Bwr İçindeki Bir Kanalda Ortaya Çıkan Kararsızlıkların Nükleer-termal Hidrolik Modellenmesi

Abstract

BWR yakıt kanallarında iki fazlı akışkanların kararlılık problemleri, önceleri termalhidrolik yönleriyle incelenmekteydi. BWR'larda da iki fazlı akışların olması ve kararsızlıkların meydana gelmesinden ötürü nükleer-termal etkileşimlerin incelenmesi gereği doğmuştur. Günümüzdeki çalışmalarda nötronik-termalhidrolik kararsızlıkların işleyişini kolayca anlamak için nötron kinetiğine nokta-reaktör modelini uygulamaktayız. Isıtılan bir kanalda iki fazlı akış kararsızlıkları genel olarak ikiye ayrılmaktadır. Bunlar statik ve dinamik kararsızlıklardır. Ledinegg kararsızlığı bir statik kararsızlıktır. Yoğunluk dalgası ve basınç düşümü kararsızlıkları da dinamik kararsızlığa güzel örnek teşkil ederler. BWR'larda kararsızlıkları şu şekilde sınıflayabiliriz: 1) Kontrol sistemi kararsızlıkları 2) Kanal termalhidrolik kararsızlıkları 3) Nötronik-termalhidrolik kararsızlıklar: a) Tüm koru etkileyen reaktivite kararsızlığı b) Zıt fazlı reaktivite kararsızlığı. Kanal kararsızlıklarının analizi için homojen-termodinamik denge modeline dayanan bir boyutlu iki fazlı akışı tasvir eden kütle, enerji ve momentum denklemleri kullanılmıştır. Üç temel denklem yardımı ile, ısıtılmış kanalın pertürbasyonlara karşı kararsızlığı araştırılmıştır. Öncelikle sürekli rejimde temel denklemlerin çözümleri yapılarak sistemin çalışma parametreleri saptanmıştır. Isıtılan kanal iki bölgeye ayrılmıştır. Birinci bölgede, soğutucu akışkanın giriş kısmından belli bir noktaya kadar kaynama olmamaktadır. Kaynamanın başladığı yere kaynama sınırı denmektedir. Sistemdeki pertürbasyonlar ile bu kaynama sınırı değişebilmektedir. Kaynamanın başladığı yerden itibaren üst kısım ikinci bölge olarak bilinir ve akışkan burada iki fazlıdır, yani buhar ve su fazındadır. Birinci bölgede akışkan bir fazlıdır ve kaynama sınırında doymuş halde olduğu varsayılır. Her iki bölgeye ait denklemler lineerleştirilerek Laplace transformlan elde edilmiştir. Basınç düşümü denklemleri önce integre edilip sonra pertürbe edilmişlerdir. Bu işlem sıralarınm değiştirilmesi durumunda iki sonuç arasında kaynama noktasından ötürü bir fark oluşmaktadır. Bu fark teriminin sistem kararlılığında pek etkili olmamaktadır. Yapılan simülasyonda bu durum gösterilmiştir. Nötron ve yakıt dinamiğinin, hacimsel boşluk oram ile ilişkisi ortaya konurken, en basit ısı transfer modeli kullanılmıştır. Sistemin karakteristik denkleminin elde edilmesi için her bölgenin basınç düşümü pertürbasyonları terimleri toplanmıştır. Kanalın giriş ve çıkışındaki basınç düşümünün sabit olması sınır koşulundan ötürü yukarıdaki toplamın sıfıra eşitliğini sağlayan frekansta sistem kararsızlığın sınırındadır. Böylece istenilen sistem transfer fonksiyonu elde edilmiştir. Bu denklemden hareketle sistemin blok diyagramı oluşturulmuştur. Sitemin kararlılık haritası Ja sayısı- Zu sayısı düzleminde elde edilmiştir. Sonuçlar daha önceleri yapılan çalışmalar ile uyum içindedir.

Stability and control problems of two-phase heat transfer system are quite involved and nuclear coupling adds a new dimension to the problem. Anybody who tried to bring a pan of milk placed on top of a stove, to a steady bubbling state, would surely experience some stability related difficulties in keeping the milk from boiling over by simply adjusting the heat. This is a natural circulation pool boiling system. A practical solution would be to stir it to improve recirculation, and heat transfer which in turn lowers voidage and hence stabilizes the system. Boiling water reactor operators have been observing and sometimes passing without reporting, some unusual power oscillations in their reactors. One of the earliest of such events is Coarso Power Plant event which was fully documented. There are various underlying mechanisms of these instability events. Sometimes even an incipient instability event may go undetected for a while until a reactor trip is announced. Such instability occurrences can be classified on the International Nuclear Event Scale (INES) of 0 to 7, between Level- 1 "Anomaly-Incident beyond the authorized operating regime with defense in depth degredational implications", and Level-0 "below scale event or deviation with no safety significance." Neutronic coupled thermal hydraulic instabilities in a boiling water reactor, are classified into three main groups: 1. Single channel instability with constant pressure drop boundary condition; 2. Inphase oscillations in which the whole core behaves as a single channel, the total loop imposing the pressure boundary condition; 3. Out of phase oscillations in which the void perturbations excite antisymmetrical an subcritical second harmonic modes of the flux accompanied by constant pressure drop boundary conditions. In out of phase oscillations the power may swing across the core in a sew-saw manner, while its average or total remaining at the set point. The stability of a given system can be studied either in the time domain in which the model equations are already built, or in the frequency domain after having those model equations linearized. In time domain approaches, numerical instabilities are always suspect. On the other hand, linearized models are good at determining the stability boundaries of the system with respect to its parameters, but they are not reliable when it comes to predicting system behavior exactly once the instabilities are excited. Another characteristics of the frequency domain approach is that, the linearization procedure consists of a series of operations, which do not commute. The main thrust of this work is to understand the physical phenomenon leading to single channel based instabilities in a boiling water reactor; and to compare the two different possible linearization procedures. In order to study the dynamic behavior of a single boiling channel, the simplest but nontrivial model is obtained by coupling the point reactor kinetic equations with the two-phase homogeneous equilibrium flow model equations. One dimensional Homogeneous equilibrium model equations (HEM) consist of a mass conservation equation, and one equation for balance of momentum and energy each. The basic model incorporates the following assumptions: 1. The pressure drop along the heated channel is very small compared to the operating pressure of the system. Therefore the saturation properties of water, namely, density and enthalpy, are taken as constant along the channel. This assumption is crucial in decoupling the momentum equation from the rest of the homogeneous equilibrium model equations. That means, mass and energy equations can be integrated along the channel to get quality or void fraction, total volumetric flux enthalpy and density, and after having these parameters at hand, the momentum equation can be integrated separately. This procedure can be amended to accommodate the local pressure dependent saturation properties of water, and drift flux model as a first order correction to homogeneous equilibrium model equations. 2. As the name implies, the two-phase mixture is in thermodynamic equilibrium, and both phases flow at the same speed. 3. Heat flux is uniform along the heated channel. 4. Subcooled boiling is ignored. 5. Effects of the compressibility, such as acoustic wave propagation is neglected. 6. Instability developing in a single channel is too weak to perturb the flow in the rest of the core. Therefore the pressure difference between the upper and lower plena can easily be assumed to remain constant, and from this assumption a stability criterion is developed; that is when the system parameters are right, the boundary condition is satisfied at a particular frequency, and then self sustained flow oscillations are likely to occur at this frequency. 7. Inlet enthalpy perturbation is ignored. 8. The model applies only under the steady state operating conditions. Unusual operating conditions bring about physical situations which are beyond the scope of this work. Prior to any linearization attempt, the system equations are solved to determine the steady state (setpoint) conditions in the channel. Integration of the energy equation in the single phase region yields the boiling boundary. The whole system is perturbed through the inlet flow fluctuations. Initially, the perturbation in the inlet flow is taken as an independent variable. Since energy and momentum equations are uncoupled, boiling boundary, void fraction, density, quality, or enthalpy perturbations are calculated directly; then single and two-phase momentum equations are perturbed and integrated. Perturbations in average void fraction are fed back into the point reactor kinetic equations through the void reactivity mechanism, which eventually connects the inlet flow perturbations to axial heat flux fluctuations.This relation, in turn can be used to eliminate heat flux perturbations in favor of flow perturbations. The stability criterion is provided by the constant pressure drop boundary condition. The boundary conditions are traced on the complex plane with the frequency being the parameter. If, with a given set of steady state operating parameters, the constant pressure drop boundary condition is satisfied exactly at any frequency, then the system is very likely to undergo self sustained flow oscillations. Because at this point the stability boundary is reached and the channel acts as if it had zero resistance against the flow. Alternatively, this statement is equivalent to the Nyquist criterion. In affecting an instability, all it takes is a special combination of system parameters. The frequency at which this instability induces oscillations, does not matter; because in reality flow fluctuations possess a white spectrum, thus it may excite perturbations at all frequencies. The main result is the stability map of a heated section given on Jacob's number ( a measure of the negative of the inlet subcooled state quality) versus Zuber number (a measure of the total quality gained by the fluid.) axes with Froude number as a parameter. For the lower end of the Zu number axis, when Ja>Zu, the system is unconditionally stable ; because, in this case the boiling boundary is beyond the exit of the heated channel. The system becomes less stable when the power is incresed and the boiling boundary recedes towards the inlet. Here are some of the main points worth observing: 1. Even if around the boiling boundary, the single-phase and two-phase frictional pressure drop coefficients were identical, the integration and perturbation operators acting on momentum equations would not commute. 2. Perturbation and averaging operators acting on the void fraction perturbation term, always commute. 3. System becomes more stable if the inlet is throttled, because the single phase pressure drop is always inphase with the inlet flow. 4. System becomes destabilized if the outlet is throttled, because the two-phase pressure drop, which is out of phase with the inlet flow perturbation, dominates the scene. 5. The system is unconditionally stable against the static instabilities. 6. As a hypothetical case, if the channel is assumed to be fed by a centrifugal pump, and the constant pressure drop boundary condition is applied between the inlet of the pump and the outlet of the heated section, the stability of the system against the flow perturbations is improved. 7. In the block diagram of the system there are three feedback loops, namely, thermal-hydraulic loop, void-reactivity loop, and the loop which appears when the interactions between the two former loops are considered. Of the three, the void reactivity loop is most dominant in determining the system behavior. Increasing void reactivity coefficient tends to destabilize the system, because it acts as the feedback gain. These observations are all in agreement with the results reported in the literature. This work is also of some tutorial value since the model is based on an elementary yet nontrivial analytical model, and calculations are all performed on MATLAB. For future work, the following points are suggested: 1. The results can be analyzed to reflect on the significance of each of the terms contributing to the stability criterion. Then the model equations can be modified by eliminating those insignificant terms. 2. With little rearrangement, the results can be used to study the stability of a boiling water reactor with natural circulation. 3. The homogeneous equilibrium model can be readily improved with the incorporation of drift flux equations. 4. The out of phase stability problem can be studied in association with the multipoint reactor kinetic equations since the same pressure drop boundary conditions apply. 5. With the formulation presented in this work, a plant model of a boiling water reactor can be prepared for control applications. 6. Simple two region lumped time domain models can be used to confirm the validity of the results. Such models mimic the nonlinear behavior of a boiling water reactor more realistically when the perturbations are no longer small. 7. The instability of the neutronics loop with void reactivity feedback, needs to be elaborated further.