Nükleer reaktörde kullanılan transformatörlerin sıcaklık dağılımının incelenmesi

Abstract

ÖZET Nükleer santrallerin elektrik üretiminin çeşitli aşamalarında elektrik akım ve gerilim dönüştürücüsü olarak transformatörler kullanılmaktadır. Kullanılma süresince hangi kısımlarının ne kadar ısındığı, bu ısınmanın transformatöre ve transformatörün izolasyonuna zarar verip vermediği bilinmesi gereken önemli noktalardır. Isınmanın hangi kısımlarda daha fazla olduğunu ve değerlerinin bilinmesi için, sıcaklık dağılımının bilinmesine ihtiyaç vardır. Transformatörün yapısından dolayı ısı iletimi denklemlerinin tamamım analitik olarak çözmek mümkün değildir. Bu nedenle böyle bir problemi çözmek için sayısal yöntemlerden birini kullanmak kaçınılmaz olur. Bu çalışmada öncelikle, çözüm için önerilebilecek sayısal yöntemler tanıtılmış, bunların probleme uygunluğu ve problem içinde denklemlerin nasıl çözüldüğü anlatılmıştır. Ayrıca tranformatörlerin ısınmasının demir gövdeye ve demir gövdenin işlevini tam olarak yapmasına etkisi, transformatör sargılarına sıcaklığın etkisi, sıcaklığının verim üzerindeki etkisi, sıcaklık dağılımının bulunmasının transformatör ve bunu kullananlar için önemi anlatılmıştır. Transformatörün karmaşık yapısından dolayı daha hassas ve daha detaylı sonuçlar verdiği için, sıcaklık dağılımı sonlu elemanlar yöntemi kullanılarak yapılmıştır. Tek fazlı, çekirdek tipli, kararlı halde, nominal şartlarda çalışan, 400 VA gücünde bir transformatör bu problem için model olarak seçilmiştir. Sonraki adımlarda; saç ve yalıtkan sayılan bulunmuş, demir nüvede üretilen ısının hesaplanması için boşta çalışma halinde gücü, sargılardaki ısı akısının tesbit edilmesi için, kısa devre halinde gücü hesaplanmıştır. Daha sonra bu model üzerinde kararlı halde ısı iletimi denklemleri sonlu eleman yöntemi ile lineer bir sisteme dönüştürülmüş ve bu denklemlerden hareketle böyle bir model problem için sıcaklık dağılımım bulan bir bilgisayar programı gehştirilmiştir.

SUMMARY The transformer is used widely at every step of production of electrical energy in Nuclear Power Plants. The transformer is a common and indispensable component of alternating current sistems where it is used to transform voltages, currents and impedances to appropriate levels for optimal use. A transformer consists of two or more windings interlinked by mutual flux. If one of these windings, the primary, is connected to an alternating voltage source, an alternating flux will be produced, whose amplitute will depend on the primary voltage and number of turns. The mutual flux will link the other winding, the secondary, and will induce a voltage whose value will depend on the number of secondary turns. The winding's flux is linked to each other by high permeability material such as, iron, stell, silicon stell etc. In this study the silicon stell is used to link to windings. This kind of transformer is called iron - core transformer. Silicon stell has the desirable properties of low cost, low core loss and high permeability at high flux densities. During the time, current exists in primary winding, thermal properties occur in the windings and the core. These thermal properties are very important for transformers. The heat of transformer at every point should be known to operate it safely and long time. High temperature makes deterioration on the transformer not only depends upon the temperature levels but olsa depends on the time. The temperatures affects firstly, the insulations which are on the windings and between the silicon steels. When the transformer is operated at higher than normal temperature its insulation is embirittled quickly and it will fail. However, since deterioration is a function of time, it is impracticable to VIfix the exact allowable temperatures which transformer should not be allowed to operate. Therefore, the insulation can be safely subjected to relatively high temperatures, provided that their duration is sufficiently short. In this study, considering steady-state operation of the core type transformer temperature distrubition is obtained by using finite element method for a model transformer. It has two zones which are formed as different structure, such as silicon steel and insulators. The insulator's thicknes is too thin compared to silicon steel. It is difficult to find an analitical solution because of this structure described above. The problem is solved by numerically. By using the finite element method to formulate a problem, first, the 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 on interpolation polynominals which is construced by the finite element method. When the equilibrium equations are written for the whole continuum, these are generally expresed in the matrix form, the actual value of the function at the nodes is found by solving this linear system. In the steady state, the heat conduction equation for x, y, z coordinates with heat source qw and conductivity coefficients k is given by [ Zt/ f&rr/ C&'rS a2T Sx + d2T + /dy J Kdz J d2T q m From the line integral of the electric field, by the induced electromotor force, a current is set up around the core. It results in an energy loss in material, it is called core loss. In units of W/ mm3 this is denoted by q*. In this study temperature distrubution of formulation are performed into two vudimensions, to determine the temperature distribution in a transformer. Thus the governing equations is taken in the form d_ dx w ^(x'y) d_ 6x J 9y flT(x,y) + q*(x,y) = 0 To get the finite element formulation of the problem, first the domain which consists of silicon steel and the insulators is divided into elements. Then, a finite element solution can be defined in each element as T(e)(x,y;a) = Sai()i(e)(x,y) where n is the number of DOF (degree of freedom) in the element and ()i{e) (x,y) are the shape functions which are functions of both x and y for two dimentional problems. Applying the (Galerkin) approximate solution method to the equations, n number of residual equations are found, then these residual equations are weighting functions. These functions are equal to the finite element shape functions for Galerkin method which is one of the method of weighted residuals. In general to construct the stifhess matrix and the load vector, the domain is divided into continuous triangle elements. In this study, the transformer is taken into five parts according to coordinates. First and second parts take x and y directions with different sizes and the same manner, third and fourth part takes whole transformer without two zones. Boundary conditions are applied diferently to the parts. If the legs are considered then a subjected thermal flux is used as a boundry conditions else crowfoots (yokes) are considered, then convective boundary condition is taken into account. vinIn this study domain is divided into C° triangle elements. C° means that functions are continuous functions accross interelement baundaries. It is considered to chose the shape functions to satisfy the completeness and continuity conditions of T(e) element solutions. Substituting the shape functions ()j(e) into the element equations and transform the integrals (stifhess and load integrals) into a form appropriate for numerical evaluation, the problem is analyzed. During the time stifhess and load integrals are used, the boundary conditions are needed. Since the transformer is cooled by air, boundary conditions which are applied to the model problem is given by ST(e) -k(e)-^- = h(T(e)-Too) The second boundary condition comes from windings as a thermal flux. Windings are around the legs of transformers only the this sides of transformer has thermal flux boundary condition. -k( 0ar^_ fîn(e) = q It The third one is, interelement 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 as ^rr(e) arr(f) L-(<0_ 1r(f>_ an(B) 5nm IXAfter evaluating the integrals for every element which are connected to each other by the above boundary condition, the stiffness matrix which represents the whole system is constructed. The most of the elements of this matrix is zero, thus only nonzero elements are stored of this sparse matrix. We then obtain a linear system which can be solved numerically. By solving this system equations, the unknown coefficients which correspond to the values of temperature for the prescribed nodes are found.