A new integral method for solving the point reactor neutron kinetics equations

A numerical integral method that efficiently provides the solution of the point kinetics equations by using the better basis function (BBF) for the approximation of the neutron density in one time step integrations is described and investigated. The approach is based on an exact analytic integration of the neutron density equation, where the stiffness of the equations is overcome by the fully implicit formulation. The procedure is tested by using a variety of reactivity functions, including step reactivity insertion, ramp input and oscillatory reactivity changes. The solution of the better basis function method is compared to other analytical and numerical solutions of the point reactor kinetics equations. The results show that selecting a better basis function can improve the efficiency and accuracy of this integral method. The better basis function method can be used in real time forecasting for power reactors in order to prevent reactivity accidents.     

Inhomogeneous point kinetics equations and the source contribution

Inhomogeneous point reactor kinetics equations with one-group of delayed neutrons are solved analytically for linear reactivity insertion as well as for step reactivity insertion in the presence of external neutron source using the prompt jump approximation. The solution is obtained as an infinite series. The methodology is found to be a promising tool for analyzing nuclear reactor kinetics with positive or negative ramp reactivity insertion on a sub-critical or a zero power delayed critical reactor, where the temperature reactivity feedback is negligibly small. To check the consistency and the accuracy of the analytical solution, the results are compared with the numerical solution for different sub-critical and delayed critical states. The comparison is found to be good for all kinds of positive and negative step and ramp reactivity insertions. The analytical solution is arranged into two terms, one as a function of source contribution the other without that. Using the newly rearranged solution, the importance of the source term and the contribution to the error while neglecting source term to the reactor kinetics analysis, can be realized. Contribution to the error is small (less than 0.1%) when the equilibrium power is more than about one megawatt for a medium sized LMFBR. Similarly, the importance of source contribution to the total reactor period as a function of initial equilibrium power is also realized with the newly rearranged analytical solution. The total reactor period is over predicted (larger period in place of smaller period) which is not conservative, if the source contribution is not considered, for considerably small initial equilibrium power. The percentage of error in not considering the source term in period calculation varies as a function of net reactivity and ramp rate. The percentage of error in period determination without considering the source is comparatively high for small ramp rates.     

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.     

Solution of the point reactor kinetics equations with temperature feedback by the ITS2 method

The ITS2 method is used to solve the point-reactor kinetics equations in the integral formulation with arbitrary number of delayed neutron groups and Newtonian temperature feedback. The method is based on low-order Taylor series expansions of neutron density and reactivity functions and uses variable time steps to control the numerical instabilities resulting from the stiff nature of the governing equations. Time steps are determined through an analytic criterion relating their magnitudes to the maximum admissible truncation error in the neutron-density expansion series. Temperature feedback is included in the reactivity as a function of the neutron density for different input types, including step change with adiabatic temperature feedback and compensated ramp functions. An iterative procedure is applied to determine the time steps while simultaneously updating the reactivity function. Numerical results show the ITS2 method is highly accurate for solving point reactor dynamics problems with temperature feedback.     

Sub-critical reactor kinetics analysis using incomplete gamma functions and binomial expansions

Point reactor kinetics equations with one group of delayed neutrons are solved analytically to determine the neutron population as a function of time for any ramp reactivity insertion in the presence of external neutron source using prompt jump approximation. With the time dependent neutron population the other important kinetic parameters such as the reactor period also can be derived. Analytical solutions are available in the literatures for any ramp reactivity insertion into a critical reactor without considering the source term. Analytical solutions available in the literature by considering the source term also to study sub-critical reactor kinetics. But such a solutions either uses constant source approximation which under predicts the solution, or the available solution is not useful for all kind of sub-critical reactivity and external ramp reactivity insertion combination due to the computer precision incompatibility. In the present work, analyses are carried out to determine the reactivity boundary to which the existing results can converge to a true solution, beyond where the precision incompatibility arises. A new series solution is recommended in the region where existing solution converges to false solution due to precision incompatibility.     

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 solution of the point reactor kinetics equations with fuel burn-up and temperature feedback

Point reactor kinetics equations are solved numerically using one group of delayed neutrons and with fuel burn-up and temperature feedback included. To calculate the fraction of one-group delayed neutrons, a group of differential equations are solved by an implicit time method. Using point reactor kinetics equations, changes in mean neutrons density, temperature, and reactivity are calculated in different times during the reactor operation. The variation of reactivity, temperature, and maximum power with time are compared with the predictions by other methods.     

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 solution of point kinetics equations for linear reactivity variation during the start-up of a nuclear reactor

The analytical solution of point kinetics equations with a group of delayed neutrons is useful in predicting the variation of neutron density during the start-up of a nuclear reactor. In the practical case of an increase of nuclear reactor power resulting from the linear insertion of reactivity, the exact analytical solution cannot be obtained. Approximate solutions have been obtained in previous articles, based on considerations that need to be verifiable in practice. In the present article, an alternative analytic solution is presented for point kinetics equations in which the only approximation consists of disregarding the term of the second derivative for neutron density in relation to time. The results proved satisfactory when applied to practical situations in the start-up of a nuclear reactor through the control rods withdraw.     

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。     

A Microprocessor Controlled Reactivity Meter for Real Time Monitoring of Reactors

Using a General Instrument 1600 microprocessor, a compact reactivity meter for real time monitoring of reactors has been developed. The microprocessor was programmed to solve the inverse kinetics equations using the digitized output of an ionization chamber neutron detector for the power history. Use of the simplest programming steps and emphasis of hardware for input/output operations enabled reactivity updates every 0.1 seconds. The device can be easily adapted to any reactor system with a power output signal once the kinetic parameters are known and incorporated into the processor program.     

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.     

An analytical solution of the point kinetics equations with time-variable reactivity by the decomposition method

In this work, we report an analytical solution for the point kinetics equations by the decomposition method, assuming that the reactivity is an arbitrary function of time. The main idea initially consists in the determination of the point kinetics equations solution with constant reactivity by just using the well-known solution results of the first-order system of linear differential equations in matrix form with constant matrix entries. Applying the decomposition method, we are able to transform the point kinetics equations with time-variable reactivity into a set of recursive problems similar to the point kinetics equations with constant reactivity, which can be straightly solved by the mentioned technique. For illustration, we also report simulations for constant, linear and sinusoidal reactivity time functions as well comparisons with results in literature.     

An analytical solution of the point kinetics equations with time-variable reactivity by the decomposition method

In this work, we report an analytical solution for the point kinetics equations by the decomposition method, assuming that the reactivity is an arbitrary function of time. The main idea initially consists in the determination of the point kinetics equations solution with constant reactivity by just using the well-known solution results of the first-order system of linear differential equations in matrix form with constant matrix entries. Applying the decomposition method, we are able to transform the point kinetics equations with time-variable reactivity into a set of recursive problems similar to the point kinetics equations with constant reactivity, which can be straightly solved by the mentioned technique. For illustration, we also report simulations for constant, linear and sinusoidal reactivity time functions as well comparisons with results in literature.     

Design & construction of a digital real time reactivity meter for Tehran research reactor

Reactivity measurement is one of the challenges of monitoring, control and investigation of nuclear reactors. In this paper design and construction of a reactivity meter for continuous monitoring of reactivity in research reactors are described. The device receives amplified output of the fission chamber, which is in mA range, as the input. Using amplifier circuits, this current is converted to voltage and then digitalized with a microcontroller to be sent to serial port of computer. The device itself consists of software, which is a MATLAB real time programming for the computation of reactivity by the solution of neutron kinetic equations. After data processing the reactivity is calculated and presented using LCD. Tehran research reactor is selected to test the reactivity meter device. The results of applying this reactivity meter in Tehran research reactor (TRR) are compared with the experimental data of control rod worth, void coefficient of reactivity and reactivity changes during approach to full power. The maximum relative error in several experiments is calculated to be 13%.     

铅铋冷却次临界堆反应性引入下的中子学动态特性分析

加速器驱动次临界系统（ADS）与临界系统相比具有不同的中子学动态特性。采用瞬跳近似导出了次临界状态下反应性扰动引起中子密度和堆功率变化的关系式,与基于RELAP5开发的次临界点堆动力学程序做了不同次临界度（keff=0.90,0.95,0.97,0.98和0.99）下1 s内引入反应性＋1β的中子学动态特性对比分析。结果表明：①有外源的瞬跳近似能够精确地描述受扰动后很短的一段时间之后的中子密度和堆功率的变化情况,能用于求解有外源的点堆动态方程渐进情况下的解;②反应性引入事故过程中,次临界堆表现出内在稳定性,次临界度越深,偏离临界越远,反应性扰动对次临界堆的影响就越小。     

Asymptotically stable solutions of 1-D space-time kinetics in the presence of delayed neutrons

The non-linear 1-D reactor kinetics equations in the presence of delayed neutrons with Newtonian feedback have been solved analytically for a slab homogeneous reactor for a step input of reactivity using a perturbation theory expansion. The asymptotic limit of the fundamental mode has been found to be the same with and without the inclusion of delayed neutrons in the analysis. However, the rate at which the limit is approached is many orders slower in the presence of delayed neutrons than in their absence. The threshold values of initial power for the occurrence of oscillations are also found to be higher in the presence of delayed neutrons for the three types of reference reactors considered.     

An analytical solution for the point reactor kinetics equations with one group of delayed neutrons and the adiabatic feedback model

The point reactor kinetics equations with one group of delayed neutrons and the adiabatic feedback model are solved analytically. The analytical solution is based on an expansion of the neutrons density in powers of the small parameter, the prompt neutrons generation time, into the second order differential equation in the neutron density. The relation between the time and the reactivity for reactor excursions near prompt critical is derived. Also, the neutron density and the average density of delayed neutron precursors as functions of reactivity are presented. The relations of reactivity, neutron density and temperature with time are calculated, drawn, and compared with other analytic method.     

三角波振荡反应性测量技术研究

堆振荡法是测量小反应性的方法之一。本工作以点堆动态方程为出发点,给出零功率反应堆传递函数的表达式。对三角波输入情况进行傅里叶分析后得到振荡反应性的表达式。在零功率反应堆上进行了振荡控制棒实验和振荡铀样品实验的反应性测量,本实验测量结果与周期法实验测量结果相符。