A nonlinear reactor model was developed taking into account several feedback effects such as the moderator and fuel temperatures, xenon absorption, and soluble boron concentration through energy balance relations in the core. The resulting equation belongs to a class of nonlinear boundary value problems, and it is shown through bifurcation theory that there may exist multiple steady-state solutions for a range of parameters which correspond to various design and operating conditions.
We obtained solutions from the nonlinear reactor model for a range of the parameters, i.e, for a set of macroscopic absorption cross sections, moderator and fuel temperature coefficients, and boron concentration. For particular values of feedback constants, the number of solutions was multiple for a range of the parameters. The solution diagram and bifurcation diagram were obtained for several feedback constants. Moreover, stability analysis was applied to each solution in order to investigate whether the solution corresponding to the parameters is stable or not.
The solution diagram consists of one bifurcation point and two limit points, one bifurcation point and one limit point, or only one bifurcation point, depending on the values of the feedback constants. In the solution diagram consisting of one bifurcation point and two limit points, the intermediate solution is unstable and the rest are stable. The solution diagram consisting of one bifurcation point and one limit point has either no solution beyond a critical value of a parameter, or two solutions below a critical value of the parameter of which the higher solution is unstable and the lower solution is stable. The solution diagram consisting of one bifurcation point has only one stable solution beyond a critical value of the parameter, or none otherwise.
Temperature feedbacks with negative moderator temperature coefficient, negative fuel temperature coefficient, and xenon feedback function in the direction of stabilizing the reactor. However, boron feedback functions in the direction of instability. Even though the moderator and fuel temperature coefficients are negative, the starting point of instability (first limit point) in the solution diagram decreases as the fuel temperature coefficient increases (approaches zero from negative values) or the boron concentration increases.
When stable and unstable regions of the steady-state solutions are plotted for a wide range of the parameters, we can determine the range of the moderator and fuel temperature coefficients and boron concentration for stable operation of the reactor. In other words, reactor design and operating conditions can be chosen such that the reactor does not encounter unstable situations at each operating power level.
감속제 온도와 핵연료 온도와 xenon과 boron에 의한 피이드백(feedback)을 노심의 에너지 보존 방정식과 결합하여 비 선형원자로 모델(model)을 유도하였다. 이렇게 얻은 식은 비 선형 경계치 문제(nonlinear boundary value problem)의 한 종류에 속하며 원자로 설계나 가동 조건에 따라 결정되는 계수들(parameters)의 어떤 범위내에서 여러 개의 해를 가질 수 있다는 것을 배가 이론(bifurcation theory)을 통해 보였다.
피이드백 상수들의 몇 몇 값에 대해서 비 선형 원자로 피이드백식(nonlinear reactor feedback equation)의 해 도형(solution diagram)과 배가 도형(bifurcation diagram)을 얻었으며 각 각의 해에 대해 안정성(stability) 분석을 하였다.
이와 같이 안정, 불안정 해는 원자로 계수들, 특히 감속제 온도계수, 핵 연료 온도 계수, boron 농도 등에 의해서 결정된다. 예를들면 음 값의 감속제 온도 계수, 음 값의 핵 연료 온도 계수, xenon 피이드백은 원자로가 안정한 해를 갖는 방향으로 작용하지만 boron 피이드백은 불안정한 해를 갖는 방향으로 작용한다.
모든 가동 출력에서 안정한 해 도형을 형성도록 하기 위해 원자로 설계나 가동 조건들에 의해 결정되는 피이드백 상수들에 유념할 필요가 있음을 알 수 있었다. 아울러 안전한(safe) 가동을 위한 감속제 온도계수, 핵 연료 온도 계수, boron 농도의 범위를 결정할 수 있음을 알았다.
