Effective point kinetic parameters calculation in Tehran research reactor using deterministic and probabilistic methods

Effective point kinetic parameters calculation in Tehran research reactor using deterministic and probabilistic methods 1 corresp first-author Corresponding author Email: Mohsen.Kheradmand@gmail.com Corresponding author Email: Mohsen.Kheradmand@gmail.com Mohsen.Kheradmand@gmail.com 1 2 Department of Nuclear Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran Department of Nuclear Engineering, Science and Research Branch, Islamic Azad University, Tehran, Iran Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran The exact calculation of point kinetic parameters is very important in nuclear reactor safety assessment and most sophisticated safety codes such as RELAP5, PARCS, DYN3D, PARET, are using these parameters in their dynamic models. These parameters include effective delayed neutron fractions as well as mean generation time. These parameters are adjoint weighted, and adoint flux is employed as a weighting function in their evaluation. Adjoint flux calculation is an easy task for most of deterministic codes, but its evaluation is cumbersome for Monte Carlo codes. However, in recent years, some sophisticated techniques have been proposed for Monte Carlo based point kinetic parameters calculation without any need of adjoint flux. The most straightforward scheme is known as the "Prompt Method" and has been used widely in literature. The main objective of this article is dedicated to point kinetic parameters calculation in Tehran Research Reactor (TRR) using deterministic as well as probabilistic techniques. WIMS-D5B and CITATION codes have been used in deterministic calculation of forward and adjoint fluxes in the TRR core. On the other hand, the MCNP Monte Carlo code has been employed in the "Prompt Method" scheme for effective delayed neutron fraction evaluation. Deterministic results have been cross checked with probabilistic ones and validated with SAR and experimental data. In comparison with experimental results, the relative differences of deterministic as well as probabilistic methods are 7.6% and 3.2%, respectively. These quantities are 10.7% and 6.4% respectively in comparison with SAR report. Point kinetic parameters Tehran research reactor Adjoint flux Prompt method Deterministic method Probabilistic method 1 Introduction 1 One of the usual simplifications, which is used widely in nuclear reactor analysis, is the neutron diffusion equation. It considers a linear angular dependency of neutron flux and scattering cross section in the neutron transport equation. The numerical solution of diffusion equation, especially in time dependent cases, is very difficult yet. However, some special efforts have been developed to simplify the time dependent neutron diffusion equation. The most famous treatment, which focused on space variable elimination, is known as the “Point Kinetic Model”. This model assumes that the spatial dependency of neutron flux can be described by a single spatial mode (fundamental mode). Moreover, this shape function is time independent and will be constant during each transient [ 1 1 ] . By this assumption, the time dependent neutron flux is divided into two independent functions: a time dependent amplitude function and a time independent shape function [ 1 1 ] , <math display="block" id="M1"><mrow><mi>φ</mi><mrow><mo>(</mo><mrow><mi>r</mi><mo>,</mo><mi>t</mi></mrow><mo>)</mo></mrow><mo>=</mo><mi>ϑ</mi><mi>n</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mstyle mathvariant="bold" mathsize="normal"><mi>ψ</mi></mstyle><mrow><mo>(</mo><mstyle mathvariant="bold" mathsize="normal"><mi>r</mi></mstyle><mo>)</mo></mrow><mo>,</mo></mrow></math> where, <math id="M2"><mi>ϑ</mi></math> is neutron velocity, n is neutron density and <math id="M3"><mtext>ψ</mtext></math> is shape function and dimensionless in this formula. This technique can be employed in one group time dependent neutron diffusion equation, including delayed neutrons and point kinetic equations, and can be derived as follows [ 1 1 ] : <math display="block" id="M4"><mtable columnalign="left"><mtr><mtd><mfrac><mrow><mi>d</mi><mi>n</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mrow><mo>[</mo><mrow><mfrac><mrow><mi>ρ</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>−</mo><mi>β</mi></mrow><mi>Λ</mi></mfrac></mrow><mo>]</mo></mrow><mi>n</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>−</mo><munderover><mstyle mathsize="140%" displaystyle="true"><mo>∑</mo></mstyle><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mn>6</mn></munderover><msub><mi>λ</mi><mi>i</mi></msub><msub><mi>C</mi><mi>i</mi></msub><mo>,</mo></mtd></mtr><mtr><mtd><mfrac><mrow><mi>d</mi><msub><mi>C</mi><mi>i</mi></msub></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><msub><mi>β</mi><mi>i</mi></msub></mrow><mi>Λ</mi></mfrac><mi>n</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>−</mo><msub><mi>λ</mi><mi>i</mi></msub><msub><mi>C</mi><mi>i</mi></msub><mtext>     </mtext><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>6</mn><mtext>                </mtext><mo>.</mo></mtd></mtr></mtable></math> These equations are a set of seven coupled ordinary differential equations in time and describe both the time dependency of the neutron population in a reactor and the decay of delayed neutron precursors. As usual, delayed neutron precursors are presented in six groups. In these equations, <math id="M5"><mrow><mi>n</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></math> is neutron population density; <math id="M6"><mrow><msub><mi>C</mi><mi>i</mi></msub><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></math> is nucleus precursor density of i th group; β is the effective delayed neutron fraction and usually presented as β eff ; Λ is the mean generation time; λ i is the decay constant of i th group, and β i is delayed neutron fraction of i th group. From these parameters, β i , β eff , and Λ are spectrum and adjoint weighted and may vary from reactor to reactor based on different compositions and spectrums. These parameters - which are known as point kinetic parameters - are very important in the analytical safety analysis of each reactor. The reactivity is directly related to the delayed neutron fraction and determines the prompt criticality. Consequently, an exact calculation of point kinetic parameters is very important in each nuclear reactor design. Most of sophisticated safety analysis codes such as RELAP5, PARCS, DYN3D, and PARET are using these parameters in their dynamic models. Traditionally, the deterministic approach has been employed to evaluate the effective point kinetic parameters. This involve an adjoint and spectrum weighting of the delayed neutron production rate and hence requires a calculation of both forward as well as adjoint fluxes and, on top of that, a suitable post-processing to calculate the weighted production rate [ 2 2 ] . Due to the free availability and versatility of WIMSD and CITATION codes, they have been used widely by developing countries for deterministic calculation of point kinetic parameters. Some efforts have been performed to evaluate the point kinetic parameters in PARR-1 as a 10 MW pool type research reactor [ 3 3 - 5 5 3-5 ] . In these calculations, forward as well as adjoint fluxes were evaluated based on the microscopic cross section approach. WIMSD/4 calculates microscopic cross sections and an auxiliary program, known as BORGES, writes them in the format of the CITATION input requirement. Calculations are in very good agreement with experimental and PARR-1 FSAR data. On the other hand, adjoint calculations are cumbersome in continuous energy Monte Carlo codes [ 6 6 ] . However, in recent years, some techniques have been proposed to evaluate the Monte Carlo based point kinetic parameters by formulations which do not require the adjoint flux. The simplest and first order method in the Monte Carlo evaluation of point kinetic parameters is known as the “Prompt Method” scheme, which has been described by Meulekamp and Van der Marck [ 7 7 ] . The results of this method are in good agreement with our experiment and other adjoint weighted methods and usually is known as an acceptable method in Monte Carlo based delayed neutron calculation [ 8 8 ] . The main objective of this article is dedicated to the calculation of point kinetic parameters in the Tehran Research Reactor (TRR). Deterministic, as well as probabilistic approaches, have been employed and compared with each other. The use of macroscopic cross sections in deterministic calculations and the Monte Carlo "Prompt Method" technique in the probabilistic approach are main innovative aspects in this work. The need of adjoint flux as a weighting function and the meaning of "effective" will be explained in Sect.2. The Tehran Research Reactor by considering different fuel element types will be described in Sect. 3. The deterministic, as well as probabilistic, methodologies in effective point kinetic parameters evaluation will be described in Sect. 4. Results and discussion will be presented in Sect. 5 and finally, the article will be concluded in Sect. 6. 2 Effective delayed neutron parameters 1 Essentially, most fission neutrons appear instantaneously after the fission event. These neutrons are referred to as prompt neutrons. However, a very few fraction of neutrons are appeared with an appreciable delayed time from the subsequent decay of radioactive fission products. Although only a few fractions of fission neutrons are delayed, they are vital for the effective control of the fission chain reaction. Delayed neutrons do not have the same properties as prompt ones released directly from fission. The averaged energy of prompt neutrons is about 2 MeV, which is much greater than the averaged energy of delayed neutrons (about 0.5 MeV). This fact that delayed neutrons are born at lower energies has two significant impacts on the way they precede through the neutron life cycle. Firstly, delayed neutrons have a much lower probability to cause fast fissions in comparison with prompt ones because their averaged energy is less than the threshold required for fast fission. Secondly, delayed neutrons have a lower probability to leak out of the core because they are born at lower energies and subsequently travel shorter distances in comparison with fast neutrons. In other words, the delayed and prompt neutrons have a difference in their effectiveness in producing a subsequent fission event. Since the energy distribution of delayed neutrons is different from group to group, different groups of delayed neutrons have a different effectiveness. To deal with this situation, it is necessary to define an important function, <math id="M7"><mrow><msup><mi>ϕ</mi><mo>†</mo></msup><mrow><mo>(</mo><mrow><mtext>r</mtext><mo>,</mo><mtext>E</mtext></mrow><mo>)</mo></mrow></mrow></math> , which is the probability that a neutron introduced at position r and energy E will ultimately results in fission. Then, the relative importance (to the production of a subsequent fission) of delayed and prompt neutrons in group i of isotope q are <math id="M8"><mrow><msubsup><mi>I</mi><mrow><mi>d</mi><mi>i</mi></mrow><mi>q</mi></msubsup></mrow></math> and <math id="M9"><mrow><msubsup><mi>I</mi><mi>p</mi><mi>q</mi></msubsup></mrow></math> , respectively [ 9 9 ] . <math display="block" id="M10"><mrow><msubsup><mi>I</mi><mrow><mi>d</mi><mi>i</mi></mrow><mi>q</mi></msubsup><mo>=</mo><mo>∫</mo><mi>d</mi><mi>V</mi><munderover><mstyle mathsize="140%" displaystyle="true"><mo>∫</mo></mstyle><mi>o</mi><mi>∞</mi></munderover><mi>d</mi><mi>E</mi><msubsup><mi>χ</mi><mrow><mi>d</mi><mi>i</mi></mrow><mi>q</mi></msubsup><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow><msup><mi>ϕ</mi><mo>†</mo></msup><mrow><mo>(</mo><mrow><mi>r</mi><mo>,</mo><mi>E</mi></mrow><mo>)</mo></mrow><munderover><mstyle mathsize="140%" displaystyle="true"><mo>∫</mo></mstyle><mn>0</mn><mi>∞</mi></munderover><mi>d</mi><msup><mi>E</mi><mo>′</mo></msup><mi>ν</mi><msubsup><mi>σ</mi><mi>f</mi><mi>q</mi></msubsup><mo stretchy="false">(</mo><msup><mi>E</mi><mo>′</mo></msup><mo stretchy="false">)</mo><msub><mi>N</mi><mi>q</mi></msub><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow><mi>ϕ</mi><mrow><mo>(</mo><mrow><mi>r</mi><mo>,</mo><msup><mi>E</mi><mo>′</mo></msup></mrow><mo>)</mo></mrow></mrow></math> <math display="block" id="M11"><mrow><msubsup><mi>I</mi><mi>p</mi><mi>q</mi></msubsup><mo>=</mo><mo>∫</mo><mi>d</mi><mi>V</mi><munderover><mstyle mathsize="140%" displaystyle="true"><mo>∫</mo></mstyle><mi>o</mi><mi>∞</mi></munderover><mi>d</mi><mi>E</mi><msubsup><mi>χ</mi><mi>p</mi><mi>q</mi></msubsup><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow><msup><mi>ϕ</mi><mo>†</mo></msup><mrow><mo>(</mo><mrow><mi>r</mi><mo>,</mo><mi>E</mi></mrow><mo>)</mo></mrow><munderover><mstyle mathsize="140%" displaystyle="true"><mo>∫</mo></mstyle><mn>0</mn><mi>∞</mi></munderover><mi>d</mi><msup><mi>E</mi><mo>′</mo></msup><mi>ν</mi><msubsup><mi>σ</mi><mi>f</mi><mi>q</mi></msubsup><mo stretchy="false">(</mo><msup><mi>E</mi><mo>′</mo></msup><mo stretchy="false">)</mo><msub><mi>N</mi><mi>q</mi></msub><mrow><mo>(</mo><mi>r</mi><mo>)</mo></mrow><mi>ϕ</mi><mrow><mo>(</mo><mrow><mi>r</mi><mo>,</mo><msup><mi>E</mi><mo>′</mo></msup></mrow><mo>)</mo></mrow></mrow></math> The main variables in these equations are as follows: <math id="M12"><mrow><msubsup><mtext>χ</mtext><mrow><mtext>di</mtext></mrow><mtext>q</mtext></msubsup></mrow></math> : Delayed neutron spectrum of the fissionable isotope, q , in group i <math id="M13"><mrow><msubsup><mtext>χ</mtext><mtext>p</mtext><mtext>q</mtext></msubsup></mrow></math> : Prompt neutron spectrum of the fissionable isotope, q <math id="M14"><mrow><msup><mi>ϕ</mi><mo>†</mo></msup></mrow></math> : Adjoint flux distribution <math id="M15"><mtext>ν</mtext></math> : Neutron fission yield <math id="M16"><mrow><msubsup><mtext>σ</mtext><mtext>f</mtext><mtext>q</mtext></msubsup></mrow></math> : Microscopic fission cross section of the fissionable isotope, q <math id="M17"><mrow><msub><mtext>N</mtext><mtext>q</mtext></msub></mrow></math> : Atom density of the fissionable isotope, q <math id="M18"><mrow><msubsup><mtext>I</mtext><mrow><mtext>di</mtext></mrow><mtext>q</mtext></msubsup></mrow></math> : Relative importance of delayed neutrons in group i of isotope q <math id="M19"><mrow><msubsup><mtext>I</mtext><mtext>p</mtext><mtext>q</mtext></msubsup></mrow></math> : Relative importance of prompt neutrons of isotope q For the fissionable isotope, q , the relative effective delayed neutrons yield in the i th group of the delayed neutrons is <math id="M20"><mrow><msubsup><mtext>I</mtext><mrow><mtext>di</mtext></mrow><mtext>q</mtext></msubsup><msubsup><mtext>β</mtext><mtext>i</mtext><mtext>q</mtext></msubsup></mrow></math> . <math id="M21"><mrow><msubsup><mtext>β</mtext><mtext>i</mtext><mtext>q</mtext></msubsup></mrow></math> is the group i of delayed neutrons yield for the fissionable isotope, q . Consequently the effective delayed neutrons fraction of isotope q in group i is <math id="M22"><mrow><mtext> </mtext><mfrac><mrow><msubsup><mtext>I</mtext><mrow><mtext>di</mtext></mrow><mtext>q</mtext></msubsup><msubsup><mtext>β</mtext><mtext>i</mtext><mtext>q</mtext></msubsup></mrow><mrow><msubsup><mtext>I</mtext><mtext>P</mtext><mtext>q</mtext></msubsup></mrow></mfrac></mrow></math> [ 9 9 ] : <math display="block" id="M23"><mrow><msubsup><mi>β</mi><mi>i</mi><mrow><mi>q</mi><mo>−</mo><mstyle mathvariant="bold" mathsize="normal"><mi>e</mi><mi>f</mi><mi>f</mi></mstyle></mrow></msubsup><mo>=</mo><mfrac><mrow><mo>∫</mo><mstyle mathvariant="bold" mathsize="normal"><mi>d</mi></mstyle><mi>V</mi><msubsup><mstyle mathsize="140%" displaystyle="true"><mo>∫</mo></mstyle><mn>0</mn><mi>∞</mi></msubsup><mstyle mathvariant="bold" mathsize="normal"><mi>d</mi></mstyle><mi>E</mi><msup><mi>ϕ</mi><mo>†</mo></msup><mrow><mo>(</mo><mrow><mi>r</mi><mo>,</mo><mi>E</mi></mrow><mo>)</mo></mrow><msubsup><mi>χ</mi><mrow><mstyle mathvariant="bold" mathsize="normal"><mi>d</mi></mstyle><mi>i</mi></mrow><mi>q</mi></msubsup><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow><msubsup><mi>β</mi><mi>i</mi><mi>q</mi></msubsup><msubsup><mstyle mathsize="140%" displaystyle="true"><mo>∫</mo></mstyle><mn>0</mn><mi>∞</mi></msubsup><mi>υ</mi><msubsup><mstyle mathvariant="bold" mathsize="normal"><mi>Σ</mi></mstyle><mstyle mathvariant="bold" mathsize="normal"><mi>f</mi></mstyle><mi>q</mi></msubsup><mrow><mo>(</mo><mrow><mi>r</mi><mo>,</mo><msup><mi>E</mi><mo>′</mo></msup></mrow><mo>)</mo></mrow><mi>ϕ</mi><mrow><mo>(</mo><mrow><mi>r</mi><mo>,</mo><msup><mi>E</mi><mo>′</mo></msup></mrow><mo>)</mo></mrow><mstyle mathvariant="bold" mathsize="normal"><mi>d</mi></mstyle><msup><mi>E</mi><mo>′</mo></msup></mrow><mrow><mo>∫</mo><mstyle mathvariant="bold" mathsize="normal"><mi>d</mi></mstyle><mi>V</mi><msubsup><mstyle mathsize="140%" displaystyle="true"><mo>∫</mo></mstyle><mn>0</mn><mi>∞</mi></msubsup><mstyle mathvariant="bold" mathsize="normal"><mi>d</mi></mstyle><mi>E</mi><msup><mi>ϕ</mi><mo>†</mo></msup><mrow><mo>(</mo><mrow><mi>r</mi><mo>,</mo><mi>E</mi></mrow><mo>)</mo></mrow><msubsup><mi>χ</mi><mi>p</mi><mi>q</mi></msubsup><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow><msubsup><mstyle mathsize="140%" displaystyle="true"><mo>∫</mo></mstyle><mn>0</mn><mi>∞</mi></msubsup><mi>υ</mi><msubsup><mstyle mathvariant="bold" mathsize="normal"><mi>Σ</mi></mstyle><mstyle mathvariant="bold" mathsize="normal"><mi>f</mi></mstyle><mi>q</mi></msubsup><mrow><mo>(</mo><mrow><mi>r</mi><mo>,</mo><msup><mi>E</mi><mtext>"</mtext></msup></mrow><mo>)</mo></mrow><mstyle mathvariant="bold" mathsize="normal"><mi>Φ</mi></mstyle><mrow><mo>(</mo><mrow><mi>r</mi><mo>,</mo><msup><mi>E</mi><mo>′</mo></msup></mrow><mo>)</mo></mrow><mstyle mathvariant="bold" mathsize="normal"><mi>d</mi></mstyle><msup><mi>E</mi><mo>′</mo></msup></mrow></mfrac><mtext> </mtext><mo>.</mo></mrow></math> The discretized format of Eq. (5) is as follow: <math display="block" id="M24"><mrow><msubsup><mi>β</mi><mi>i</mi><mrow><mi>q</mi><mo>−</mo><mstyle mathvariant="bold" mathsize="normal"><mi>e</mi><mi>f</mi><mi>f</mi></mstyle></mrow></msubsup><mo>=</mo><mfrac><mrow><msubsup><mstyle mathsize="140%" displaystyle="true"><mo>∑</mo></mstyle><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><msub><mi>N</mi><mrow><mstyle mathvariant="bold" mathsize="normal"><mi>m</mi><mi>e</mi><mi>s</mi><mi>h</mi></mstyle></mrow></msub></mrow></msubsup><msub><mi>V</mi><mi>n</mi></msub><msubsup><mstyle mathsize="140%" displaystyle="true"><mo>∑</mo></mstyle><mrow><mi>g</mi><mo>=</mo><mn>1</mn></mrow><mi>G</mi></msubsup><msubsup><mi>ϕ</mi><mrow><mi>g</mi><mo>,</mo><mi>n</mi></mrow><mo>†</mo></msubsup><msubsup><mi>χ</mi><mrow><mstyle mathvariant="bold" mathsize="normal"><mi>d</mi></mstyle><mi>i</mi></mrow><mi>g</mi></msubsup><msubsup><mi>β</mi><mi>i</mi><mi>q</mi></msubsup><msubsup><mstyle mathsize="140%" displaystyle="true"><mo>∑</mo></mstyle><mrow><msup><mi>g</mi><mo>′</mo></msup><mo>=</mo><mn>1</mn></mrow><mi>G</mi></msubsup><mi>υ</mi><msub><mi>Σ</mi><mrow><mi>f</mi><mo>,</mo><msup><mi>g</mi><mo>′</mo></msup><mo>,</mo><mi>n</mi></mrow></msub><msub><mi>Φ</mi><mrow><msup><mi>g</mi><mo>′</mo></msup><mo>,</mo><mi>n</mi></mrow></msub></mrow><mrow><msubsup><mstyle mathsize="140%" displaystyle="true"><mo>∑</mo></mstyle><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><msub><mi>N</mi><mrow><mstyle mathvariant="bold" mathsize="normal"><mi>m</mi><mi>e</mi><mi>s</mi><mi>h</mi></mstyle></mrow></msub></mrow></msubsup><msub><mi>V</mi><mi>n</mi></msub><msubsup><mstyle mathsize="140%" displaystyle="true"><mo>∑</mo></mstyle><mrow><mi>g</mi><mo>=</mo><mn>1</mn></mrow><mi>G</mi></msubsup><msubsup><mi>ϕ</mi><mrow><mi>g</mi><mo>,</mo><mi>n</mi></mrow><mo>†</mo></msubsup><msubsup><mi>χ</mi><mi>p</mi><mi>g</mi></msubsup><msubsup><mstyle mathsize="140%" displaystyle="true"><mo>∑</mo></mstyle><mrow><msup><mi>g</mi><mo>′</mo></msup><mo>=</mo><mn>1</mn></mrow><mi>G</mi></msubsup><mi>υ</mi><msub><mi>Σ</mi><mrow><mi>f</mi><mo>,</mo><msup><mi>g</mi><mo>′</mo></msup><mo>,</mo><mi>n</mi></mrow></msub><msub><mi>Φ</mi><mrow><msup><mi>g</mi><mo>′</mo></msup><mo>,</mo><mi>n</mi></mrow></msub></mrow></mfrac><mtext>  </mtext><mo>.</mo></mrow></math> G is the total number of groups and V n is the volume of the n th mesh point. In a mixture of fissionable isotopes, the effective delayed neutron fraction in the i th group ( <math id="M25"><mrow><msubsup><mtext>β</mtext><mtext>i</mtext><mrow><mtext>eff</mtext></mrow></msubsup></mrow></math> ) becomes the average of different <math id="M26"><mrow><msubsup><mtext>β</mtext><mtext>i</mtext><mrow><mtext>q</mtext><mo>−</mo><mtext>eff</mtext></mrow></msubsup></mrow></math> weighted by amount of fission due to each isotope [ 10 10 ] . The total effective delayed neutron fraction will be the sum of these fractions at different groups: <math display="block" id="M27"><mrow><msub><mi>β</mi><mrow><mstyle mathvariant="bold" mathsize="normal"><mi>e</mi><mi>f</mi><mi>f</mi></mstyle></mrow></msub><mo>=</mo><munderover><mstyle mathsize="140%" displaystyle="true"><mo>∑</mo></mstyle><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mn>6</mn></munderover><msubsup><mi>β</mi><mi>i</mi><mrow><mstyle mathvariant="bold" mathsize="normal"><mi>e</mi><mi>f</mi><mi>f</mi></mstyle></mrow></msubsup><mo>.</mo></mrow></math> Such a manner should be repeated for mean generation time calculation, and finally, the following formula will be obtained [ 10 10 ] : <math display="block" id="M28"><mrow><mstyle mathvariant="bold" mathsize="normal"><mi>Λ</mi></mstyle><mo>=</mo><mfrac><mrow><mo>∫</mo><mstyle mathvariant="bold" mathsize="normal"><mi>d</mi></mstyle><mi>V</mi><msubsup><mstyle mathsize="140%" displaystyle="true"><mo>∫</mo></mstyle><mn>0</mn><mi>∞</mi></msubsup><mstyle mathvariant="bold" mathsize="normal"><mi>d</mi></mstyle><mi>E</mi><msup><mi>ϕ</mi><mo>†</mo></msup><mrow><mo>(</mo><mrow><mi>r</mi><mo>,</mo><mi>E</mi></mrow><mo>)</mo></mrow><mrow><mo>[</mo><mrow><mfrac><mn>1</mn><mrow><mi>v</mi><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow></mrow></mfrac></mrow><mo>]</mo></mrow><mi>ϕ</mi><mrow><mo>(</mo><mrow><mi>r</mi><mo>,</mo><mi>E</mi></mrow><mo>)</mo></mrow></mrow><mrow><mo>∫</mo><mstyle mathvariant="bold" mathsize="normal"><mi>d</mi></mstyle><mi>V</mi><msubsup><mstyle mathsize="140%" displaystyle="true"><mo>∫</mo></mstyle><mn>0</mn><mi>∞</mi></msubsup><mstyle mathvariant="bold" mathsize="normal"><mi>d</mi></mstyle><mi>E</mi><msup><mi>ϕ</mi><mo>†</mo></msup><mrow><mo>(</mo><mrow><mi>r</mi><mo>,</mo><mi>E</mi></mrow><mo>)</mo></mrow><msub><mstyle mathvariant="bold" mathsize="normal"><mi>χ</mi></mstyle><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle></msub><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow><msubsup><mstyle mathsize="140%" displaystyle="true"><mo>∫</mo></mstyle><mn>0</mn><mi>∞</mi></msubsup><mstyle mathvariant="bold" mathsize="normal"><mi>υ</mi></mstyle><msub><mstyle mathvariant="bold" mathsize="normal"><mi>Σ</mi></mstyle><mstyle mathvariant="bold" mathsize="normal"><mi>f</mi></mstyle></msub><mrow><mo>(</mo><mrow><mi>r</mi><mo>,</mo><msup><mi>E</mi><mtext>"</mtext></msup></mrow><mo>)</mo></mrow><mstyle mathvariant="bold" mathsize="normal"><mi>Φ</mi></mstyle><mrow><mo>(</mo><mrow><mi>r</mi><mo>,</mo><msup><mi>E</mi><mo>′</mo></msup></mrow><mo>)</mo></mrow><mstyle mathvariant="bold" mathsize="normal"><mi>d</mi></mstyle><msup><mi>E</mi><mo>′</mo></msup></mrow></mfrac></mrow></math> The discretized format of Eq. (8) is as follow: <math display="block" id="M29"><mrow><mstyle mathvariant="bold" mathsize="normal"><mi>Λ</mi></mstyle><mo>=</mo><mfrac><mrow><msubsup><mstyle mathsize="140%" displaystyle="true"><mo>∑</mo></mstyle><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><msub><mi>N</mi><mrow><mstyle mathvariant="bold" mathsize="normal"><mi>m</mi><mi>e</mi><mi>s</mi><mi>h</mi></mstyle></mrow></msub></mrow></msubsup><msub><mi>V</mi><mi>n</mi></msub><msubsup><mstyle mathsize="140%" displaystyle="true"><mo>∑</mo></mstyle><mrow><mi>g</mi><mo>=</mo><mn>1</mn></mrow><mi>G</mi></msubsup><msubsup><mi>ϕ</mi><mrow><mi>g</mi><mo>,</mo><mi>n</mi></mrow><mo>†</mo></msubsup><mrow><mo>[</mo><mrow><mfrac><mn>1</mn><mrow><msup><mi>v</mi><mi>g</mi></msup></mrow></mfrac></mrow><mo>]</mo></mrow><msub><mi>ϕ</mi><mrow><mi>g</mi><mo>,</mo><mi>n</mi></mrow></msub></mrow><mrow><msubsup><mstyle mathsize="140%" displaystyle="true"><mo>∑</mo></mstyle><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><msub><mi>N</mi><mrow><mstyle mathvariant="bold" mathsize="normal"><mi>m</mi><mi>e</mi><mi>s</mi><mi>h</mi></mstyle></mrow></msub></mrow></msubsup><msub><mi>V</mi><mi>n</mi></msub><msubsup><mstyle mathsize="140%" displaystyle="true"><mo>∑</mo></mstyle><mrow><mi>g</mi><mo>=</mo><mn>1</mn></mrow><mi>G</mi></msubsup><msubsup><mi>ϕ</mi><mrow><mi>g</mi><mo>,</mo><mi>n</mi></mrow><mo>†</mo></msubsup><msubsup><mi>χ</mi><mi>p</mi><mi>g</mi></msubsup><msubsup><mstyle mathsize="140%" displaystyle="true"><mo>∑</mo></mstyle><mrow><msup><mi>g</mi><mo>′</mo></msup><mo>=</mo><mn>1</mn></mrow><mi>G</mi></msubsup><mi>υ</mi><msub><mstyle mathvariant="bold" mathsize="normal"><mi>Σ</mi></mstyle><mrow><mstyle mathvariant="bold" mathsize="normal"><mi>f</mi></mstyle><mo>,</mo><mstyle mathvariant="bold" mathsize="normal"><msup><mi>g</mi><mo>′</mo></msup></mstyle><mo>,</mo><mstyle mathvariant="bold" mathsize="normal"><mi>n</mi></mstyle></mrow></msub><msub><mstyle mathvariant="bold" mathsize="normal"><mi>Φ</mi></mstyle><mrow><mstyle mathvariant="bold" mathsize="normal"><msup><mi>g</mi><mo>′</mo></msup></mstyle><mo>,</mo><mstyle mathvariant="bold" mathsize="normal"><mi>n</mi></mstyle></mrow></msub></mrow></mfrac><mo>⋅</mo><mtext> </mtext></mrow></math> 3 Tehran research reactor 1 The Tehran Research Reactor (TRR) is a 5 megawatt pool-type light-water moderated, heterogeneous solid fuel reactor in which the water is also used for cooling and shielding [ 11 11 ] . The TRR core is immersed in either section of a two-section, concrete pool filled with water. The utilization of the reactor is essentially for research, training, and production of radioisotopes [ 11 11 ] . The reactor core is composed of MTR-type fuel assemblies inserted in the grid plate. The assemblies may be arranged in a variety of lattice patterns depending on the experimental requirements. The main characteristics of the TRR core have been shown in Table 1 Table 1 . A series of fifty-four holes, capable of accommodating the end fittings of fuels, are arranged in a 9 × 6 rectangular lattice. The first core configuration is a 19 fuel element arrangement reflected by water, as shown in Fig.1 Fig.1 . The original core of TRR was High Enriched Uranium (HEU) fuels. Upon procurement of the next cores, Low Enriched Uranium (LEU) fuels had to be considered [ 11 11 ] . There are two types of LEU fuel elements named as the Standard Fuel Element (SFE) and Control Fuel Element (CFE). Each fuel type and the core physical characteristics are described in following subsections. 1 1 Fuel elements: 1 1   1 1 U 235 per Standard Fuel Element (SFE) 1 1 214 gr 1 1 U 235 per Control Fuel Element (CFE) 1 1 76 gr 1 1 U per fuel plate 1 1 290 gr 1 1 Number of plates per fuel element 1 1 19 for SFE 1 1   1 1 14 for CFE 1 1 Meat: 1 1 Enriched U 3 O 8 1 1  20% in weight of U 235 1 1 U density 1 1 2.9617 gr/cm 3 1 1 Meat density 1 1 4.76 gr/cm 3 1 1 Void fraction 1 1 10.0% 1 1 Weight percentage 1 1 U 235 12.45%, U 238 49.78%, O 11.18%, Al 26.59% 1 1 Aluminum meat 1 1 Purity 99.6% 1 1   1 1 Density=2.7 gr/cm 3 1 1 Shim and safety rods absorber 1 1 Four Ag-In-Cd Alloy (80%,15%,5% in weight respectively)Density 10.17 gr/cm 3 1 1 Control rods’ cladding material 1 1 AISI-316/L stainless steel 1 1   1 1 Density= 7.95 gr/cm 3 1 1 Regulating rod 1 1 One AISI-316/L stainless steelDensity= 7.95 gr/cm 3 1 1 Grid plate 1 1 Grid array X-Y 1 1   1 1 Pitch: 7.71×8.1 cm 1 1 Coolant/Moderator 1 1 Light water/ Light water 1 1 Reflectors 1 1 Light water/Graphite 3.1 Standard fuel element (SFE) 2 The SFE is 20% enriched in weight of 235 U and has 19 flat fuel plates inserted in two grooved side plates (lateral walls), as shown in Fig.2 Fig.2 and Fig.3 Fig.3 . The meat is made of U 3 O 8 powder dispersed in a pure aluminum matrix. Each fuel plate is made up of fuel meat and cladding which seals it off hermetically while isolating it from the coolant. Cladding consists of a frame and two covers in an annealed aluminum Al-6061 alloy. The gap between the fuel plates and vertical through hole in the end fitting assures proper cooling water flow during the operation. Side plates and external plates are fastened to the end fitting at the assembly's lower end by means of TIG welding. The main and geometric data of SFE is summarized in Table 2 Table 2 . 1 1 Enrichment 1 1 20 % 1 1 Number of fuel plates 1 1 19 1 1 SFE dimensions 1 1 8.01×7.6×91.8 cm 1 1 Plate thickness 1 1 0.15 cm 1 1 Clad thickness 1 1 0.04 cm 1 1 Water channel thickness 1 1 0.27 cm 1 1 Meat thickness 1 1 0.07 cm 1 1 Meat width 1 1 6.0 cm 1 1 Meat length 1 1 61.5 1 1 Lateral wall thickness (side plate) 1 1 0.45 cm 1 1 Inlet/exit channel entrance length 1 1 4.55 cm 1 1 Inner diameter of inlet/exit Nozzle 1 1 5.30 cm 1 1 Outer diameter of inlet/exit Nozzle 1 1 6.16 cm 1 1 Coolant flow area 1 1 33.92 cm 2 1 1 Heat transfer area 1 1 14022.0 cm 2 1 1 Meat material 1 1 U 3 O 8 -Al 1 1 Fuel plate cladding and side wall material 1 1 Al-6061 3.2 Control fuel element (CFE) 2 The control fuel element (CFE) has 14 flat fuel plates and accommodates fork-type control rods in a lateral position of fuel assembly as shown in Fig. 4 Fig. 4 and Fig. 5 Fig. 5 . Positioning four shim safety rods and one regulating rod into the reactor core controls the reactor. The shim safety control rod is composed of two reactivity control plates (absorbing plates) and knuckle sub-assembly. The absorber plate is an alloy of Silver, Indium, and Cadmium (80%, 15%, and 5% wt. respectively), while the regulating rod is made of stainless steel. All absorbing rods are fork type. Shim rods facilitate, the start-up and operation of the TRR and ensure safe shut down of the reactor at any moment, while the regulating rod is used for fine reactivity insertion either automatically or manually. Control rods are dropped into the core upon receiving a trip signal from various safety channels. The main and geometrical data of the CFE is summarized in Table 3 Table 3 . 1 1 Enrichment 1 1 20 % 1 1 Number of fuel plates 1 1 14 1 1 CFE dimension 1 1 8.01×7.6×161.4 cm 1 1 Plate thickness 1 1 0.15 cm 1 1 Clad thickness 1 1 0.04 cm 1 1 Water channel thickness 1 1 0.27 cm 1 1 Meat thickness 1 1 0.07 cm 1 1 Meat width 1 1 6.0 cm 1 1 Meat length 1 1 61.5 1 1 Lateral wall thickness (side plate) 1 1 0.45 cm 1 1 Inlet/exit channel entrance length 1 1 4.55 cm 1 1 Inner diameter of inlet/exit Nozzle 1 1 5.30 cm 1 1 Outer diameter of inlet/exit Nozzle 1 1 6.16 cm 1 1 Coolant flow area 1 1 25.81 cm 2 1 1 Heat transfer area 1 1 10332 cm 2 1 1 Fuel plate cladding and side walls material 1 1 Al-6061 1 1 Meat material 1 1 U 3 O 8 -Al 1 1 Fuel plate cladding and side walls material 1 1 Al-6061 1 1 Absorber material for shim safety rods 1 1 Ag-In-Cd 1 1 Absorber material for fine regulating rod 1 1 AISI-316L SS 4 Calculation methodologies 1 4.1 Deterministic approach 2 Deterministic calculations of point kinetic parameters in the TRR have been performed using WIMS-D5B [ 12 12 ] and CITATION-LDI2 [ 13 13 ] codes for cell and core calculations, respectively. In order to distinguish between prompt and delayed neutron spectrums, the total number of energy groups in these calculations should be more than the usual two groups' structure. Since in later one fractions of neutrons, which are born in fast and thermal groups, are one and zero, respectively. In other words, in the two groups calculation, all of the neutrons (either prompt or delayed) are born in fast group, and hence, it's not possible to distinguish between prompt and delayed neutrons. The six groups' structure has been considered for cell calculation. The bounds of each energy group - based on the 69 groups' structure of WIMS library - as well as the prompt fission spectrum are shown in Table 4 Table 4 . First two groups are based on the fast group structure of the WIMS library. The first group is above the threshold fission of 238 U and the second one is considered as a reminder. Third and fourth groups are considered for the resonance region of the library. And finally, the last two groups are based on the thermal group structure. 1 1 Energy Group 1 1 Energy boundariesbased on WIMS library 1 1 Prompt fission spectrum 1 1 1 1 1 10.000 Mev - 2.231 Mev 1 1 3.564777E-01 1 1 2 1 1 2.231 Mev - 9.118 kev 1 1 6.431600E-01 1 1 3 1 1 9.118 kev - 906.898 ev 1 1 3.488916E-04 1 1 4 1 1 906.00 ev - 4.0000 ev 1 1 1.339680E-05 1 1 5 1 1 4.0000 ev - 0.0250 ev 1 1 0.000000E+00 1 1 6 1 1 0.02500 ev - 0.0000 ev 1 1 0.000000E+00 If one assumes a Maxwellian distribution, the spectrum of delayed neutrons ( <math id="M30"><mrow><msub><mi>χ</mi><mstyle mathvariant="bold" mathsize="normal"><mi>d</mi></mstyle></msub></mrow></math> ) is calculated by the following expression [ 14 14 ] : <math display="block" id="M31"><mrow><msubsup><mi>χ</mi><mrow><mstyle mathvariant="bold" mathsize="normal"><mi>d</mi></mstyle><mi>i</mi></mrow><mi>g</mi></msubsup><mo>=</mo><munderover><mstyle mathsize="140%" displaystyle="true"><mo>∫</mo></mstyle><mrow><msub><mi>G</mi><mn>1</mn></msub></mrow><mrow><msub><mi>G</mi><mn>2</mn></msub></mrow></munderover><mn>2.07296489</mn><msup><mrow><mrow><mo>(</mo><mover accent="true"><mi>E</mi><mo>¯</mo></mover><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>1.5</mn></mrow></msup><msqrt><mi>E</mi></msqrt><msup><mi>e</mi><mrow><mrow><mo>(</mo><mrow><mo>−</mo><mfrac bevelled="true"><mrow><mn>1.5</mn><mi>E</mi></mrow><mover accent="true"><mi>E</mi><mo>¯</mo></mover></mfrac></mrow><mo>)</mo></mrow></mrow></msup><mstyle mathvariant="bold" mathsize="normal"><mi>d</mi></mstyle><mi>E</mi><mo>,</mo></mrow></math> where <math id="M32"><mrow><msubsup><mi>χ</mi><mrow><mtext>d</mtext><mi>i</mi></mrow><mi>g</mi></msubsup></mrow></math> is the fraction of the i th group of delayed neutrons in the g th energy group, <math id="M33"><mrow><mover accent="true"><mi>E</mi><mo>¯</mo></mover><mtext> </mtext></mrow></math> is the Mean energy of the i th group of delayed neutrons, and G 1, G 2 are the energy boundaries of the g th energy group. The delayed neutron spectrum of each group has been calculated and shown in Table 5 Table 5 . The mean energy of each delayed neutron group has been taken from Ref.[ 14 14 ]. On the other hand, bounding energies are based on six groups' structure in cell calculation and taken from Table 4 Table 4 . Also, delayed neutron parameters of 235 U and 238 U have been presented in Table 6 Table 6 and Table 7 Table 7 , respectively. 1 1 Family number 1 1 <math id="M34"><mover accent="true"><mi>E</mi><mo>¯</mo></mover></math> (MeV) [ 14 14 ] 1 1 Energy group 1 1 Spectrum 1 1 1 1 1 0.25 1 1 1 1 1 6.5722e-06 1 1   1 1   1 1 2 1 1 9.9070e-01 1 1   1 1   1 1 3 1 1 9.0000e-03 1 1   1 1   1 1 4 1 1 3.0087e-04 1 1   1 1   1 1 5 1 1 8.8402e-08 1 1   1 1   1 1 6 1 1 4.3702e-11 1 1   2 1 1   0.46 1 1 1 1 1 7.6000e-03 1 1   1 1   1 1 2 1 1 9.8960e-01 1 1   1 1   1 1 3 1 1 2.7000e-03 1 1   1 1   1 1 4 1 1 8.9908e-05 1 1   1 1   1 1 5 1 1 2.6369e-08 1 1   1 1   1 1 6 1 1 1.3036e-11 1 1   3 1 1   0.405 1 1 1 1 1 1.4000e-03 1 1   1 1   1 1 2 1 1 9.9440e-01 1 1   1 1   1 1 3 1 1 4.1000e-03 1 1   1 1   1 1 4 1 1 1.3356e-04 1 1   1 1   1 1 5 1 1 3.9189e-08 1 1   1 1   1 1 6 1 1 1.9373e-11 1 1   4 1 1   0.45 1 1 1 1 1 1.2900e-02 1 1   1 1   1 1 2 1 1 9.8470e-01 1 1   1 1   1 1 3 1 1 2.4000e-03 1 1   1 1   1 1 4 1 1 7.7188e-05 1 1   1 1   1 1 5 1 1 2.2635e-08 1 1   1 1   1 1 6 1 1 1.1190e-11 1 1   5 1 1   0.42 1 1 1 1 1 1.2000e-03 1 1   1 1   1 1 2 1 1 9.9450e-01 1 1   1 1   1 1 3 1 1 4.2000e-03 1 1   1 1   1 1 4 1 1 1.3835e-04 1 1   1 1   1 1 5 1 1 4.0597e-08 1 1   1 1   1 1 6 1 1 2.0070e-11 1 1   6 1 1   0.18 1 1 1 1 1 4.2082e-08 1 1   1 1   1 1 2 1 1 9.8490e-01 1 1   1 1   1 1 3 1 1 1.4600e-02 1 1   1 1   1 1 4 1 1 4.9185e-04 1 1   1 1   1 1 5 1 1 1.4470e-07 1 1   1 1   1 1 6 1 1 7.1532e-11 1 1 Delayed neutron group 2 1 Fast Neutrons 2 1 Thermal Neutrons 1 1   1 1 Decay constant λ i (S -1 ) 1 1 Delayed neutron yield ( β =0.0064) 1 1 Decay constant λ i (S -1 ) 1 1 Delayed neutron yield ( β =0.0067) 1 1 1 1 1 0.0127 1 1 0.0002432 1 1 0.0124 1 1 0.0002211 1 1 2 1 1 0.0317 1 1 0.0013632 1 1 0.0305 1 1 0.0014673 1 1 3 1 1 0.115 1 1 0.0012032 1 1 0.111 1 1 0.0013132 1 1 4 1 1 0.311 1 1 0.0026048 1 1 0.301 1 1 0.0026465 1 1 5 1 1 1.40 1 1 0.0008192 1 1 1.14 1 1 0.0007705 1 1 6 1 1 3.87 1 1 0.0001664 1 1 3.01 1 1 0.0002814 1 1 Delayed Neutron Group 2 1 Fast Neutrons 1 1   1 1 Decay Constant λ i (S -1 ) 1 1 Delayed Neutron Yield ( β  =0.0164) 1 1 1 1 1 0.0132 1 1 0.0002132 1 1 2 1 1 0.0321 1 1 0.0022468 1 1 3 1 1 0.139 1 1 0.0026568 1 1 4 1 1 0.358 1 1 0.0063632 1 1 5 1 1 1.41 1 1 0.00369 1 1 6 1 1 4.02 1 1 0.00123 Since the BORGES auxiliary program was not available for authors, a macroscopic cross section approach has been selected for criticality calculations. In this technique, macroscopic cross sections are calculated using the WIMS code with resonance shielding being normally automatically calculated [ 12 12 ] . These macroscopic cross sections will be used in the "008" section of CITATION code. WIMS input files have been prepared for different basic cells including the SFE and CFE fuel plates, Absorber plate, and reflectors (Top, Bottom, and Lateral) based on the six groups of energy structure. Macroscopic cross sections have been extracted from the WIMS output files and implemented in the whole core CITATION model. Important neutronic parameters, including effective multiplication factor as well as forward and adjoint flux distributions, were calculated. Each fuel assembly has been divided into 21 axial mesh points and 5×5 mesh points in the X - Y plane. The first five mesh points and the last five mesh points in axial direction are reserved for top and bottom reflectors, respectively. The reminded 11 mesh points are allocated to active core height. 4.2 Probabilistic approach 2 In the probabilistic approach, one can rewrite the Eq. (5) as follow [ 8 8 ] : <math display="block" id="M35"><mrow><msub><mi>β</mi><mrow><mi>e</mi><mi>f</mi><mi>f</mi></mrow></msub><mo>=</mo><mfrac><mrow><msup><mi>ϕ</mi><mo>†</mo></msup><msub><mover accent="true"><mi>F</mi><mo>^</mo></mover><mi>d</mi></msub><mi>Φ</mi></mrow><mrow><msup><mi>Φ</mi><mo>†</mo></msup><mover accent="true"><mi>F</mi><mo>^</mo></mover><mi>ϕ</mi></mrow></mfrac><mo>,</mo></mrow></math> where <math id="M36"><mover accent="true"><mi>F</mi><mo>^</mo></mover></math> is the creation operator that takes in to account all neutrons (prompt and delayed) created in the phase space and <math id="M37"><mrow><msub><mover accent="true"><mi>F</mi><mo>^</mo></mover><mi>d</mi></msub></mrow></math> is the delayed neutron creation operator that takes into account only delayed neutrons. Brackets indicate the integration over the whole phase space. Once again <math id="M38"><mi>ϕ</mi></math> and <math id="M39"><mrow><msup><mi>ϕ</mi><mo>†</mo></msup></mrow></math> are forward and adjoint fluxes, respectively, and obtained from eigenvalue equations [ 8 8 ] : <math display="block" id="M40"><mrow><mover accent="true"><mi>M</mi><mo>^</mo></mover><mi>ϕ</mi><mo>=</mo><mfrac><mn>1</mn><mrow><msub><mi>k</mi><mrow><mstyle mathvariant="bold" mathsize="normal"><mi>e</mi><mi>f</mi><mi>f</mi></mstyle></mrow></msub></mrow></mfrac><mover accent="true"><mi>F</mi><mo>^</mo></mover><mi>ϕ</mi><mo>,</mo></mrow></math> <math display="block" id="M41"><mrow><mover accent="true"><mrow><msup><mi>M</mi><mo>†</mo></msup></mrow><mo stretchy="true">^</mo></mover><msup><mi>ϕ</mi><mo>†</mo></msup><mo>=</mo><mfrac><mn>1</mn><mrow><msub><mi>k</mi><mrow><mstyle mathvariant="bold" mathsize="normal"><mi>e</mi><mi>f</mi><mi>f</mi></mstyle></mrow></msub></mrow></mfrac><msup><mi>F</mi><mo>†</mo></msup><msup><mi>ϕ</mi><mo>†</mo></msup><mo>,</mo></mrow></math> where <math id="M42"><mover accent="true"><mi>M</mi><mo>^</mo></mover></math> is loses operator which takes in to account all neutrons leaving the phase space (for example capture or out-scattering) and <math id="M43"><mrow><mover accent="true"><mrow><msup><mi>M</mi><mo>†</mo></msup></mrow><mo stretchy="true">^</mo></mover></mrow></math> has the same meaning for adjoints. Some techniques have been proposed to evaluate Eq. (11) based on the Monte Carlo method. The most straightforward method is based on the assumption that <math id="M44"><mrow><mo>≃</mo><msub><mi>ϕ</mi><mtext>p</mtext></msub></mrow></math> , which means the total flux is approximated by the prompt one. However, this assumption is not crucial since more than 99% of all neutrons are the prompt one. By this assumption, the following eigenvalue equations are obtained instead of Eqs (12) and (13) [ 8 8 ] : <math display="block" id="M45"><mrow><mover accent="true"><mi>M</mi><mo>^</mo></mover><msub><mi>ϕ</mi><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle></msub><mo>=</mo><mfrac><mn>1</mn><mrow><msub><mi>k</mi><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle></msub></mrow></mfrac><msub><mover accent="true"><mi>F</mi><mo>^</mo></mover><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle></msub><msub><mi>ϕ</mi><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle></msub><mo>,</mo></mrow></math> <math display="block" id="M46"><mrow><mover accent="true"><mrow><msup><mi>M</mi><mo>†</mo></msup></mrow><mo stretchy="true">^</mo></mover><msubsup><mi>ϕ</mi><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mo>†</mo></msubsup><mo>=</mo><mfrac><mn>1</mn><mrow><msub><mi>k</mi><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle></msub></mrow></mfrac><msubsup><mover accent="true"><mi>F</mi><mo>^</mo></mover><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mo>†</mo></msubsup><msubsup><mi>ϕ</mi><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle><mo>†</mo></msubsup><mo>,</mo></mrow></math> where <math id="M47"><mrow><msub><mover accent="true"><mi>F</mi><mo>^</mo></mover><mtext>p</mtext></msub></mrow></math> is the prompt neutron creation operator. The creation operator <math id="M48"><mover accent="true"><mi>F</mi><mo>^</mo></mover></math> can be decomposed into the sum of prompt and delayed ones: <math id="M49"><mrow><mo> </mo><mover accent="true"><mi>F</mi><mo>^</mo></mover><mo>=</mo><msub><mover accent="true"><mi>F</mi><mo>^</mo></mover><mtext>p</mtext></msub><mo>+</mo><msub><mover accent="true"><mi>F</mi><mo>^</mo></mover><mtext>d</mtext></msub></mrow></math> . By these definitions the Eq. (10) is written as follows [ 8 8 ] : <math display="block" id="M50"><mtable columnalign="left"><mtr><mtd><msub><mi>β</mi><mrow><mstyle mathvariant="bold" mathsize="normal"><mi>e</mi><mi>f</mi><mi>f</mi></mstyle></mrow></msub><mo>=</mo><mfrac><mrow><mrow><mo>〈</mo><mrow><msup><mi>ϕ</mi><mo>†</mo></msup><msub><mover accent="true"><mi>F</mi><mo>^</mo></mover><mstyle mathvariant="bold" mathsize="normal"><mi>d</mi></mstyle></msub><mi>Φ</mi></mrow><mo>〉</mo></mrow></mrow><mrow><mrow><mo>〈</mo><mrow><msup><mi>Φ</mi><mo>†</mo></msup><mover accent="true"><mi>F</mi><mo>^</mo></mover><mi>ϕ</mi></mrow><mo>〉</mo></mrow></mrow></mfrac><mo>=</mo><mfrac><mrow><mrow><mo>〈</mo><mrow><msup><mi>ϕ</mi><mo>†</mo></msup><mo stretchy="false">(</mo><mover accent="true"><mi>F</mi><mo>^</mo></mover><mo>−</mo><msub><mover accent="true"><mi>F</mi><mo>^</mo></mover><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle></msub><mo stretchy="false">)</mo><mi>Φ</mi></mrow><mo>〉</mo></mrow></mrow><mrow><mrow><mo>〈</mo><mrow><msup><mi>Φ</mi><mo>†</mo></msup><mover accent="true"><mi>F</mi><mo>^</mo></mover><mi>ϕ</mi></mrow><mo>〉</mo></mrow></mrow></mfrac></mtd></mtr><mtr><mtd><mo>=</mo><mn>1</mn><mo>−</mo><mfrac><mrow><mrow><mo>〈</mo><mrow><msup><mi>ϕ</mi><mo>†</mo></msup><msub><mover accent="true"><mi>F</mi><mo>^</mo></mover><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle></msub><mi>Φ</mi></mrow><mo>〉</mo></mrow></mrow><mrow><mrow><mo>〈</mo><mrow><msup><mi>Φ</mi><mo>†</mo></msup><mover accent="true"><mi>F</mi><mo>^</mo></mover><mi>ϕ</mi></mrow><mo>〉</mo></mrow></mrow></mfrac><mo>≃</mo><mn>1</mn><mo>−</mo><mfrac><mrow><mrow><mo>〈</mo><mrow><msup><mi>ϕ</mi><mo>†</mo></msup><msub><mi>k</mi><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle></msub><mover accent="true"><mi>M</mi><mo>^</mo></mover><msub><mi>Φ</mi><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle></msub></mrow><mo>〉</mo></mrow></mrow><mrow><mrow><mo>〈</mo><mrow><msup><mi>Φ</mi><mo>†</mo></msup><msub><mi>k</mi><mrow><mstyle mathvariant="bold" mathsize="normal"><mi>e</mi><mi>f</mi><mi>f</mi></mstyle></mrow></msub><mover accent="true"><mi>M</mi><mo>^</mo></mover><msub><mi>Φ</mi><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle></msub></mrow><mo>〉</mo></mrow></mrow></mfrac></mtd></mtr><mtr><mtd><mo>=</mo><mn>1</mn><mo>−</mo><mfrac><mrow><msub><mi>k</mi><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle></msub><mrow><mo>〈</mo><mrow><msup><mi>ϕ</mi><mo>†</mo></msup><mover accent="true"><mi>M</mi><mo>^</mo></mover><msub><mi>Φ</mi><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle></msub></mrow><mo>〉</mo></mrow></mrow><mrow><msub><mi>k</mi><mrow><mstyle mathvariant="bold" mathsize="normal"><mi>e</mi><mi>f</mi><mi>f</mi></mstyle></mrow></msub><mrow><mo>〈</mo><mrow><msup><mi>Φ</mi><mo>†</mo></msup><mover accent="true"><mi>M</mi><mo>^</mo></mover><msub><mi>Φ</mi><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle></msub></mrow><mo>〉</mo></mrow></mrow></mfrac><mo>=</mo><mn>1</mn><mo>−</mo><mfrac><mrow><msub><mi>k</mi><mstyle mathvariant="bold" mathsize="normal"><mi>p</mi></mstyle></msub></mrow><mrow><msub><mi>k</mi><mrow><mstyle mathvariant="bold" mathsize="normal"><mi>e</mi><mi>f</mi><mi>f</mi></mstyle></mrow></msub></mrow></mfrac><mo>.</mo></mtd></mtr></mtable></math> This technique is known as the "Prompt Method", and its results usually are accepted as a first approximation in Monte Carlo based point kinetic parameters evaluation [ 15 15 ] . However, the “Prompt Method” has a weak theoretical justification since it does not use the adjoint weighted parameter in the calculation of <math id="M51"><mrow><msub><mtext>β</mtext><mrow><mtext>eff</mtext></mrow></msub></mrow></math> [ 8 8 ] . 5 Results and discussion 1 At first, cell and core calculations, as well as six groups of energy structure, have been verified via neutronic calculations of the TRR core at different states. The SAR report and MCNP4C probabilistic code have been employed for verification. The deterministic calculation of excess reactivity at each operational state is compared with SAR and cross checked with the MCNP calculation in Table 8 Table 8 . Deterministic excess reactivity calculations in the six groups are in very good agreement with SAR and MCNP results. In the deterministic approach, the maximum relative error is about 6.3% with respect to the SAR report. On the other hand, maximum relative error in the probabilistic approach is only 1.7%. These results confirm input files as well as six groups of energy structure, and they can be used in point kinetic parameters calculation, confidently. R.E: Relative Error 1 1 Case 1 1 Core State 1 1 SAR [ 11 11 ] 1 1 Six GroupWIMS & CITATION(R.E %) 1 1 MCNP (R.E %) 1 1 1 1 1 Cold 1 1 6916 1 1 6481(6.3) 1 1 6934(0.26) 1 1 2 1 1 0 MW 1 1 6549 1 1 6398(2.3) 1 1 6647(1.5) 1 1 3 1 1 5 MW (without Xe) 1 1 6469 1 1 6316(2.4) 1 1 6577(1.7) Equation (6) has been employed in the deterministic approach to evaluate effective delayed neutron fractions. Forward and adjoint fluxes of each mesh point have been extracted from the CITATION output file and implemented in this equation. As was mentioned earlier, the total delayed neutron fraction ( <math id="M52"><mrow><msub><mtext>β</mtext><mrow><mtext>eff</mtext></mrow></msub></mrow></math> ) depends on the core isotopic content, especially relative to 235 U and with minor contributions from fission of 238 U. WIMS results showed that 99.5% of all fissions take place in 235 U whilst the fast fission of 238 U is responsible for only 0.5% of fissions. The MCNP4C code [ 16 16 ] is employed in the probabilistic approach to evaluate the effective delayed neutron fraction based on the “Prompt Method” scheme. In this method, the MCNP results of the effective multiplication factor have been calculated in two cases. In the first one, the effective multiplication factor is calculated by considering the TOTNU card of the MCNP code with No as the entry ( K P ). Once again, the effective multiplication factor is calculated without the TOTNU card ( K eff ). In this regards, the prompt effective delayed neutron fraction is calculated based on Eq. (16). Calculated values of the effective delayed neutron fractions have been shown in Table 9 Table 9 . The “Prompt Method”, as well as SAR results and experimental data, have been employed for validation. NA: Not Available 1 1 Groups 1 1 Six GroupWIMS & CITATION 1 1 Prompt Method 1 1 SAR [ 11 11 ] 1 1 Experiment [ 17 17 ] 1 1 1 1 1 0.00024  1 1 NA 1 1 NA 1 1 NA 1 1 2 1 1 0.00158  1 1 NA 1 1 NA 1 1 NA 1 1 3 1 1 0.00142  1 1 NA 1 1 NA 1 1 NA 1 1 4 1 1 0.00286  1 1 NA 1 1 NA 1 1 NA 1 1 5 1 1 0.00085  1 1 NA 1 1 NA 1 1 NA 1 1 6 1 1 0.00031  1 1 NA 1 1 NA 1 1 NA 1 1 Total 1 1 0.00726 1 1 0.00761 1 1 0.00813 1 1 0.00786 Deterministic, as well as probabilistic, calculations of the total effective delayed neutron fraction are in good agreement with the experiment and SAR report. In comparison with the experimental data, the relative differences of deterministic as well as probabilistic methods are 7.6% and 3.2%, respectively. These quantities are 10.7% and 6.4% respectively in comparison with the SAR report. On the other hand, the deterministic approach has an advantage to provide the delayed neutron fraction in each group beside the total one. The group wise delayed neutron fraction is required in the computational model of some sophisticated dynamic codes, such as PARCS and PARET [ 18 18 ] . The mean generation time ( <math id="M53"><mtext>Λ</mtext></math> ) and consequently the prompt neutron life time ( <math id="M54"><mrow><mi mathvariant="script">l</mi><mo>=</mo><mtext>Λ</mtext><mi>k</mi></mrow></math> ) has been evaluated using Eq. (9). According to this equation, inverse velocity in each energy group is required to calculate the mean generation time. In the WIMS code, the inverse velocity calculation is activated by including the ( <math id="M55"><mrow><mfrac bevelled="true"><mn>1</mn><mi>ν</mi></mfrac></mrow></math> ) absorber (MAT ID 1000) in NREACT and REACT cards. Then, the inverse velocity of each group is calculated according to the following formula: <math display="block" id="M56"><mrow><msub><mrow><mrow><mo>(</mo><mrow><mfrac bevelled="true"><mn>1</mn><mi>ϑ</mi></mfrac></mrow><mo>)</mo></mrow></mrow><mi>g</mi></msub><mo>=</mo><mfrac><mrow><msubsup><mstyle mathsize="140%" displaystyle="true"><mo>∑</mo></mstyle><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>N</mi><mstyle mathvariant="bold" mathsize="normal"><mi>m</mi><mi>e</mi><mi>s</mi><mi>h</mi></mstyle></mrow></msubsup><mi>R</mi><msubsup><mi>R</mi><mrow><mstyle mathvariant="bold" mathsize="normal"><mi>a</mi><mi>b</mi><mi>s</mi><mi>o</mi><mi>r</mi><mi>p</mi><mi>t</mi><mi>i</mi><mi>o</mi><mi>n</mi></mstyle></mrow><mi>n</mi></msubsup><mi>V</mi><mi>O</mi><msub><mi>L</mi><mi>n</mi></msub></mrow><mrow><mn>2.2</mn><mi>e</mi><mo>+</mo><mn>5</mn><msubsup><mstyle mathsize="140%" displaystyle="true"><mo>∑</mo></mstyle><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>N</mi><mstyle mathvariant="bold" mathsize="normal"><mi>m</mi><mi>e</mi><mi>s</mi><mi>h</mi></mstyle></mrow></msubsup><msub><mi>ϕ</mi><mrow><mi>g</mi><mo>,</mo><mi>n</mi></mrow></msub><mi>V</mi><mi>O</mi><msub><mi>L</mi><mi>n</mi></msub></mrow></mfrac><mo>.</mo></mrow></math> The prompt neutron life time of the Tehran Research Reactor has been shown in Table 10 Table 10 . The calculated result is in very good agreement with the SAR one. 1 1   1 1 Six GroupWIMS & CITATION 1 1 SAR 1 1 Prompt neutron life time (µ Sec) 1 1 48 1 1 45 6 Conclusion 1 Effective point kinetic parameters in the Tehran Research Reactor core have been calculated using deterministic as well as probabilistic schemes. These parameters include the effective delayed neutron fraction and prompt neutron life time. In the deterministic approach, WIMS-D5B and CITATION-LDI2 codes have been implemented efficiently for cell and core calculations, respectively. In order to distinguish between prompt and delayed neutrons, an energy structure in six groups has been selected and group constants have been computed. The six groups' structure was verified via excess reactivity comparison in different states. In the probabilistic approach, the well-known MCNP4C code is employed in the "Prompt Method" scheme. 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Kozlowski , et al ., PARCS v2.7 U.S.NRC Core Neutronics Simulator User Manual , ( Purdue University and RES/U.S. NRC, USA , 2006 ) PARCS v2.7 U.S.NRC Core Neutronics Simulator User Manual 10.1007/s41365-017-0323-7 2017-02-16 2017-04-09 2017-04-13 2017-11-16 2017-12-01 2022-08-18 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2017 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-017-0323-7 http://dx.doi.org/10.1007/s41365-017-0323-7 . 2017 Nuclear Science and Techniques 12 28