Kullanılmış bir nükleer yakıt çubuğundaki sıcaklık dağılımının sonlu eleman yöntemi ile belirlenmesi

Abstract

ÖZET Bir nükleer yakxt çubuğu, reaktörde bulunduğu süre içerisinde, yapısında bir takım değişiklikler meydana gelmekte ve nonhomojen bir yapıya sahip olmaktadır. Bu yapısal değişikliklerin sonucu olarak, yakıttaki sıcaklık dağılımına ait nonlineer denklemlerin analitik çözümleri tam olarak yapılamamaktadır. Bu nedenle, böyle bir problemi çözmek için sayısal yöntemlerden birini kullanmak kaçınılmaz olmaktadır. Bu çalışmada öncelikle, çözüm için önerilebilecek sayısal yöntemler tanıtılmış ve bunların probleme uygunluğu tartışılmıştır. Ayrıca, yakıtın yapısında meydana gelen değişiklikler ve bunların nedenleri, başka bir deyişle, nonlineer lig in ve nonhomojenliğin oluşumunun sebepleri hakkında bilgi verilmiş, bu durumun yakıtın sıcaklık dağılımı üzerindeki sonuçlarından bahsedilmiştir. Reaktörde, genellikle üç farklı yapıya sahip olacak şekilde yeniden yapılanan ve merkezi boşalan, ileri safhalarda içinde yeryer çatlakların oluştuğu bir yakıt çubuğundaki nonlineer sıcaklık dağılımının belirlenmesi problemini esas alarak bir model problem seçilmiştir. Bu model, üç farklı yapıda şekillenmiş, merkezi boşalmış ve içinde halka şeklinde bir kırığın bulunduğu bir yakıt çubuğudur. Problem için, sonlu eleman yöntemi il© kararlı halde ısı iletimi denklemleri türetilmiş ve bu denklemlerden hareketle, böyle bir model problem için sıcaklık dağılımını bulan bir bilgisayar programı geliştirilmiştir.

SUMMARY During the period of time that the fuel element remains in the core, substantial alterations in the morphology of the fuel material take place because of the high temperature levels and steep temperature levels gradients in reactor fuel pins. The ceramic UOz fuel elements are used in LWRs*. These are generally fabricated by sintering* The fuel of any desired density can be produced by controlling the sintering conditions. The conductivity of a solid decreases as the amount of the pores increase within its structure. On th© other hand, these pores are necessary to accomodate the fission gases produced during operation. As the sintering temperature is approached, pores migrate to the center of the fuel, then forms the central void of the fuel. As a result of migration of the pores, a dense region which is characterized by large columnar grains occurs adjacent to the void. The boundary of these grains are formed by the pores and fission gase bubbles which migrate to the center of the fuel. Large equiaxed grains take place as a band outside of columnar grain. Equiaxed grain region, temperature is lower than the sintering temperature then, a large amount of pores still occur there. Moving outward from the equiaxed grain region and adjacent to the clad, fuel has its original structure, since the temperatures are too low to cause any alteration of its morphology. VIIn addition to all of these morphologic alterations, volumetric increase of fission gases increases the internal gas pressure in the fuel pin, and cause the swelling. Du© to the thermal stresses radial cracks or circumf erencial cracks occur in the fuel. Because of the material heterogenity, the equations which give the temperature distribution within the fuel, become nonlineer. Hence, it is difficult to find an analytical solution for this type of problem. Therefore, to solve the problem numerically is unavoidable. In this study, considering a typical fuel element, steady-state temperature distribution is obtained by using the finite element method for a model problem. It is a fuel element in which has a void and two or three zones formed as different structure and a random selected circumf erencial crack. By using the finite element method to formulate a problem, first, domain is divided into elements which are considered to be interconnected at specified joints. These joints are called nodes and can be selected on the boundary of the element. Then, an approximate function is defined instead of the actual field variable, these functions are generally an interpolation polinomial which is constructed by the finite element method. When the equilibrium equations are written for the whole continuum, these are generally expressed in the matrix form, the actual value of the function at the nodes is found by solving this linear system. VIIIn the steady state, the heat conduction equation for a /// cylindrical fuel pin with a volumetric heat source q in which axial heat conduction can be ignored is given by; III -V. k,7 T( r) =qC r) If the cracks in the fuel pin are considered, the axial symmetry will be destroyed, and the temperature expression will become © dependent. Although, the model which is developed in this study doesn't contain an asymmetric temperature distribution, the formulations are performed in two dimensions, to determine the temperature distribution in a fuel element which has radial cracks. Then the governing equation in the fuel region is taken in the form? d dx kC x, y) a rr x,y) <?x dy k( x, y) d TC x, y) dy in + q( x,y) =0 Since there is no heat generation in crack, gas gap and clad, the governing equation will be d d-x. kC x, y) d T( x,y) #x ay k( x, y) a T( x, y> To get the finite element formulation of the problem, first the domain which consists of fuel, crack, gas gap and clad is divided into elements. Then, a finite element solution can be defined in each element as; V11I<x,y;a)=) a 0.(x,y) L. J J j = l Applying the approximate solution to the equations, n number of residual equation are found, then, these residual equations are weighted over element by multiplying the weighting functions. These functions are equal to the finite element shape functions for Galerkin method which belongs to the method of Weighted Residuals. In general, to construct the element shape functions, a parametric mapping is carried out, first the shape functions are constructed on a parent elenssmt which has a simple shape, then mapped on to the real element. In this study, since domain is divided in to triangle elements, C quadratic isoparametric triangels and C lineer triangels can be chosen as parent element to construct the shape functions. C means that functions are continuous functions across interelement boundaries. It is considered to chose the shape functions to satisfy the completeness and continuity conditions of T element trial solution. The equations which depend on global coordinates (x,y) are transformed to the local coordinates CÇ,?}} of parent element, by the described procedure above. Then, the transformed integrals are calculated by using an appropriate quadrature formulas. The quadrature formulas require evaluating the integrands at various quadrature points CÇ,17 }. IXThe boundary conditions which are applied to the model problem are chosen as follows since the fuel is cooled outside of the clad, the boundary condition can be chosen for the boundary of the element which occupies outer surface of the clad < © > a T < « > - k a n - = h ( T - TJ < e > 0° The second boundary condition come from the inner surface of the fuel which har. a central void. Along the boundary of this void the heat flux is zero, namely. c © > < e > _ _ k & T = o d n The third one is, inter element boundary condition which applies to the boundaries of internal elements. Assuming continuous heat flux is along the internal boundary of elements. It can be written asj < © > ` < f > Je> d T <f><? T k _ k _ < a >~ _ < f > an an after calculate the numerical integrations for every element they are connected with each other by using the above boundary condition and the stiffness matrix which represents the whole system is constructed. The most of the elements of this matrix are zero, thus only nonzero elements are stored. By solving these system equations, the unknown coefficients which correspond to the values of the temperature for the selected nodes are found. x