核电计算中刚性常微分方程的数值解

在核电计算中,核岛中的反应堆动态学方程和常规岛中的湿蒸汽汽轮机甩负荷时各级压力变化方程都是典型的刚性常微分方程。本文介绍用通用性强的Gear方法求解这些刚性方程的有效性。在解点堆动态学方程时,Gear方法的计算效率与分段多项式逼近法相当;在对矩阵求逆的算法作了改进后,该方法的计算效率就高于分段多项式逼近法。     

基于指数变换的对角隐式龙格库塔法求解中子点堆动力学方程

点堆动力学对于反应堆安全运行有着重要作用,但点堆动力学方程是刚性的,通常使得数值求解所采用的步长很小。本文研究了基于指数变换的对角隐式龙格库塔(DIRK)方法用来求解点堆动力学方程。基于指数变换的DIRK保留了DIRK方法适合求解刚性方程的特点,同时在反应性引入较大的情况下,它比对角隐式库塔方法表现更好。若干算例,如反应性阶跃、线性或者正弦变化等,表明基于指数变换的DIRK方法具有很高的计算精度。     

求解反应堆动态方程的插值多项式法

本文利用三阶 Hermite 插值多项式法求解点堆动态方程,衰变热方程和反应性反馈方程。由于考虑了反应堆连续运行史对衰变热的影响,所给出的公式可精确计算反应堆运行瞬态和停堆后剩余功率随时间的变化。对例题的核算表明,上述方法允许很大的时间步长并保证很高的计算精度。     

求解点堆中子动力学方程的三角插值法

在求解点堆中子动力学方程组中,对中子密度N(t)使用三角函数近似方法,并使用全隐式格式保证A稳定性,给出一个求解的方法。计算结果表明,此方法可以灵活应用于各种反应性输入,计算时间短,计算精度较高,可以满足实际工程的需要,是求解点动态方程的一种较好方法。     

点堆中子动力学方程的指数基函数法求解

给出了一个求解点堆中子动力学方程组的指数基甬数法.该方法通过将点堆中子动力学方程组变成矩阵形式,利用指数函数为基甬数的特点将其显式化,并根据初始条件求得各项系数,进而获得方程组的解.对阶跃、线性和正弦等不同反应性输入进行了计算.结果表明,指数基函数法过程简捷明了、易于编程,是一种计算速度较快、精度较高、适用性较强的求解点堆中子动力学方程的方法.     

代中子时间法求解点堆中子动力学方程

由于点堆中子动力学方程是个刚性方程,因此准确、快速、稳定地求解方程是困难的。得益于现代计算机技术的进步,本文直接采用代中子时间计算法求解点堆中子动力学方程,并用C++语言编制了计算程序。经过基准例题和动态-逆动态对比计算,验证了模型、程序计算的准确性和稳定性,而计算时间也是可接受的。     

基于反应性温度系数的金属型脉冲堆波形计算

金属型脉冲堆的反应性反馈效应主要由热膨胀引起,本文在反应性温度系数的基础上建立了波形计算方法,该方法由蒙特卡罗中子输运程序、热力学计算程序和点堆方程3部分组成。首先由三维中子输运程序和热力学计算程序计算出热功率和反应性的耦合关系,然后将耦合关系代入点堆方程,即可求解出波形。采用该方法计算了Lady Godiva的波形,计算结果与LANL的实验结果一致。     

基于六角形节块法的快堆燃耗计算程序NDHEXB

描述了快堆燃耗计算程序ＮＤＨＥＸＢ的理论模型，并给出了中国实验快堆（ＣＥＦＲ）的计算结果。结果表明ＮＤＨＥＸＢ具有良好的计算效率与精度。在ＮＤＨＥＸＢ程序中，采用六角形几何下的节块展开法求解中子扩散方程，利用常微分方程的一种数值方法——梯形法求解燃耗方程。     

点堆随机动力学方程伊藤解与中子数概率分布解析解的对比分析

为了研究反应堆弱中子源启动过程中的中子数密度和缓发中子先驱核随机涨落现象,我们推导和建立了点堆随机动力学方程组,在传统的点堆动力学方程组中引入了伊藤随机项。为了验证方程组的伊藤解方法和计算精度,我们在简化物理条件和方程形式下,对定态系统的中子数密度分别用随机动力学方程伊藤解和中子数概率分布函数解析解进行了对比分析。结果表明,伊藤解是一种有效、具有较高计算精度的方法,计算精度满足sigma的标准,置信水平在95%以上。     

全隐式龙格库塔法求解点堆动力学方程

强刚性问题时数值求解点堆中子动力学方程组的难点之一。该文用基于高斯-勒让特求积公式节点的全隐式龙格库塔法(简称GLFIRK)求解点堆动力学方程组。该方法是B稳定的,而且计算精度高,对于E级GLFIRK,其计算精度为2E阶。该文在阶跃、线性和正弦等不同反应性加入条件下对点堆动力学方程组进行了计算,计算结果表明,该方法计算精度高、计算速度较快、适应能力较好,可满足一定的工程应用要求。     

Analytical solution to solve the point reactor kinetics equations

New analytical solution for solving the point reactor kinetics equations with multi-group of delayed neutrons is presented. This solution is based on the roots of inhour equation, eigenvalues of the coefficient matrix. The inhour equation presents in new sample formula. The analytical solution represents the exact analytical solution for the point kinetics equations of multi-group of delayed neutrons with constant reactivity. Also, it represents the accurate solution for solving the point kinetics equations of multi-group of delayed neutrons with ramp and temperature feedback reactivities. This method are applied to different types of reactivity and compared to the traditional methods.     

Power series solution method for solving point kinetics equations with lumped model temperature and feedback

Point kinetics equations are stiff differential equations, and their solution by the conventional explicit methods will give a stable consistent result only for very small time steps. Since the neutron lifetime in a LMFBR is very short, the point kinetics equations for LMFBRs become even stiffer. In this study the power series solution (PWS) method is applied for solving the point kinetics equations for a typical LMFBR. A Fortran program is developed for accident analysis of LMFBRs with the PWS method for solving the point kinetics and a lumped model for solving the heat transfer equations. A new technique is developed with fixing factor to find out the average temperature at the peak power node (PPN) without performing temperature calculations at all axial nodes in a reactor fuel pin. The temperature at PPN also decides whether the reactor is within the design safety limit (DSL) or it has entered a serious transient that may lead to an accident. The coupled heat transfer and point kinetics models for a peak power node give the average fuel, clad and coolant temperatures. For the transient over power accidents (TOPA), this is the best way for calculating the temperature, with minimum amount of computations. TOPA analyses are carried out with PWS method. It is found that the PWS methodology uses a small number of numerical operations, while the computational time and the accuracy are comparable with the available fast computational tools. This methodology can be used in nuclear reactor simulation studies and accident analysis.     

Analytical exponential model for stochastic point kinetics equations via eigenvalues and eigenvectors

The stochastic point kinetics equations with a multi-group of delayed neutrons, which are the system of a couple of stiff stochastic differential equations, are presented.The analytical exponential model is used to solve the stochastic point kinetics equations in the dynamical system of the nuclear reactor. This method is based on the eigenvalues and corresponding eigenvectors of the coefficient matrix. The analytical exponential model calculates the mean and standard deviations of neutrons and precursor populations for the stochastic point kinetics equations with step, ramp, and sinusoidal reactivities.The results of the analytical exponential model are compared with published methods and the results of the deterministic point kinetics model. This comparison confirms that the analytical exponential model is an efficient method for solving stochastic stiff point kinetics equations.     

Generalized power series method with step size control for neutron kinetics equations

Based on the power series method (PWS), a generalized power series method (GPWS) has been introduced for solving the point reactor kinetics equations. The stiffness of the kinetics equations restricts the time interval to a small increment, which in turn restricts the PWS method within a very small constant step size. The traditional PWS method has been developed using a new formula that can control the time step at each step while transient proceeds. Two solvers of the PWS method using two successive orders have been used to estimate the local truncation errors. The GPWS method has employed these errors and some other constraints to produce the largest step size allowable at each step while keeping the error within a specific tolerance. The proposed method has resolved the stiffness point kinetics equations in a very simple way with step, ramp and zigzag ramp reactivities. The generalized method has turned out to represent a fast and accurate computational technique for most applications. The method is seemed to be valid for a time interval that is much longer than the time interval used in the conventional numerical integration, and is thus useful in reducing computing time. The method constitutes an easy-to-implement algorithm that provides results with high accuracy for most applications where, the reactor kinetics equations are reduced to a differential equation in a matrix form convenient for explicit power series solution. Results obtained by GPWS method: attest the power of the theoretical analysis, they demonstrate that the convergence of the iteration scheme can be accelerated, and the resulting computing time can be greatly reduced while maintaining computational accuracy. The point kinetics equations have been solved as a preliminary simple case aimed at testing the applicability of the GPWS method to solve point kinetics equations with feedback or, space kinetics problems.     

Numerical treatment for the point reactor kinetics equations using theta method,eigenvalues and eigenvectors

The point reactor kinetics model is a stiff system of linear/nonlinear ordinary differential equations. In fact, the numerical solutions of this stiff model need a smaller time step intervals within various computational schemes. The aim of this work is an accurate numerical solution without need to the smaller time step intervals. Theta method is the most popular, simplest and widely used method for solving the first order ordinary differential equations. In light of this fact, theta method is treated for solving the matrix form of this model via the eigenvalues and corresponding eigenvectors of the coefficient matrix. In this work, the matrix form of the stiff point kinetics equations with multi-group of delayed neutrons is introduced. The treatment theta method is applied to solve the stiff point kinetics equations with six groups of delayed neutrons. The performance of the treatment theta method is evaluated in several case studies involving step, ramp, sinusoidal and pulse reactivities. The results of the treatment theta method are more accurate than the theta method comparing with the conventional methods.     

An efficient technique for the point reactor kinetics equations with Newtonian temperature feedback effects

The point reactor kinetics equations of multi-group of delayed neutrons in the presence Newtonian temperature feedback effects are a system of stiff nonlinear ordinary differential equations which have not any exact analytical solution. The efficient technique for this nonlinear system is based on changing this nonlinear system to a linear system by the predicted value of reactivity and solving this linear system using the fundamental matrix of the homogenous linear differential equations. The nonlinear point reactor kinetics equations are rewritten in the matrix form. The solution of this matrix form is introduced. This solution contains the exponential function of a variable coefficient matrix. This coefficient matrix contains the unknown variable, reactivity. The predicted values of reactivity in the explicit form are determined replacing the exponential function of the coefficient matrix by two kinds, Backward Euler and Crank Nicholson, of the rational approximations. The nonlinear point kinetics equations changed to a linear system of the homogenous differential equations. The fundamental matrix of this linear system is calculated using the eigenvalues and the corresponding eigenvectors of the coefficient matrix. Stability of the efficient technique is defined and discussed. The efficient technique is applied to the point kinetics equations of six-groups of delayed neutrons with step, ramp, sinusoidal and the temperature feedback reactivities. The results of these efficient techniques are compared with the traditional methods.     

克服点堆中子动力学方程刚性的新方法—端点浮动法

点堆中子动力学方程是一个刚性微分方程,本文提出一个简洁有效地克服点堆中子动力学方程刚性的方法——端点浮动法。该方法采用端点浮动,用少量多项式来分段近似积分中的中子密度,并采用全隐离散格式,以有效地克服刚性,从而保证解的必要精度。根据几种实例的验算证明:该方法是有效可靠的,时间步长不受最大(绝对值)负时间常数的限制,而计算公式简单,便于使用。