cladding temperature of about 573 о K. The dominant thermal resistance is formed by a nuclear fuel and a gas gap filled with helium. Under the nominal conditions of the reactor operation the significant non-uniformity of the radial distribution of nuclear fuel temperature and helium temperature in the gap (temperature differences are 873 and 1473 о K, respectively) is determined by the relatively low thermal conductivity of helium and fuel being λ UO 2 = 2 and λ He = 0.2 W/(m о K), respectively. The relatively low thermal conductivity of UO 2 means that the thermal fuel resistance is several orders of magnitude higher than the thermal resistance of the heat transfer convective barrier during the normal reactor operation. Hence, the maximum fuel temperature T max f is more sensitive to temperature mode changes than the maximum fuel oil temperature T max c . In emergency conditions, the convective barrier thermal resistance can increase significantly. Therefore, in the general case, the ratio between T max f and T max c depends not only on the heat transfer in the fuel rod, but also on the thermohydraulic processes in the primary circuit. But at the reactor nominal operation even a simple extrapolation of the temperature distribution in the fuel rod to the emergency mode allows to conclude that in the cladding central part the temperature reaches 1473 C. Also, the maximum allowable fuel temperature is affected by characteristics such as fuel burnout and its chemical composition changes. Thus, when uranium oxide burns out and plutonium accumulates, the nuclear fuel melting point can be lowered by 20 -40 K. The measures implementation for improving the thermal and neutron-physical properties of nuclear fuel by changing its chemical composition (as exampled in (Shimkevich, et al., 2011)) can also lead to a change in relations between T max f and T max c even in the stationary conditions of a nuclear reactor operation.
Another limitation for the application of the safety probabilistic assessment results is the accidents mod-eling with the deterministic method use only for the most probable scenarios. The main disadvantage of the probabilistic approach is that such an approach to the accident events ranking in fact excludes from the consideration the events versions that have relatively low probabilities and, consequently, the final status of most accident sequences is determined a priori without any sufficient research. Now, the main methods for the severe accidents full-scale modeling are the calculation methods, which under given initial and boundary conditions ensure the numerical implementation of processes mathematical modelling based on calculation software (codes).
The codes traditionally used for the severe accidents analysis are divided into integrated and detailed ones. The detailed codes are characterized by the relatively narrow application fields (they model individual processes, phases, stages) and relatively high modeling realism probability. The integrated codes have the wider application spheres (they model more than one stage of the severe accident development) and relatively low realism probability of individual processes modeling. However, this division is quite conditional, as the modern models for severe accidents usually contain the features of both detailed and integrated codes. Thus, the integrated code ASTES for severe accidents analysis is in fact composed of detailed codes that model individual processes and stages, and the detailed code SCDAP/RELAP5-3D for the first stage of a major accident also uses a «built-in» code to model the accumulation and decay of radionuclides. To model all the main stages of severe accidents development the computer code ASTEC use is promising (Skalozubov, Kochneva, 2010).
Although, it should be recognized that the accident modeling in full requires a lot of efforts, most of which will be ineffective. Therefore, the express method elaboration for nuclear safety analysis is an urgent problem.

Research methods and results
The main provisions of the nuclear safety analysis method. The proposed method of nuclear safety analysis is based on the following assumptions: • the fuel rod is modeled as a cylindrical system with concentrated parameters; • the fuel temperature T ƒ corresponds to the maximum value of the spatial temperature distribution in the fuel rod; • the external conditions of heat exchange with the fuel rod are usually determined by the specific scenarios of the accident transition process.
Within the assumptions the heat balance equation can be presented as (Skalozubov, 2012): where ρ ƒ and h ƒ are the density and specific enthalpy of the nuclear fuel, respectively; T f , T c and T cool are the temperatures of fuel, cladding and coolant, respectively; α(t) is the heat transfer coefficient on the outer surface; t is the time; r is the fuel segment radius; is the specific heat flow.
The value is determined by non stationary neutron-physical processes in the nuclear fuel (Bartholomew, 1989;Feinberg, 1978;Semenov, 2015): (4) where Ф is the specific neutron flux density; N j is the nuclear concentration for nuclide j; is the neutron distribution cross-section averaged over the energy spectrum for nuclide j; q j is the fission energy for the nuclide j.
The following scales for temperature and time can be introduced, respectively: Then after the transformations the equations (1) -(3) shall be written as follows: where is the Nusselt criterion; Then the analytical solutions (5) -(7) and the corresponding fuel cell safety criteria will be obtained in the following: (9) In generalized form, the nuclear safety criteria of the proposed method are presented as follows: (10) (11) The limit values mean that the conditions that are critical to nuclear safety can be identified using the critical values of heat flow and heat transfer.
Thus, the criterion proposed in the nuclear safety analysis method does not require any detailed modeling of all possible emergency sequences but allows to obtain the conservative estimates for the critical values of heat flow in the nuclear fuel and heat transfer on the cladding surface for specific scenarios.
Applied aspects of the method application. A partial analysis of the method possible application in a version comparable with the results presented in the materials (Semenov at al., 2015;Muhamedzhanova, Akatiev, 2017;Markov, 2014;Gruzincev, Shelegov, 2014) is performed.
One of the factors determining the critical safety parameters is the maximum temperature of the fuel claddings, which is formed under the influence of various engineering uncertainties. These uncertainties have the various causes including the errors occurring in the reactor units manufacture and assembly, used calculation formulas, experimentally obtained dependences and constants, accuracy of operating parameters maintaining during operation and data processing, as well as methodological and metrological errors etc.
To conduct the pre-design calculations, a procedure was developed for the statistical assessment of the active zone parameters random deviations influence on the fuel rods temperature (Kartashov, Bogoslovskaja, 2011). It introduces the overheating factor F as a random variable that characterizes the maximum deviation of a certain parameter P determining the temperature difference from its nominal value. As a rule, the parameters random deviations have significant values and arbitrary distribution laws.
In general, the maximum value of the nominal temperature of the fuel rod cladding inner surface is correctly calculated as the superposition of the cool-ant average temperature in the surrounding channels; average «wall-liquid» temperature pressure (Δt а ) along the cladding perimeter; half of the local temperature non-uniformity for the cladding perimeter; cladding overheating (Δt p ) under the remote wire (if any) and the temperature difference (Δt об ) on the fuel cell cladding: In this case, the temperature deviation due to the overheating factors influences can be determined for correlated values by the following ratio: Here, δt i is the temperature deviation preceding in the calculation chain of the desired one; Δt i is the nominal value of the i-th temperature difference; Δt j is the nominal value of the j-th temperature difference, the deviation of which under the overheating factors influences is associated by a linear relationship with the deviation Δt i ; is the sum of the squares of the products of the relative scattering (K m ), coefficients a m (a m is an indicator of the degree, to which the m-th parameter of the i-th temperature difference is reduced) and overheating factors that determine the temperature difference Δt i (F m ); is the sum of the squares of the products of the relative scattering, coefficients a m and overheating factors that simultaneously influence the temperature differences Δt i and Δt j (j<i).
Based on the method proposed for fast reactors, the overheating factors influence on the deviation of the maximum temperature of the fuel rod cladding in the WWER-SKD reactor was determined considering the values obtained by the express method.
It is known that the reactor reactivity in rapid transient processes depends on the fuel and coolant temperatures. These temperatures, in turn, are determined based on the solution of the heat and mass transfer equations, also recorded for the «point» reactor. The validity of the point kinetics equations application is substantiated in several theoretical works. The legitimacy of the equations use is considered in guidance documents at the level of intuitive approach and the relationship between the fuel and coolant temperatures is determined declaratively, based on the assumption (8) of a linear law of the average temperature distribution across the core. The law on the dependence of the average temperature of the moving heat carrier at the height of the cross section of the fuel channel on the vertical coordinate is determined by the law of the dependence of the volumetric heat release in the heater on the specified coordinate. The reason for this is that all the heat from the rod is removed through the side surface and coolants with the convective heat transfer dominating if compared to the thermal conductivity along the channel. Since the dependence q (z) is symmetric with respect to the core middle, the average coolant temperature is presented as the arithmetic mean of the inlet and outlet temperatures of the coolant. In real reactors, this distribution is asymmetric, which may introduce some error in the results of processing the experimental data on the reactivity coefficients determination. Numerous experiments performed by the paper authors with various offset distributions show that the differences between the obtained values of temperature differences (Semenov, Volman, 2015) compared to the resulting values of average and maximum fuel temperatures are 4%.
The operation safety of nuclear power reactors and other various purposes radiation-hazardous installations is one of the most important problems of modern energy and nonlinear dynamics. The understanding and research of their dynamic characteristics is of great importance in solving this problem. Ultimately, the dynamics study purpose is to confirm the stability of nonlinear system stationary regimes or to find the conditions, in which this stability is ensured. Regarding these conditions, almost the only possibility is the creation of the adequate mathematical models and their subsequent analysis. Here the role of various effects associated with the systems distribution and heterogeneity and ongoing complex processes nonlinearity is growing sharply. At the same time, the very simplified mathematical models used for the analysis of such nonlinear systems take these effects into account very approximately, for instance, within linearized models. First, this is because the complex distributed systems in inhomogeneous nonlinear media are usually described by an equations system in partial derivatives or an integro-differential equations system and must be considered in general Banach spaces. It should be noted that for complex nonlinear systems the linearization procedure itself, as a rule, requires strict justification.
Thus, the development of adequate distributed nonlinear models of complex objects and processes, as well as the studies of emerging modes, stability issues and nonlinear phenomena, including asymptot-ic and irregular (chaotic) behavior, are the very important tasks. In addition, the solution of the issues of existence, unity, spectrum properties, root vectors completeness, positivity of solutions and some others is of a considerable interest. This circumstance is associated with the solution of a new class of problems that describe much more complex processes.
For several nonlinear dynamical systems, a sequence of period doubling bifurcations is observed with parameter changes from values in which it has only fixed points to values in which there are many periodic orbits. These cascades of period doubling bifurcations have a rich structure (Feigenbaum universality). There are properties associated with these cascades that are universal in the sense that they do not depend on the choice of a particular dynamic system. At the same time, the task of establishing the mechanism of transition to «chaos» through the period doubling bifurcation for a specific distributed model transfer problem, which was observed experimentally, is very important. It is important to note that for the nonlinear distributed dynamical systems a space-time chaos is possible that is characterized by the fact that in the process of oscillations not only the temporary process realizations, but also the spatial field distributions (radiation, temperature) become random. This mode of space-time chaos in the theory of nonlinear dynamical systems has a special name of dissipative structures. This mode is characterized by a corresponding attracting set (quasi-attractor). In recent years, some successes are achieved in the studies of such sets structure and properties based on new ideas and concepts (inertial diversity and inertial forms, order parameters, fractal sets etc.), as well as of the main provisions (paradigms) of synergetic. The significant achievements in the development of the appropriate methods and approaches for a wide class of nonlinear systems have been obtained by the research teams of Applied Mathematics, RSC KI, MSU, MEPhI, UNN, SSU etc., as well as by all the rest researchers, the results of studies of nonlinear distributed transfer models and their various approximations are rarely published, and the number of publications on the studies of invariant diversities, attractive sets structures and related issues is even less.
A mathematical model (family of models) is presented below, within which the function of radiation (neutrons) distribution satisfies the gas-kinetic equation of transfer, the nuclear-physical characteristics of which functionally depend on the medium temperature; the medium temperature, in turn, is subject to the equation written in general, so that all the options that occur in practice are its special cases: with the following initial and boundary conditions: Here the coefficients and nuclei are given by the expressions: F u n c t i o n s are characterized by macroscopic neutron cross-sections; p is the type of the radiation (neutrons) interaction with the nuclei of the l-th nuclide, which is a part of the medium.
The system determines the evolution in time of the spatial-energetic (velocity) distribution of radiation density (neutrons) N(x,ʋ,t), the spatial distribution of M groups of delayed radiation (neutrons) C k (x,t), k=1,...,M and the medium temperature T(x,t) in a certain (multiplicative) volume G with a boundary ∂G(≡Г). The temperature T(x,t) is deducted from the medium temperature; according to the physical content of the problem the expression T(x,t)≥0∀х∈ G,t≥0 is obtained. It is further assumed that T(x,t)=0 corresponds to the system state with N(x,ʋ,t) ≡ 0.
It should be noted that the operator equation for the temperature in the system is written in enough general form, which allows to cover almost all cases of heat transfer in the multiplying and moderating media. Generally speaking, the concretization of the linear unlimited operator А^0 corresponds to the concretization of the method and nature of heat transfer in different parts of the volume G; the concretization of the nonlinear operator corresponds to the concretization of the heat release (energy release) nature in the process of distribution of fissile nuclears of materials and (or) in other acts of heat release.
For an example of concretization of a nonlinear operator it is possible to assume that The thermal conductivity operator (diffusion type) can be considered as a simple, but important, example of the linear unlimited operator А^0 concretization: A more complicated example of the operator А^0 concretization is provided a distributed active medium consisting of technological channels, in which heat is released due to the nuclear fuel distribution process. At the same time, the technological channels system is cooled by a liquid coolant, which is pumped through the active medium in the space between the fuel elements. It should be noted that the definition of the linear operator А^0 with the definition domain D (А^0) includes boundary conditions of the latter type; they can be more complicated.
The system is the most common family of models for describing the radiation (neutrons) transfer and temperature distributions in inhomogeneous multiplying media.
It is worth to consider the properties of a nonlinear stationary problem corresponding to the system: The parameter λ in this system corresponds to the «effective multiplication factor k eff », which is used in practical calculations of multiplying systems. It should be noted that this method of the spectral parameter λ introducing is typical, but not the only one. Everything is determined by the convenience of studying the emerging spectral problems.

Conclusions
1. It is shown that the safety criteria use with respect to the cladding temperature range with selective deterministic accident modeling is usually insufficient for an adequate assessment of the nuclear reactors safety. 2. The express method of nuclear safety analysis based on an adequate interdependence of the fuel matrix and cladding safety criteria, as well as the conservative criteria of heat flow in nuclear fuel and heat exchange on the cladding surface is proposed. 3. The proposed express method of nuclear safety analysis does not require detailed modeling of all possible sequences of accidents, which significantly reduces the ineffective studies quantity, expands this method application for express assessment of nuclear reactor safety across the entire operation spectrum. 4. The further research shall focus on the specific application of the proposed method for the analysis of nuclear power plants nuclear safety.