Measurement of k_eff with an improved neutron source multiplication method based on numerical analysis

Measurement of k eff with an improved neutron source multiplication method based on numerical analysis 1 2 first-author 1 1 2 1 corresp Corresponding author, zhengchun@caep.cn Corresponding author, zhengchun@caep.cn zhengchun@caep.cn Institute of Nuclear Physics and Chemistry, China Academy of Engineering Physics, Mianyang621900, China Institute of Nuclear Physics and Chemistry, China Academy of Engineering Physics, Mianyang621900, China College of Physical Science and Technology, Sichuan University, Chengdu610064, China College of Physical Science and Technology, Sichuan University, Chengdu610064, China In this work, we developed a numerical analysis-associated experiment method to determine the effective multiplication factor k eff , which is difficult to obtain directly from conventional neutron source multiplication (NSM) method. The method is based on the relationship between k eff , subcritical multiplication factor k s and external neutron source efficiency Φ ∗ in the subcritical system. On basis of the theoretical analysis, the dependence of k s and Φ ∗ on subcriticality and source position was investigated at the Chinese Fast Burst Reactor-II (CFBR-II). A series of k s were measured by NSM experiments at four subcritical states ( k eff = 0.996; 0.994; 0.991; 0.986) with the 252 Cf neutron source located at different positions (from the system center to outside) at each subcritical states. The Φ ∗ was obtained by Monte-Carlo simulation for each condition. With the measured k s and calculated Φ ∗ , k eff of the subcritical system was evaluated with a relative difference of < 1% between values obtained by the improved method and by positive period method. Especially, the relative difference of < 0.18% with the source located at the system center. k eff Neutron source multiplication Monte-Carlo Fast critical system I. INTRODUCTION 1 Neutron source multiplication (NSM) method, a simple measurement technique that uses an external neutron source and an ordinary neutron detector, is suitable for real-time measurement. With the recent development of accelerator-driven system (ADS) [ 1 1 ], an increasing attention has been paid to NSM experiments associated with various subcritical systems [ 2 2 , 3 3 ]. Implementing this kind of experiments can acquire subcritical multiplication factor k s and external neutron source efficiency Φ ∗ , which can be used to design the ADS systems, and its neutron multiplication [ 4 4 , 5 5 ]. The subcritical multiplication factor k s is related to the neutron multiplication M [ 6 6 ] by k s =1-1/ M , but in practice one needs to know the effective multiplication factor k eff . In this paper, in order to overcome the difficulties of getting k eff directly from the conventional neutron source multiplication method, we propose a numerically associated experiment method. In Sec.II, the NSM method and the relationship between effective multiplication factor k eff , subcritical multiplication factor k s and neutron source efficiency Φ ∗ are reviewed briefly. The experiment apparatus is described in Sec. III. In Sec.IV, for verifying the proposed NSM method, the numerical analysis and a series of NSM experiments were carried out at the Chinese Fast Burst Reactor-II (CFBR-II) at various subcritical states with 252 Cf neutron source located at different positions in each subcritical states (from center of the system to outside). II. THEORETICAL BASIS 1 For the critical systems composed of the fissile materials and other materials, the neutron balance equation in fundamental mode is expressed as: <math display="block" id="M1"><mrow><mi>L</mi><msub><mi>φ</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mi>r</mi><mn>,</mn><mi>E</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mrow><msub><mi>k</mi><mrow><mtext>eff</mtext></mrow></msub></mrow></mfrac><mi>P</mi><msub><mi>φ</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mi>r</mi><mn>,</mn><mi>E</mi><mo stretchy="false">)</mo><mn>,</mn></mrow></math> where, φ 0 (r, E ) is the neutron flux distribution at position r ( x , y , z ) at energy E , L is the destruction operator, and P is the production operator. By taking the inner product < ⋅ > of Eq. (1), which indicates the integration over the space and energy variables, k eff in Eq. (1) can be expressed as: <math display="block" id="M2"><mrow><msub><mi>k</mi><mrow><mtext>eff</mtext></mrow></msub><mo>=</mo><mfrac><mrow><mo><</mo><mi>P</mi><msub><mi>φ</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mi>r</mi><mn>,</mn><mi>E</mi><mo stretchy="false">)</mo><mo>></mo></mrow><mrow><mo><</mo><mi>L</mi><msub><mi>φ</mi><mn>0</mn></msub><mo stretchy="false">(</mo><mi>r</mi><mn>,</mn><mi>E</mi><mo stretchy="false">)</mo><mo>></mo></mrow></mfrac><mn>.</mn></mrow></math> For the subcritical system with an external source at steady state, the neutron balance equation is: <math display="block" id="M3"><mrow><mi>L</mi><msub><mi>φ</mi><mi>s</mi></msub><mo stretchy="false">(</mo><mi>r</mi><mn>,</mn><mi>E</mi><mo stretchy="false">)</mo><mo>=</mo><mi>P</mi><msub><mi>φ</mi><mtext>s</mtext></msub><mo stretchy="false">(</mo><mi>r</mi><mn>,</mn><mi>E</mi><mo stretchy="false">)</mo><mo>+</mo><mi>s</mi><mo stretchy="false">(</mo><mi>r</mi><mn>,</mn><mi>E</mi><mo stretchy="false">)</mo><mn>.</mn></mrow></math> Similarly, the subcritical multiplication factor k s can be expressed as: <math display="block" id="M4"><mrow><msub><mi>k</mi><mtext>s</mtext></msub><mo>=</mo><mfrac><mrow><mo><</mo><mi>P</mi><msub><mi>φ</mi><mi>s</mi></msub><mo stretchy="false">(</mo><mi>r</mi><mn>,</mn><mi>E</mi><mo stretchy="false">)</mo><mo>></mo></mrow><mrow><mo><</mo><mi>L</mi><msub><mi>φ</mi><mi>s</mi></msub><mo stretchy="false">(</mo><mi>r</mi><mn>,</mn><mi>E</mi><mo stretchy="false">)</mo><mo>></mo></mrow></mfrac><mn>,</mn></mrow></math> where, φ s (r, E ) is the neutron flux in the subcritical system with the external neutron source s (r, E ). The external source efficiency, Φ ∗ , is the ratio of average importance between the source neutron and fission neutron. The Φ ∗ implies that a neutron emitted from source, with a certain position and energy distribution, will produce certain "important" neutrons that can lead to higher or lower neutron multiplication for a given proliferation system. Deduced from properties of the operator and Eqs. (1)– (4), we have Eq. (5): <math display="block" id="M5"><mrow><mn>1</mn><mo>−</mo><mfrac><mn>1</mn><mrow><msub><mi>k</mi><mrow><mtext>eff</mtext></mrow></msub></mrow></mfrac><mo>=</mo><msup><mi>Φ</mi><mo>∗</mo></msup><mrow><mo>(</mo><mrow><mn>1</mn><mo>−</mo><mfrac><mn>1</mn><mrow><msub><mi>k</mi><mtext>s</mtext></msub></mrow></mfrac></mrow><mo>)</mo></mrow><mo>⇒</mo><msub><mi>k</mi><mrow><mtext>eff</mtext></mrow></msub><mo>=</mo><mfrac><mrow><msub><mi>k</mi><mtext>s</mtext></msub></mrow><mrow><msub><mi>k</mi><mtext>s</mtext></msub><mo>−</mo><msup><mi>Φ</mi><mo>∗</mo></msup><mo stretchy="false">(</mo><msub><mi>k</mi><mtext>s</mtext></msub><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac><mn>.</mn></mrow></math> Then, the multiplication factors k s and k eff can be theoretically correlated with the neutron source efficiency Φ ∗ . Obtaining k eff of the system through Eq. (5) needs to know the k s and Φ ∗ . As mentioned above, the k s can be measured by neutron multiplication experiment, but the quantity of neutron source efficiency Φ ∗ cannot be obtained from experiments directly. In the following sections, we acquired Φ ∗ from numerical analysis and evaluated k eff successfully. III. EXPERIMENT APPARATUS REVIEW 1 Experiments were carried out at the Chinese Fast Burst Reactor-II (CFBR-II) [ 7 7 ], an ellipsoidal assembly, which consists of, from inside to outside, the enriched-uranium spherical shells, inner brass reflector, depleted uranium reflector and outer brass reflector. The reactor can be divided into the upper, middle and lower parts ( Fig. 1 Fig. 1 ). The middle part is a 5.2 cm thick stainless steel plate, which has three horizontal holes for three control rods to pass through, i.e. an auto- adjustment rod, a composition rod and a pulse rod. The three rods are made of casted enriched-uranium. A 10 cm×4.2 cm×14.5 cm ( w × h × d ) irradiation chamber and two horizontal experimental holes are located in the plate. The neutron source can be put at any position in the horizontal holes by a neutron source transmission device. Reactivity of the assembly can be controlled by adjusting insert depth of the three control rods. The assembly works at subcritical, delayed critical or super prompt critical states. A 252 Cf spontaneous fission neutron source (in activity of 1.41×10 -5 s -1 ) sealed in an aluminum box of Φ 7.8 mm×11 mm can be sent into a horizontal hole of the stainless steel plate by the transmission device when the system is in deep subcritical state. Then, the control rods are inserted to different depths to operate the reactor at different subcritical power states. Two BF 3 proportional counters, of the SZJ-1 type, produced by Beijing Nuclear Instrument Factory, are used to record the neutrons (which is proportional to the neutron density in the system). They consist of the paraffin barrel (Hanson long counter), preamplifier, HV power supply, main amplifier and multi-scalar. The detectors are 180 cm away from the system center. IV. NUMERICAL ANALYSIS AND EXPERIMENTS 1 A. Numerical calculations 2 Before a series of experiments are carried out, Monte-Carlo numerical method [ 8 8 ] was used to build the subcritical system model that is identical to CFBR-II in Fig. 1 Fig. 1 . Four subcritical states of k eff =0.996;0.994;0.991;0.986 were selected, which are conditions for our multiplication experiments. The source positions were changed along the r-axis at each subcritical state (i.e. 0 mm, 20 mm, 40 mm, 60 mm, 80 mm, 100 mm, 120 mm and 150 mm). The spontaneous 252 Cf neutron source, described by Watt fission spectrum [ 9 9 ], is positioned the same way as that in the planned experiments. The k s of all the experimental conditions were calculated by MCNP5 with statistic difference of < 0.7%. The neutron source efficiency Φ ∗ is closely connected with neutron importance, which is proportional to the detector response to a unit source. In other words, neutron importance is a measure of the "importance" of a neutron in contributing to the detector response, or the expected counts per neutron in certain position, direction and energy. The detector counting rates to the system with the 252 Cf neutron source in the core could be obtained, and the counting rate to the eigen neutron source distributed in the system could be recorded, too, by using MCNP code. The ratio of the two counting rates should be the external neutron source-related average efficiency Φ ∗ [ 10 10 , 11 11 ]. In terms of eigen-distribution neutron source acquisition, we used the KCODE and SSW cards in MCNP5 critical calculation to achieve the goal. Thus, we tallied Φ ∗ in MCNP5 calculation following certain regulation according to MCNP rules. By calculating the k s , Φ ∗ and k eff inside and outside of the neutron source, we were able to investigate the effect of system subcriticality and source position. For the subcritical states ( k eff =0.994), the effect of the source positions along the r-axis are demonstrated by the calculation results in Fig. 2 Fig. 2 . The source efficiency Φ ∗ and subcritical multiplication factor k s at different source positions indicate that the smallest discrepancy between k s and k eff is achieved when the source is at the center of the core. Table 1 Table 1 shows the subcriticality effect on k s and Φ ∗ at k eff =0.996; 0.994; 0.991; 0.986. One sees that for the four subcritical states, when the source is at the core center, k s is very close to k eff , with a relative difference of < 0.8%, and the Φ ∗ was greater than 0.91. The results indicate that the closer the system reaches to the critical state, the more likeliness the relative values of φ s and φ 0 become equal to each other, and so do the values of k s and k eff . Inversely, placing the source outside the core, the discrepancy between k s and k eff becomes much larger, with the maximum relative difference being 22.1340% at k eff = 0.986 and the source being 150 mm away from the core center. 1 1 case 1 1 k eff 1 1 Source position (mm) 1 1 ( k s / k eff -1) (%) 1 1 Φ ∗ 1 1 1 1 1 0.9960 1 1 0 1 1 -0.6436 1 1 0.9599 1 1 1 1 1 1 150 1 1 -12.2234 1 1 0.0251 1 1 2 1 1 0.9940 1 1 0 1 1 -0.5851 1 1 0.9272 1 1 1 1 1 1 150 1 1 -15.2628 1 1 0.0887 1 1 3 1 1 0.9910 1 1 0 1 1 -0.7513 1 1 0.9471 1 1 1 1 1 1 150 1 1 -18.6502 1 1 0.0939 1 1 4 1 1 0.9860 1 1 0 1 1 -0.7852 1 1 0.9176 1 1 1 1 1 1 150 1 1 -22.1340 1 1 0.1036 In order to check the source position effects on the neutron leakage rate, we did another numerical analysis to investigate the detector response to non-collided neutrons emitted from the source, and to total neutrons. The results in Fig. 3 Fig. 3 show that the relative difference is below 0.25%. Placing the neutron source closer to the system edge, the counting rate for non-collided neutrons from the source becomes greater, while the counts of total neutrons decrease. In other words, the detector counting rate decreases for the eigen-distribution neutron (fission neutron inside the system). This explains why the discrepancy between k s and k eff becomes large while moving the 252 Cf neutron source away from the system center. B. Experiments Results and Discussion 2 On the basis of the numerical investigation in Sec.IV.A, the NSM experiments were carried out at CFBR-II to measure k s and evaluate Φ ∗ . The subcritical states were experimentally achieved by adjusting insert depth of the control rods. All the experiment apparatus and conditions were completely identical to the numerical ones. The results are shown in Fig. 4 Fig. 4 . The k s decreased with increasing distance of the source from the core center and with deepening subcriticality. This is identical to the calculation results in Fig. 2 Fig. 2 . In order to test the feasibility of this improved NSM method, we considered four subcritical states ( k eff = 0.996; 0.994; 0.991; 0.986) and eight neutron source positions according to each subcritical states. For the sake of the calculated source efficiency Φ ∗ and the measured k s , we obtained the effective multiplication factor k eff according to Eq. (5) at 32 studied conditions. For comparison, we used the positive periodic method to measure the reactivity as the reference method in the work. The specific process is that we measured reactivity in super-criticality, and extrapolated to subcriticality with calibrated reactivity curve of control rods. As shown in Table 2 Table 2 , a comparison between the reference method and the results in this work at the four subcritical conditions with the source located at the core center revealed fairly good agreement with a maximum relative difference of 0.18%. For a certain subcriticality ( k eff =0.991), the discrepancy between k eff measured by positive period method and the evaluated ones in this work is the smallest of all. When k eff > 0.9910, the evaluated k eff are a bit larger than experimental measured ones. However, at a very deep subcriticality, e.g. k eff =0.9860, the evaluated k eff is a little less than the measured one. This is because the source neutrons could not propagate efficiently in deep subcriticality. The results demonstrated the good precision of this improved method and the validation of the Monte-Carlo calculation model. a Positive periodic method. b Calculated in this work. c Neutron Source Multiplication experiments and Eq. (1). d Numerical analyzed method described in Sec.IV.A. 1 1 Case 1 1 k eff a a 1 1 k eff b b 1 1 k s c c 1 1 Φ ∗ d d 1 1 ( k eff a a / k eff b b -1) / % 1 1 1 1 1 0.9960 1 1 0.9977 1 1 0.9936 1 1 0.9599 1 1 0.1707 1 1 2 1 1 0.9940 1 1 0.9948 1 1 0.9899 1 1 0.9272 1 1 0.0805 1 1 3 1 1 0.9910 1 1 0.9915 1 1 0.9847 1 1 0.9471 1 1 0.0505 1 1 4 1 1 0.9860 1 1 0.9844 1 1 0.9757 1 1 0.9176 1 1 -0.1623 In general, k eff of the system for all studied cases were well approximated as shown in Fig. 5 Fig. 5 . The comparison between the k eff results in references (positive period method, marked as k eff-exp ) and our method ( k eff-eva ) is done by Δ ϵ =[( k eff-eva - k eff-exp )/ k eff-exp ]×100%. As shown in Fig. 6 Fig. 6 , the space effect on k eff can be seen in a large range of the neutron position. Generally, when the neutron source is inside the subcritical system, the relative difference is small, comparing to that when the source is outside the system; and the relative difference increase rapidly with distance of the neutron source to the system center, being irregularly large at the distance of > 85 cm, because this is where the brass and depleted uranium reflectors locate, causing uncertainty factor related to complicated neutron behavior. Therefore, to reduce the relative difference in experiments implemented, the neutron source should be put inside the subcritical system. V. CONCLUSION 1 A methodology to evaluate the effective multiplication factor k eff from measured k s and numerical analyzed Φ ∗ was investigated via NSM experiments on CFBR-II. The evaluated results were approximately equal to the reference method (positive period method). 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Nuclear Reactor Theory 10.13538/j.1001-8042/nst.25.020602 Supported by Natural Science Foundation of China (No.11175164) 2013-05-29 2013-11-08 2014-03-20 2014-04-20 2021-04-11 2014 Nuclear Science and Techniques 2 25