MATHEMATICAL AND COMPUTER MODELING OF CONVECTIVE HEAT TRANSFER IN FUEL CARTRIDGES OF FUEL ELEMENTS WITH DIFFERENT SHAPES AND PACKING OF RODS

Ukraine The paper consists of three sections and is of an informational and generalizing nature, indicating promising areas for further research. The first section "R-functions method in mathematical modeling of convective heat transfer in fuel cartridges with fuel elements" is devoted to the use of new constructive tools of the R-functions method for mathematical and computer modeling of fuel elements packings with different types of symmetry, as well as the study of convective heat transfer in fuel elements grids and the effect of the type of packing on the distribution of velocity and temperature. An octahedral cartridge with 37 fuel elements packed according to three patterns (cyclic, checkerboard and in-line) is considered. It is noted that when constructing the equations of a cartridge with bundles of fuel elements using the new method, the number of R-operations and, accordingly, the calculation time are significantly reduced. An analysis of the obtained results allows to conclude that the maximum temperature is obtained with cyclic packing. The scheme of the reactor, the cartridges of which are hexagonal casings, with 91 fuel elements placed in each of them both with checkerboard and cyclic packing, is also considered. In the second section "Ther-mal-hydraulic calculation of fuel elements cartridges in case of violation of the rods packing symmetry", a hexagonal fuel cartridge with 169 fuel elements and checkerboard packing is considered. An increase in temperature is analyzed in case of violation of the packing symmetry while maintaining the parallelism of the rods, as well as in case of a curvature of one of them. The third section "R-functions, fuel element with polyzonal finning of the shell and heat transfer during fluid motion" is focused on the construction of equations for various finning surfaces of fuel elements and the study of hydrodynamic and temperature fields in case of polyzonal finning of the shell. At the same time, using the apparatus of tensor analysis, a transition to a curvilinear non-orthogonal (helical) coordinate system was made. It is noted that mathematical modeling and the associated computer experiment are indispensable in cases where a full-scale experiment is impossible or difficult to conduct for one reason or another. In addition, working with mathematical model of the process and the computational experiment make it possible to investigate the properties and behavior of the process in various situations relatively quickly and without significant expenses. The reliability of the methods, results and conclusions is confirmed by comparison with the information given in the references, the results of the analysis of the numerical convergence of solutions and the calculation of the


Introduction
Currently, the world pays great attention to green energy, which is explained by the desire to replace dirty coal generation with renewable energy sources. It should be noted that many people consider nuclear energy to be the second component of replacement. With this in mind, Ukraine has set a goal for the next seven years -to increase nuclear capacity by three power units.
Due to the growth in the number of nuclear power plants, as well as in the number of models and modifications of nuclear reactors, the advantages of certain plants become of particular importance. At the same time, designers face a number of problems, the optimal solutions of which have not yet been found. There is the largest turnover of financial resources at nuclear power plants, and the slightest gain in effi- ciency brings huge profits, but one must not forget about the reliability and expenses during the construction of the plant. Naturally, this task, which is complex, is solved at the design stage. Calculation of the reactor at the design stage involves the determination of the main parameters of the core, temperature values, etc. In addition, one of the cornerstones in justifying the safe operation of a nuclear power plant is the thermalhydraulic calculation of the reactor core. Calculation of coolant parameters and temperatures of fuel elements is carried out at all stages of designing and justifying the safety of nuclear power plants.
As is known, the processes of hydrodynamics and heat transfer in reactor fuel assemblies used in the power industry are complex, but their exact description is necessary to achieve optimal design characteristics under nominal operation modes and to analyze reliability in case of deviations from nominal modes and in emergency situations. Thus, an accurate calculation of the temperature regimes of fast reactor fuel elements makes it possible to refine the calculated estimates of the dimensions, swelling and bending of elements, etc., which determines the efficiency of the reactor core and the level of permissible fuel burnup. Increase of requirements for the thermal-hydraulic calculation of fuel elements cartridges, in turn, necessitated the development of new methods for the theoretical study of processes in rod bundles. The currently developed methods and programs [1][2][3] differ: in the initial equations that have to be solved and the methods for their solution; in taking into account various factors; in relations for the initial constants and, accordingly, in the accuracy of the calculation and the classes of problems that have to be solved. However, a common feature for all methods is a closed relation between the velocity and temperature fields. It is impossible to accurately calculate heat transfer without knowing the velocity field. For example, in [3], the conjugate problem of heat transfer in case of a rod flow of a coolant in fuel elements grids was reduced to a solution for one translational element. At the same time, the author considered the symmetry of the system. However, the analysis of the nature of the velocity and temperature distribution carried out in [4][5][6] allows to conclude that consideration of the velocity field for a cell, in case of its remoteness from the boundary, is appropriate. The temperature field in this case will be very far from reality, as evidenced by the results obtained for the cartridge as a whole. The papers [7][8][9] laid the foundations for new constructive tools of the R-functions theory for the analytical description of geometric objects with different types of symmetry.
The aim of the paper is: -the use of new constructive tools of the R-function method for mathematical and computer modeling of fuel elements packings with checkerboard, in-line and cyclic symmetry, as well as the study of convective heat transfer in fuel elements grids and the effect of the packing type on the velocity and temperature distribution; -the use of methods and the construction of equations for various surfaces of the fuel elements finning, as well as the study of hydrodynamic and temperature fields in case of polyzonal finning of the shell; -study of the possibilities of joint application of the structural R-functions theory and variational methods for solving conjugate problems of convective heat transfer.
The paper is of an informational and generalizing nature; it indicates promising tasks for further research. At the same time, using the apparatus of tensor analysis, a transition to a curvilinear non-orthogonal (helical) coordinate system was made.

R-functions method in mathematical modeling of convective heat transfer in fuel cartridges with fuel elements
The reactor core, which is assembled from a large number of fuel cartridges, is considered [4,5,10]. An equation for a model octahedral fuel cartridge with 37 fuel elements packed according to three wellknown patterns (cyclic, checkerboard, and in-line) is constructed. In this case, the octahedral cartridge was chosen due to convenience in the position of all types of fuel elements packings with the same number of the said packings, which is a necessary condition for assessing the effect of the type of packing on the velocity and temperature distribution.
To construct the equation for an octahedral casing, the method developed in [4,5,7,8] Let's dwell on the construction of the function   0 , , when fuel element is translated with cyclic symmetry n 1 times along a circle of radius R 1 , n 2 times along a circle of radius R 2 and n 3 times along a circle of radius R 3 .
It should be noted that here and below we used the simplest R-operations ; ; With a central fuel element, we obtain , and the total number of fuel elements becomes equal to 37 ( Fig. 1, а).
To construct a triangular (checkerboard) packing of fuel elements, we set 0 , and 0 2 1 Then the equation of the fuel cartridge has the form . The construction of the function   y x,  is performed with the following values of the literal parameters: R tv =0.2, h x =2.75, h y =1.52 ( Fig. 1, b). It should be noted that, during the construction of the cartridge equation with the new method, the R-operations were used only twice.
To construct a rectangular (in-line) packing of fuel elements, we set 0 . In this case, the fuel cartridge is performed with the following values of the literal parameters: is a substantial (or total) derivative; µΦ is a dissipative function; p c a    is a thermal diffusivity; c p is a heat capacity of the medium; q V is a power of internal heat sources.
In case of stationary processes, the body temperature does not depend on time, and the thermal conduction equation for longitudinal flow around fuel elements takes the form , and the mathematical model of the velocity field in laminar flow has the form C l pressure drop along the pipe in an arbitrarily chosen section of length l.
In the region of thermal stabilization, when const . Thus, the mathematical model of heat transfer in case of laminar motion of fluid along the cartridge with fuel elements is reduced to the system of equations Knowing the equations of the cartridge and fuel elements, we can rewrite the statement of problem (2) in the form The R-functions method was used in combination with the Ritz variational method to obtain the solution. The structure of the solution of the problem of laminar flow in case of longitudinal flow around fluid fuel elements has the form is a boundary equation of the cartridge cross section, and an indefinite component will be calculated by minimizing the func- It should be noted that the solution V z is obtained in an analytical form and used without any further processing (approximation, interpolation). Therefore, the resulting velocity distribution is substituted into the right side of the thermal conduction equation. The structure of the solution to the problem of determining the temperature field was used as exactly satisfying the boundary conditions on , as well as in the form T=p 2 , where, as before, . As approxima- , Schoenberg's cubic splines were used at N=6400, 10000. Computational experiments were carried out under the operating conditions of the POLYE system, developed in the Department of Applied Mathematics and Computational Methods of IPMach NAS of Ukraine. The results of studies for various packings of fuel elements are shown below (Fig. 2-3). Each packing contains 37 rods, with all other conditions being equal λ 1 =1, λ 1 =10, h=1, q V =10. By changing the values of the literal parameters, it is possible to obtain different distributions of the studied fields. 1 -cyclic, 2 -checkerboard, 3 -in-line packings By the analysis of the obtained results, one can conclude that with cyclic packing, the maximum temperature is 9.8% higher than with checkerboard packing, and 18.6% higher than with in-line packing.
In real structures, a fastening in the form of a rod is located in place of the central fuel element. Then In this case, with cyclic packing, the maximum temperature is 9% higher than with checkerboard packing, and 20% higher than with inline packing. If we compare the results of the solution with central fuel element and central rod fastening, then during cyclic packing, the maximum temperature drops by 27%, during checkerboard packing it drops by 27%, and with inline packing -by 29%. In a similar setting and using the same methodology, we will consider a typical structural diagram of a reactor whose cartridges are hexagonal casings, with 91 fuel elements are placed in each of them, both with an expanded triangular packing, which is sometimes called a checkerboard one, and during translation with cyclic symmetry. Fig. 6, a shows the pattern of the level lines of the function ω(x, y) with 91 fuel elements arranged in a checkerboard pattern. Fig. 6, b shows the pattern of the level lines of the function ω(x, y) with 91 fuel elements during translation with cyclic symmetry. Analyzing the results, one can conclude that the presence of rods in the central zone leads to a higher temperature. Therefore, by changing the structural nature of the packing and the types of symmetry, it is possible to control the nature of the flow and the temperature distribution over the cartridge, achieving the required value, conditioned by the technical task. Thermal-hydraulic calculation of fuel elements cartridges in case of violation of the rods packing symmetry A typical structural diagram of the reactor, the active zone of which is assembled from a large number of fuel cartridges, is considered [2,4]. Cartridges are hexagonal casings in which fuel elements are placed. To construct the equation for a hexagonal fuel cartridge with 169 fuel elements and an expanded triangular (checkerboard) packing, we will use the methodology from the previous paragraph.
As a result, we get 0 The construction of displaced fuel elements was carried out as follows [11]. For displacement in the central zone For displacement in the far zone The statement of the problem and the solution method are similar to the previous ones. The results of studies for symmetrical packing, packing with symmetry violation of a straight and curved fuel element in the central zone and on the periphery are shown in Fig. 9. Each packing contains 169 rods. From the analysis of the obtained results, it follows that in case of the packing symmetry violation, while maintaining the parallelism of the rods, the local temperature increases by 2%. In case of rod bending, the local temperature increases by 7%.
In this case, only one rod, which violates the packing symmetry, is considered. With several "nonstandard" rods, the calculation of the temperature field for the cartridge as a whole is all the more important. Mathematical modeling and the associated computer experiment are indispensable in cases where a full-scale experiment is impossible or difficult to conduct for one reason or another. In addition, working with mathematical model of the process and the computer experiment make it possible to investigate the properties and behavior of the process in various situations freely, relatively quickly and without significant expenses. At the same time, computational experiments with object models make it possible to study them in detail and in depth, based on modern numerical methods. The reliability of the analytical identification of the cartridges was confirmed by their visualization. The reliability of the calculation methods, results and conclusions is confirmed by the analysis of the numerical convergence of solutions and the calculation of the residual.

R-functions, fuel element with polyzonal finning of the shell and heat transfer during fluid motion
Finning of heat transfer surfaces is widely used to increase the effective heat transfer coefficient in engineering. The shape of the edge is very diverse. Longitudinal, transverse, spiral edges, edges in the form of spikes, etc. are used. Detailed information about various methods of finning and technology for manufacturing finned surfaces can be found in [10,12]. The finning not only increases the heat exchange surface, but also has a great influence on the hydrodynamics of the flow, and thus on the heat transfer coefficient. In the course of experiments with various methods of fuel elements shells finning, more advantageous forms of finning, the so-called polyzonal and chevron ones, were developed. With the latter, for example, the entire surface of the shell is divided into four, six, or eight sectors, and on neighboring sectors, the spirals are arranged symmetrically about the longitudinal axis. a b c Fig. 9. Pictures of the velocity field (a) and temperature field (b) distribution, including in case of violation of the rods packing symmetry, and a comparative analysis of graphs in the section 0≤x≤R v , y=0 (c) The problem of heat transfer in the laminar fluid flow for the fuel element with polyzonal shell finning is considered. Let's build a mathematical model of such a fuel element shell (Fig. 10)   The invariant form of writing equations (1) allows, using the apparatus of tensor analysis, to move to a new coordinate system, in our case, to a curvilinear non-orthogonal system In the helical coordinate system, we obtain (1) in the form [8]   , and for the temperature field on the wall it can be specified in different ways: In the curvilinear non-orthogonal coordinate system, the normal derivatives included in the boundary conditions, taking into account the normalization of the function   x and the equations obtained in [8] take the following form Thus, from three-dimensional boundary problems we come to two-dimensional problems, for which the Ritz method in combination with the R-functions method can be applied [7,8]. The positive definiteness of the operators of problems (5) was proved in [8].
We minimize the functionals equivalent to the boundary problems (4) and (5), , on the inside wall - The structure of the solution to problem (4) has the form , and to problem (5) it has the form ; . For different values of the swirl parameter  , we obtain the results shown in Fig. 11.
The results shown on Fig. 11 indicate that with an increase in the value of the swirl parameter α, heating in the intercostal zone increases.
The conducted studies make it possible in the future to effectively continue work related to mathematical and computer modeling of convective heat transfer in fuel cartridges of fuel elements with various shapes and packing of rods, as, for example, in the following problems. At present, conceptual designs for a supercritical water reactor (SCWR) have been implemented in various countries. In this case, both square and hexagonal grids of fuel elements with cavities filled with a moderator, for example, water or a solid moderator, are used. With the help of R-functions, it has already been possible to develop a methodology and construct an equation for a bundle of fuel elements with hexagonal grid (Fig. 12) [15].
In addition, it is noted that finning of heat transfer surfaces is widely used in engineering. It was possible to build a number of mathematical models of multi-zone fuel elements with longitudinal finning and helical plates (Fig. 13) [15].
The obtained mathematical models can be used by designers in research work during computational experiments.

Conclusions
It has been established that the R-functions method is an effective method for solving problems of calculating physical fields in structural elements of nuclear power plants of complex shape. The developed constructive tools for constructing equations of the boundaries of regions with translational and cyclic types of symmetry made it possible to significantly reduce the number of operations with subsequent automation of this process, and, consequently, reduce the time for solving problems. The conducted studies allow designers to choose certain types of packing depending on technical requirements. The paper considers only one rod that breaks the symmetry of the packing. With several "non-standard" rods, the calculation of the temperature field for the cartridge as a whole is all the more important. The application of the R-functions theory to mathematical and computer modeling of heat transfer during fluid flow for the fuel element with polyzonal finning of the shell and the analytical recording of the designed objects makes it possible to use alphabetic geometric parameters, complex superpositions of functions, which, in turn, allows to quickly change their structural elements. Reducing the pitch of the helical spiral leads to an increase in resistance, outstripping the increase in heat transfer. It is noted that an increase in the thickness of the edges increases their efficiency. In the future, it is of interest to study the influence of the edge thickness and channel length on heat transfer and hydrodynamic resistance. Mathematical modeling and the computer experiment associated with it are irreplaceable in cases where a full-scale experiment is impossible or difficult to conduct for one reason or another. In addition, working with the mathematical model of the process and the computational experiment make it possible to investigate the properties and behavior of the process in various situations freely, relatively quickly and without significant expenses. At the same time, computational experiments with object models make it possible to study them in detail and in depth, based on modern numerical methods. The reliability of the analytical identification of cartridges is confirmed by their visualization, and the calculation methods, results and conclusions are confirmed by the analysis of the numerical convergence of solutions and the calculation of the residual.