A Method for Estimating Criticality Lower Limit Multiplication Factor. Yoshitaka NAITO NAIS Co., Ltd.

Transcription

1 A Method for Estimating Criticality Lower Limit Multiplication Factor Yoshitaka NAITO NAIS Co., Ltd.

2 Progress Grade up of computer performance Sub-criticality becomes to be decided using computed results Necessity to determine K limit (Criticality Lower Limit Multiplication Factor) If a computed keff is less than K limit, the system is determined to be sub-critical. The keff value 0.95 has been often used for K limit. In Japanese Criticality Safety Handbook published in 1988, the value K limit s are shown for several nuclear fuel types. Here, I will show you the way to obtain the K limit and the possibility.

3 Introduction of Fundamental Eq. Luebill Eq. (Conservation low of particle density in phase space) Q.M. Area Assumption: a) Truncation of collision area, b) binary collisions, c) molecular chaos, d) slow variation of distribution function Mass density distribution of the Boltzmann Eq. f Approximate the collision term with neutron cross section, 1 f ( E, Ω) = Ω gradf ( E, Ω) f ( E, Ω) { s( E) + a( E) } + S( E, Ω) where + ξ f t Born Appr. = x1 1 m v t de dω f, { f ( ξ1) f ( ξ 2) f ( ξ1) f ( ξ 2)} Vdωdξ 2 ( E, Ω ) s( E E Ω Ω) ( E, Ω) = X ( E) keff v f ( E Ω ) S f, de dω (1) Eigen value Keff is a function of macroscopic cross sections,which are composed of microscopic cross sections and atomic number densities. + (2) (3)

4 Dependency of computed Keff s of ICSBEP-THERM-004 on nuclear data files and computer codes C / E (Keff) MVP(JENDL-3.3) MVP(JEFF-3.0) MVP(ENDF/B-VI) C / E (Keff) MVP (ENDF/B-VI) MCNP4C (ENDF/B-VI) H / (heavy metal) H / (heavy metal) Fig.1 Comparison of computed Keff s with three kinds of nuclear data files, and the MVP code Fig.2 Comparison of computed Keff s with MVP and MCNP, and a nuclear data file ENDF/B-VI Computed Keff s depend strongly upon nuclear data files and weakly upon computer codes

5 Sample Calculation Identification Main purpose Benchmark keff Benchmark keff uncertainty (1 sigma) Confidence interval Code Name Library Calc. Keff Keff uncertainty (1 sigma) LEU-COMP-THERM Transport unknown MCNP ENDF/B-V Continuous LEU-COMP-THERM Transport unknown MCNP ENDF/B-V Continuous LEU-COMP-THERM Transport unknown MCNP ENDF/B-V Continuous LEU-COMP-THERM Transport unknown MCNP ENDF/B-V Continuous LEU-COMP-THERM Transport unknown MCNP ENDF/B-V Continuous LEU-COMP-THERM Transport unknown MCNP ENDF/B-V Continuous LEU-COMP-THERM Transport unknown MCNP ENDF/B-V Continuous LEU-COMP-THERM Transport unknown MCNP ENDF/B-V Continuous LEU-COMP-THERM Transport unknown MCNP ENDF/B-V Continuous LEU-COMP-THERM Transport unknown MCNP ENDF/B-V Continuous LEU-COMP-THERM Transport unknown MCNP ENDF/B-V Continuous LEU-COMP-THERM Transport unknown MCNP ENDF/B-V Continuous LEU-COMP-THERM Transport unknown MCNP ENDF/B-V Continuous LEU-COMP-THERM Transport unknown MCNP ENDF/B-V Continuous LEU-COMP-THERM Transport unknown MCNP ENDF/B-V Continuous LEU-COMP-THERM Transport unknown MCNP ENDF/B-V Continuous LEU-COMP-THERM Transport unknown MCNP ENDF/B-V Continuous LEU-COMP-THERM Transport unknown MCNP ENDF/B-V Continuous LEU-COMP-THERM Transport unknown MCNP ENDF/B-V Continuous LEU-COMP-THERM Transport unknown MCNP ENDF/B-V Continuous LEU-COMP-THERM Transport unknown MCNP ENDF/B-V Continuous LEU-COMP-THERM Transport unknown MCNP ENDF/B-V Continuous LEU-COMP-THERM Transport unknown MCNP ENDF/B-V Continuous LEU-COMP-THERM Transport unknown MCNP ENDF/B-V Continuous LEU-COMP-THERM Transport unknown MCNP ENDF/B-V Continuous LEU-COMP-THERM Transport unknown MCNP ENDF/B-V Continuous LEU-COMP-THERM Transport unknown MCNP ENDF/B-V Continuous LEU-COMP-THERM Transport unknown MCNP ENDF/B-V Continuous LEU-COMP-THERM Transport unknown MCNP ENDF/B-V Continuous LEU-COMP-THERM Transport unknown MCNP ENDF/B-V Continuous LEU-COMP-THERM Transport unknown MCNP ENDF/B-V Continuous LEU-COMP-THERM Transport unknown MCNP ENDF/B-V Continuous LEU-COMP-THERM Transport unknown MCNP ENDF/B-V Continuous LEU-COMP-THERM Transport unknown MCNP ENDF/B-V Continuous LEU-COMP-THERM Transport unknown MCNP ENDF/B-V Continuous LEU-COMP-THERM Transport unknown MCNP ENDF/B-V Continuous averaged Keff standard deviation of Keff number of cases

6 A sample of computed results on ICSBEP data by MCNP with ENDF/B-V (Benchmark Keff=1.0) Identification Bechmark Keff Calc. Keff Keff uncertainty (1σ) uncertainty LEU-COMP-THERM LEU-COMP-THERM : LEU-COMP-THERM LEU-COMP-THERM : LEU-COMP-THERM average Keff standard deviation of Keff number of cases 36

7 Histogram on benchmark calculation results number of cases sigma number of cases : 36 averaged keff : standard deviation : calculated keff K limit 1.02 Fig.3 Histogram of benchmark calculation results Calculated by MCNP with ENDF/B-V

8 Concluding Remarks Standard deviation of the histogram is affected by the following four factors, (1) errors of microscopic cross sections of nuclides, (2) errors of benchmark data such as uncertainties of atomic number density and geometrical dimension, (3) errors from Monte Carlo calculation such as statistical deviation and source convergence criterion, (4) errors of measurement on criticality conditions such as critical water level.

9 Characteristics of the error (1),(2),(3) and (4) are as follow. The errors of (1) are expressed in covariance files of nuclear data and affect to reactivity according to neutron energy spectrum in each benchmark problem. That is, the reactivity error of each benchmark problem depends upon nuclide compositions and neutron energy spectrum. The reactivity effect may be expressed by some sensitivity coefficients such as those of the TSUNAMI code. With these sensitivity coefficients, benchmark data will be grouped. In each group, the quantity of reactivity error due to cross section errors is almost same and the dependency on spectrum index will be weak.. The errors of (2) are random errors and reactivity errors due to them may be expressed statistically. The errors of (3) and (4) are specified by input data for the calculation and by reports on benchmark data

10 I propose a following procedure to obtain criticality lower limit multiplication factor in each group. Step1. Analyze many benchmark problems. Step2. Group them into several categories. Step3. Obtain an average keff and a standard deviation in each group. Step4. Obtain a criticality lower limit multiplication factor

11 Procedure for sub-criticality assessment for an object Step1. Identify the object to a group and find the criticality lower limit multiplication factor. Step2. Calculate the multiplication factor and, obtain the criterion. statistical deviation and source convergence With these results, an estimated lower limit of calculation multiplication factor. Step3. Assess calculated keff of the object comparing it with the lower limit of calculated multiplication factor of