Benchmark Comparison of the Oregon State TRIGA^® Reactor Between MCNP^® and Serpent 2

Abstract

The results of a recently developed Serpent 2 model of the Oregon State TRIGA^® Reactor (OSTR) are compared to the results from the OSTR MCNP^® model and measured values for reactor steady state behavior. This benchmark comparison is performed using fresh fuel isotopic data and measured reactivity values at the beginning of the current core life in 2008 to negate burnup uncertainties in calculated values. Reactivity bias, integral control rod reactivity worths, core excess reactivity, shutdown margin, the fuel temperature coefficient of reactivity, and kinetic parameters calculated by Serpent 2 and MCNP^® are compared to the measured values. The results from the Serpent 2 model strongly agree with both MCNP^® results and measured values and are within one standard deviation of each other in all cases, with the exception of the Serpent 2 calculated total control rod reactivity worth, which slightly under-predicts the total rod worth when compared to the measured value despite the MCNP^® and Serpent 2 calculated total rod worth values being within each other’s 1$\sigma$ standard deviations.

1. Introduction

Oregon State University (OSU) houses the Oregon State TRIGA^® Reactor (OSTR), a light water reactor with the ability to pulse. Specifically, it is an original Mark II Training, Research, Isotope, General Atomics (TRIGA^®) reactor developed by General Atomics, utilizing uranium/zirconium hydride fuel elements. The reactor has several different radiation facilities used for a variety of purposes. OSTR was originally fueled with standard TRIGA^® fuel elements from 1967 to 1974 and fueled with HEU FLIP fuel from 1974 to 2008. In 2008, OSTR underwent conversion from HEU FLIP fuel to the modern TRIGA^® LEU 30/20 fuel. As of 2021, the OSTR core consists of 89 standard LEU TRIGA^® fuel elements, three fuel-followed control rods, an air-followed control rod, and 25 graphite reflector elements arranged in a circular array. The safety, shim, and regulating control rods are fuel-followed while the transient control rod is air-followed to enable reactor pulsing via pneumatic ejection of the transient rod. The core assembly is surrounded by an aluminum clad graphite reflector. Renderings of the core assembly from the MCNP^® model are shown in Figure 1 and Figure 2.

The reflector assembly is composed of an 8-inch-thick, 28.9-inch-tall graphite annulus with 2 inches of lead at the periphery to reduce nuclear heating of the bioshield concrete. The lead is not present at the beam port locations or the thermalizing and thermal column locations. The entire reflector assembly is clad in 0.25 inches of aluminum. Cylindrical air voids exist through the graphite and lead annulus for the penetrating beam port (beam port 4), the tangential beam port (beam port 3), and beam port 1. The reflector assembly rests freely on an aluminum platform at the bottom of the primary tank, and the reflector assembly supports the upper grid plate, the lower grid plate, and the safety plate. The lower grid plate supports the entire weight of the fuel, reflector elements, control rod guide tubes, and in-core experiments. The safety plate lies below the lower grid plate and is present to prevent control rods from accidentally falling out of the core. The safety plate also supports adapters that enable the placement of fuel elements in locations that allow for control rods to pass through the lower grid plate.

The OSTR is licensed to operate at steady state up to 1.1 MW_th by the U.S. Nuclear Regulatory Commission, based on, in part, the Safety Analysis Report (SAR) [1] written in support of the license application. The approach for characterizing the neutronics behavior of the OSTR in the SAR involved utilizing the radiation transport code Monte Carlo N-Particle (MCNP^®) 5 [2]. The important context is that the methodology for the current license and future changes to the facility under 10 CFR 50.59 was established using MCNP^®. It has proved to be a valuable tool for analyzing changes to the facility while staying consistent, as required, with the licensing basis. Examples include the removal or addition of fuel or the movement of large reactivity worth experiments or experimental facilities. As the MCNP^® model evolved during its development, the computational time increased significantly. The complexity of the model increased, and the distances from the core required to calculate flux tallies grew to predict beam port and thermal column performance. MCNP^® version 6.2 [3] was used to perform calculations for the OSTR model at the time of this work.

In 2008, OSTR underwent conversion from HEU TRIGA^® fuel to LEU 30/20 TRIGA^® fuel under the U.S. Reduced Enrichment for Research and Test Reactors (RERTR) program. Illustrated in Figure 3 is the core configuration of the core in 2008 during the loading of new fuel. Each fuel rod is composed of a mixture of zirconium hydride, 30 weight% uranium enriched 20% in ²³⁵U, and includes approximately 1.1 weight% erbium. The rod ends are capped by 3.5-inch graphite slugs, and the entire assembly is contained by stainless steel cladding. The erbium acts as a burnable poison. It is used to smooth the reactivity changes in the reactor over the life of the fuel since it unproductively absorbs neutrons during the early period of fuel life, which offsets the excess positive reactivity due to the extra uranium loading in LEU 30/20 TRIGA^® fuels. There are four control rods controlling the reactivity of the system, identified as the shim, safe, regulating, and transient rods. They all have a maximum withdrawal height of 15 inches. The first three are all composed of, from top to bottom, a graphite slug, 15 inches of a graphite and powdered boron carbide mixture, 15 inches of fuel follower, and a graphite slug. These three are electric stepper motor driven, unlike the transient rod, which has a pneumatic–electromechanical drive. As such, the transient rod has an air-filled follower. The CLICIT core configuration of 2008 had four in-core irradiation facilities for sample irradiation, three of which were usable as the central thimble in position A1 is voided with an aluminum rod. The B1 position contains an air-filled, cadmium-lined aluminum tube, the G14 position contains an air-filled aluminum tube, and the pneumatic transfer system known as the “Rabbit” in position G2 is an air-filled aluminum tube. All irradiation facilities are included in the model. All aluminum for the irradiation facilities is modeled as 6061-T6 aluminum alloy, the cadmium for the B1 irradiation tube is modeled as pure cadmium at natural abundance, and all air is modeled as dry air at STP.

A Serpent 2.1.31 model of the OSTR has been developed to replace the current MCNP^® 6.2 [3] model for neutronic analysis performed in support of regulatory activities due to Serpent 2′s multiphysics interface for the coupling of thermal hydraulic codes and to facilitate reactor transient analysis using Serpent 2′s dynamic external source simulation mode. This work investigates the use of Serpent 2.1.31 [4] as an equivalent neutronic analysis methodology to that found in the SAR using MCNP^® 5 [2]. The most significant difference between the MCNP^® and Serpent 2 k-eigenvalue calculation algorithms is Serpent 2 uses the delta-tracking method developed by Woodcock et al. 1965 [5]. MCNP^® tracks particles as a piecewise function in a surface-tracking algorithm, whereas Serpent uses a rejection sampling algorithm. Serpent 2 uses the maximum or majorant cross section from all the materials in the problem space [6]. The neutron is then traced using that cross section to a virtual collision, disregarding any surfaces it may cross. After an interaction location is determined, the algorithm determines if it is a true collision by comparing a pseudo-random number to the ratio of the true cross section to the majorant cross section. If the pseudo-random number is less than or equal to the fraction, a collision occurs [6]. For this reason, and due to the presence of heavy neutron absorbing materials in the core such as boron carbide and cadmium, the Serpent 2 model takes significantly more time to run than the MCNP^® model. However, no timing data was kept for the calculations presented in this work as the motivation for this work was to show that both models produce the same results.

Castagna et al. 2018 [7] developed a Serpent 1 model of the TRIGA^® Mark II reactor in Pavia, which was based on their own MCNP^® model. In comparison to the OSTR, the reactor in Pavia only significantly differs in irradiation facilities and the cluster array geometry. The OSTR has two additional irradiation positions and a safety rod. Castagna et al. 2018 [7] performed several tests to determine the accuracy of their Serpent model. The benchmarking started with an analysis of a low-power critical reactor. In 26 different configurations, determined experimentally, they determined the average bias of the model to be $ 0.26 with a deviation of $ 0.10 [7]. Secondly, they performed a control rod calibration by taking a critical configuration, where a single control rod fully inserted, and then the rod was gradually raised over several trials. The worth of each rod was determined to be within 10% accuracy of the experimental worth. Lastly, they performed a direct comparison to MCNP^® using the same cross section libraries. This showed that Serpent was consistently $ 0.06 above the MCNP^® calculated reactivity bias. Their work suggests that there should be little difference between MCNP^® and Serpent calculations. The conversion for the OSTR should be similar.

Called the “beginning of life” Cadmium-Lined In-Core Irradiation Tube (CLICIT) core, this core configuration of OSTR was selected for analysis and was completely composed of the fresh 1.1% erbium loaded 30 w/%, 19.75% enriched TRIGA^® fuel at the time. This core was selected because the material characteristics of the fuel were well known, there was only one fuel type (no mixed fuel types), and burnup did not have to be considered. Additionally, there were several measurements made of the core during both low and high temperatures, and it was to be the desired core configuration going forward.

The structural components of the OSTR included in both models include the upper and lower grid plates, the safety plate, the reflector assembly and core barrel, the four horizontal beam port tubes, and the reactor tank liner. All structural materials are 6061-T6 aluminum alloy, and the reflector assembly consists of 6061-T6 aluminum cladding, the 8-inch-thick graphite reflector, and 2-inch-thick lead shield that reduces gamma heating of the bioshield concrete. The fuel material is modeled as zero burnup UZrH based on isotopic analysis data provided by the manufacturer. The cladding material is 304 stainless steel, and the upper and lower cladding endcaps are modeled as homogenous mixtures of de-ionized light water and 304 stainless steel to avoid defining the complex fluted geometry of the endcaps. The graphite reflector slugs in the fuel elements and the graphite reflector elements are modeled as pure ¹²C at a density of 1.75 $\mathrm{g}/{\mathrm{c}\mathrm{m}}^3$. The graphite in the reflector assembly is modeled as pure ¹²C at a density of 1.60 $\mathrm{g}/{\mathrm{c}\mathrm{m}}^3$.

2. Comparison of MCNP^® and Serpent 2 Model Results

2.1. Reactivity Bias at Beginning of Core Life

A series of MCNP^® 6.2 and Serpent 2.1.31 k-eigenvalue calculations were performed with the control rod geometries set to 23 combinations of heights that the OSTR was observed to be critical at during control rod calibrations in 2008. These control rod configurations are listed below in

Table 1. Critical core control rod height cases.

Critical Case	Control Rod Heights [%]
	Transient	Safety	Shim	Regulating
1	24.4	58.7	58.6	59.1
2	36.9	55.7	55.7	54.0
3	49.2	52.0	52.0	49.4
4	61.6	48.0	48.0	46.6
5	72.2	44.5	44.5	46.5
6	61.0	0.0	61.0	61.7
7	57.3	26.4	57.4	57.3
8	53.7	40.0	53.7	53.8
9	50.2	51.7	50.2	50.6
10	46.9	65.8	46.9	46.8
11	63.6	63.6	0.0	64.0
12	59.7	59.7	24.7	59.7
13	55.5	55.5	37.0	55.4
14	51.5	51.5	47.7	52.0
15	48.1	48.0	58.7	48.1
16	44.9	44.9	69.6	45.0
17	68.3	67.8	67.8	0.0
18	62.9	62.8	62.9	22.1
19	58.3	58.2	58.3	33.5
20	53.9	53.8	53.8	43.3
21	49.7	49.7	49.6	52.9
22	45.8	45.8	45.8	63.4
23	42.3	42.4	42.4	75.3

. All four control rods have 15 inches of travel from the fully inserted to fully withdrawn position and the control rod heights are expressed as a percentage of the 15-inch distance of travel, with 0% being fully inserted and 100% being fully withdrawn.

The ENDF71x continuous energy neutron data ACE library [8] was used for all neutron cross sections and the ENDF71Sab discrete energy and angle thermal scattering library [9] was used for thermal neutron scattering for light water, graphite, and zirconium hydride. For reactivity bias and control rod calibrations, all materials in the models use cross sections evaluated at 293 K as these operations are performed at low reactor powers and temperatures ranging from 20 to 30 °C. For power-per-element tallies at 1 MW_th, fuel materials use cross sections evaluated at 600 K and all other materials use the 293 K cross sections as there is sufficient convective coolant flow to maintain a bulk coolant temperature of 30 °C. For the fuel temperature coefficient of reactivity calculation, fuel material cross sections evaluated at 293, 400, 500, and 600 K were used while all other materials use the 293 K cross sections.

Each critical control rod configuration determined above was run to determine their respective deviation from criticality. The mean $k_{eff}$ was then calculated and converted to reactivity worth, in dollars, per Equation (1):

$$\rho \left[\$\right]= {\displaystyle \frac{k-1}{k{\beta }_{eff}}}$$

(1)

where $\beta$ is the delayed neutron fraction. In the OSTR, $\beta$ is approximately 0.0075 [1]. Serpent v2.1.31 was used to perform calculations, but changes to the thermal scattering were applied. This was due to the limitations of this version of Serpent. This version did not allow for the use of continuous spectrum thermal scattering cross sections. This could contribute to the error because the cross sections of discrete energy ranges are averaged. The MCNP^® and Serpent 2 calculated reactivity bias are listed below in

Table 2. Calculated eigenvalues for critical core control rod height cases.

Critical Case	keff
	MCNP	Serpent 2
1	0.99999 ± 0.00032	1.0003 ± 0.00035
2	0.99977 ± 0.00035	0.9998 ± 0.00027
3	1.00098 ± 0.00033	1.0003 ± 0.00030
4	1.00022 ± 0.00038	1.0007 ± 0.00029
5	1.00111 ± 0.00030	1.0001 ± 0.00031
6	1.00131 ± 0.00030	1.0014 ± 0.00035
7	1.00006 ± 0.00033	1.0004 ± 0.00032
8	1.00025 ± 0.00034	1.0009 ± 0.00032
9	1.00035 ± 0.00032	1.0001 ± 0.00029
10	1.00030 ± 0.00034	1.0005 ± 0.00036
11	1.00098 ± 0.00030	1.0003 ± 0.00031
12	1.00041 ± 0.00030	1.0003 ± 0.00032
13	0.99846 ± 0.00029	1.0003 ± 0.00037
14	1.00042 ± 0.00029	1.0006 ± 0.00031
15	1.00017 ± 0.00038	1.0000 ± 0.00030
16	1.00023 ± 0.00035	1.0004 ± 0.00031
17	1.00150 ± 0.00031	1.0020 ± 0.00037
18	1.00184 ± 0.00030	0.9999 ± 0.00033
19	1.00116 ± 0.00031	1.0005 ± 0.00031
20	1.00055 ± 0.00029	1.0004 ± 0.00033
21	0.99955 ± 0.00030	1.0003 ± 0.00035
22	0.99970 ± 0.00036	1.0002 ± 0.00032
23	0.99966 ± 0.00030	1.0001 ± 0.00033

.

The MCNP^® model’s average reactivity bias was calculated to be $ 0.05 ± 0.11. Similarly, the Serpent 2 model’s average reactivity bias was calculated to be $ 0.06 ± 0.08. Both models are found to be well within one standard deviation of each other for a critical system. In terms of the k-eigenvalue, the Serpent 2 model only differs from the MCNP^® model by approximately 0.004%.

2.2. Control Rod Reactivity Worths

Control rod calibrations at OSTR are carried out using the positive period or “rod-pull” method. In the rod-pull method the reactivity worth of a segment of a control rod is determined by withdrawing the rod being calibrated to induce a power rise. The OSTR is initially brought critical by withdrawing three of the control rods to a banked critical height with the rod that is to be measured remaining fully inserted. The reactor power is held constant at a low power level of approximately 10 W. A low power level is required to ensure there are no temperature effects as the OSTR has a strong negative fuel temperature coefficient of reactivity. The control rod being measured is then withdrawn some distance to induce a positive reactor period between 6 and 15 s. A timer connected to the control console receives signal from the linear power channel and measures the time of voltage rise from 2 V to 8 V, which corresponds to a power rise from 200 to 800 W when the linear range switch is in the 1 kW position.

These power levels of 200 and 800 W were chosen to ensure the reactor period is as constant as possible. Previous experience over the operating life of OSTR has shown that reactor period is constant and an asymptotic flux rate of increase is achieved before 200 W for a reactor period of approximately 6 s. The reactor operator then manually SCRAMs one of the other three control rods after passing 800 W to bring the reactor subcritical. The initial and final heights of the control rods and the time of power rise from 200 to 800 W in milliseconds are recorded. The control rod being measured is left at its previous height and the reactor is brought back to critical at approximately 10 W by re-banking the other three control rods. This process is repeated until the control rod being measured is completely withdrawn.

The reactivity insertion associated with each control rod perturbation can be calculated from the Inhour equation where the reactor period term is expressed in terms of the time of power rise from one power to another. The average period during the power rise is calculated from the time of rise, and curves are fitted to the recorded data to determine the differential and integral control rod worths. The reactivity insertion associated with the control rod perturbation can be calculated from the Inhour equation using six delayed neutron precursor groups:

$${\displaystyle \frac{\rho }{{\beta }_{eff}}}={\displaystyle \frac{\Lambda }{{\beta }_{eff}T}}+\sum _{i=1}^6{\displaystyle \frac{{\beta }_i}{1+{\lambda }_{i}T}}$$

(2)

where $\rho$ is the reactivity insertion, ${\beta }_{eff}$ is the effective delayed neutron fraction, $\Lambda$ is the prompt neutron lifetime, ${\beta }_i$ is the delayed neutron fraction of the ith delayed neutron precursor group, ${\lambda }_i$ is the decay constant of the ith delayed neutron precursor group, and $T$ is the reactor period. The reactor period is the time for reactor power to increase by a factor of e. The reactor power as a function of time and the reactor period is

$$P\left(t\right)=P_1{e}^{\frac{t}{T}}$$

(3)

where power at some time $t$ can be expressed as $P\left(t\right)$ and initial power as $P_1$. Solving for reactor period yields the following:

$$T={\displaystyle \frac{\Delta t}{ln \left(\frac{P_2}{P_1}\right) }}$$

(4)

where $\Delta t$ is the time for reactor power to rise from $P_1$ to $P_2$. The reactivity insertion due to a control rod perturbation can then be expressed as follows:

$$\rho \left[\$\right]={\displaystyle \frac{\Lambda }{\frac{\Delta t}{ln \left(\frac{P_2}{P_1}\right) }}}+{\beta }_{eff}\cdot \sum _{i=1}^6 {\displaystyle \frac{{\beta }_i}{1+{\lambda }_i\left(\frac{\Delta t}{ln \left(\frac{P_2}{P_1}\right) }\right)}}$$

(5)

The result is that the reactivity insertion can be known from measurable quantities using reactor power detectors to measure the time of power rise $\Delta t$. $\Delta t$ is measured by a timer connected to the fission chamber power measurement channel and the time it takes for power to rise from 200 W to 800 W is displayed in milliseconds. The uncertainty or error in $\Delta t$ using this method is very low as the electronic timer directly receives signal from the fission chamber power detector. The timer measures in milliseconds with a resolution of 1 ms, where the typical values for $\Delta t$ are in the order of tens of seconds, and the control rod height is measured from 0 to 100% withdrawn with a 0.1% resolution. We estimate the uncertainty to be in the order of 1% for the measured total rod worth, assuming no uncertainty in the values in

Table 3. OSTR six-group delayed neutron fraction and decay constant data.

Delayed Neutron Group (i)	Delayed Neutron Fraction (β)	Decay Constant (λ) [s−1]
1	0.038	0.012716
2	0.213	0.031738
3	0.188	0.115525
4	0.407	0.310828
5	0.128	1.397474
6	0.026	3.872331

.

The six-group delayed neutron data in Table 3 was generated in 2008 for the LEU conversion licensing effort by normalizing the six-group decay constant data for ²³⁵U to an effective delayed neutron fraction of 0.0075 for fresh LEU 30/20 TRIGA^® fuel [10]. This six-group delayed neutron data is used for annual control rod calibrations at OSTR because it was the data used in the 2008 LEU Conversion SAR approved by the U.S. NRC. Differential and integral reactivity tables are produced by fitting a fourth-order polynomial curve to the data points that are a summation of the individual reactivity insertions, calculated from the time of rise measurements, as the control rod being measured is withdrawn progressively further out of the core in discrete steps.

The calculated integral reactivity worths of each control rod is shown below in Figure 4, Figure 5, Figure 6 and Figure 7.

The control rod integral reactivity calculations for the MCNP^® and Serpent 2 models agree well with one another with all values being within one standard deviation of each other and measured values. The largest relative error for the MCNP^® results was $ 0.12 and the largest relative error for the Serpent 2 results was $ 0.09. The results suggest that both MCNP^® and Serpent 2 models accurately predict control rod behavior. This provides a high degree of confidence that the models can accurately predict reactivity changes due to changes in the core geometry.

2.3. Core Excess Reactivity and Shutdown Margin

Two values that are measured and calculated at OSTR prior to each day of operation are core excess reactivity and shutdown margin reactivity. The core excess reactivity is measured by observing the control rod heights the reactor is critical at and summating the known reactivity in the remainder of the control rod lengths from control rod calibration tables. The shutdown margin is calculated from the core excess reactivity measurement. The shutdown margin is the reactivity difference between a critical reactor and a completely shut down system. For the MCNP^® and Serpent 2 models, this is determined by calculating the $k_{eff}$ of the reactor system with all control rods fully inserted. Similarly, core excess is the reactivity insertion that would result from all the control rods being fully withdrawn from a critical state. The MCNP^® and Serpent 2 calculated core excess reactivity and shutdown margin, as well as the measured core excess reactivity, are compared in

Table 4. Core excess and shutdown margin.

Evaluation	Method	Reactivity Worth [$]
Core Excess	MCNP®	4.68 ± 0.12
	Serpent	4.64 ± 0.09
	Measured	4.58
Shutdown Margin	MCNP®	−6.60 ± 0.12
	Serpent	−6.72 ± 0.09

.

The MCNP^® and Serpent 2 calculated values for core excess reactivity and shutdown margin are within each other’s 1$\sigma$ standard deviation, and the core excess measurement performed for the CLICIT core on 31 October 2008 is within the standard deviation of both. Another property of the reactor that is quantified for regulatory purposes is the sum of the integral reactivity worths for all control rods or the “total rod worth.” This value is typically determined by summing the total integral reactivity values of each control rod after they are measured in the control rod calibration process but can also be determined by adding the core excess reactivity value and the negative of the shutdown margin value.

However, the modeling method of calculating core excess and shutdown margin by fully ejecting or inserting all control rods has been shown by Spoerer et al., 2024 [11] to not be an equal reactivity insertion compared to the summation of the total control rod reactivity worth, as determined by the rod-pull method. This is due to differences in the thermal neutron flux distribution in the core for the two different three-dimensional core geometries. Since it is not possible to fully eject all control rods of the OSTR without violating the steady state licensed power limit of 1.1 MW_th or the pulse reactivity insertion limit of $ 2.30, only the summation of the total integral reactivity worth of the control rods determined by the rod-pull method is used to determine the total rod worth. Core excess and shutdown margin as compared to the total rod worth are shown below in

Table 5. Core excess and shutdown margin as compared to total rod worth.

Evaluation	Method	Reactivity Worth [$]
Core Excess/Shutdown Margin	MCNP®	11.28 ± 0.16
	Serpent 2	11.36 ± 0.12
Total Rod Worth	MCNP®	10.51 ± 0.24
	Serpent 2	10.34 ± 0.18
	Measured	10.75

.

The summation of the core excess and negative shutdown margin calculated by MCNP^® and Serpent 2 using the all rods ejected and inserted method are well within each other’s 1$\sigma$ standard deviation. The sum of the calculated total integral rod worths for MCNP^® and Serpent 2 using the rod-pull method are well within each other’s 1$\sigma$ standard deviation. The measured total rod worth is within the 1$\sigma$ standard deviation of the MCNP^® calculated total rod worth. The Serpent 2 model appears to under-predict the total rod worth as the measured value is outside of the 1$\sigma$ standard deviation of the calculation and the 2$\sigma$ standard deviation of the two calculated average values. However, the MCNP^® and Serpent 2 calculated total rod worths are within each other’s 1$\sigma$ standard deviation.

2.4. Fuel Temperature Coefficient of Reactivity

The fuel temperature coefficient of reactivity $\left(-{\alpha }_T\right)$ is an important parameter for TRIGA^® reactors as it governs pulse behavior and the inherent operational safety TRIGA^® reactors were designed to achieve, as defined in Technical Foundations of TRIGA^® [12]. For the MCNP^® and Serpent 2 models, the fuel temperature coefficient of reactivity is calculated from the negative reactivity insertion induced using the pre-packaged cross sections for various fuel temperatures from the ACE libraries provided with MCNP^® 6.2 [8,9]. The resulting slope of the linear line fit to these values is the fuel temperature coefficient of reactivity. Core reactivity at various fuel temperatures are shown below in

Table 6. Core reactivity at various temperatures.

Temperature [K]	Reactivity Worth [$]
	MCNP®	Serpent 2
293	−0.06 ± 0.04	−0.02 ± 0.05
400	−0.87 ± 0.04	−0.78 ± 0.05
500	−2.03 ± 0.05	−1.87 ± 0.04
600	−3.37 ± 0.04	−3.21 ± 0.04

and Figure 8. A linear fit is used because the fuel temperature coefficient of reactivity over this temperature range has been shown to be linear for TRIGA^® fuels [12].

Values calculated by MCNP^® and Serpent 2 for each fuel temperature are all within two standard deviations of each other. The MCNP^® model predicts a negative reactivity insertion of $ −0.0108 ± 0.0009 per K and Serpent 2 predicts $ −0.0104 ± 0.0010 per K. These results are in strong agreement with one another and the measured value of $ −0.01 per °C reported in the SAR [1].

2.5. Power-Per-Element Calculations

The differences in the fission power distribution between the MCNP^® and Serpent 2 models were assessed by tallying the fission power-per-element in each model at a total reactor power of 1 MW_th. MCNP^® calculates the fission energy deposition in a cell by estimating the neutron flux through a direct tally and applying the macroscopic fission cross section and the fission heating response for a specified material. Serpent 2 uses a combination of surface tracking and collision estimations to determine the flux within a system. The collision estimator of neutron flux is a summation of all collisions inside a region over the total macroscopic material cross section. The power-per-element tally for the MCNP^® model is shown in Figure 9 and the tally for the Serpent 2 model is shown in Figure 10. The average difference between the two models is shown in Figure 11 and the maximum difference between the two based on relative errors is shown in Figure 12.

The element that had the maximum power generated for the Serpent 2 model was in grid location B4 at a power of 14.92 kW and the minimum element was in grid location F13 at a power of 8.28 kW. The largest difference between the Serpent 2 and MCNP^® models for maximum power difference was 0.7 kW at grid location E7 and the largest average difference occurred at grid location E7 with an average power difference of 0.4 kW.

2.6. Point-Kinetic Parameters

Kinetic parameters such as the delayed neutron fraction ($\beta$) and the prompt neutron lifetime ($l_p$) are important parameters that govern the time-dependent behavior of a reactor. Thus, they are important inputs for Point Reactor Kinetics Equations (PRKE) models that predict the time-dependent behavior of the OSTR and are used in the pulse safety analysis that was performed in support of the SAR. The OSTR prompt neutron lifetime ($l_p$) has been historically calculated using the ${\displaystyle \frac{1}{v}}$ absorber method using the OSTR MCNP^® model [13] and MCNP^® 5. This work was performed prior to updates in MCNP^® 6 that enabled the Iterated Fission Probability (IFP) technique of calculating kinetic parameters through a combination of forward and adjoint-weighted tallies. This was achieved by injecting a small amount of boron into the system. Using the k-eigenvalues from the un-borated system and the borated system, the prompt neutron lifetime can be expressed as follows:

$$l_p={\displaystyle \frac{1}{N_{10\mathrm{B}}{\sigma }_a^{10\mathrm{B}}{\nu }_0}}\left({\displaystyle \frac{k_{ref}-k_P}{k_P}}\right)$$

(6)

where $k_{ref}$ is the neutron multiplication factor of the unperturbed system, $k_P$ is the multiplication factor of the perturbed system, ${\nu }_0$ is the speed of the thermal neutron, $N_{10\mathrm{B}}$ is the ¹⁰B atom density, and ${\sigma }_a^{10\mathrm{B}}$ is the ¹⁰B thermal neutron absorption cross section. Additionally, Hartman 2013 [13] calculated the delayed neutron fraction using MCNP^® 5. The ratio of the neutron multiplication factor due to prompt neutrons alone and the multiplication factor due to both prompt and delayed neutrons is used to determine the delayed neutron fraction. Serpent 2 estimates the delayed neutron fraction and prompt neutron lifetime using three different methods: Nauchi-Kameyama, IFP, and perturbation.

The delayed neutron fraction and prompt neutron lifetime as predicted by the various methods are listed below in

Table 7. Kinetic parameter results.

Method	Delayed Neutron Fraction (β)	Prompt Neutron Lifetime (lp) [μs]
MCNP® 6 IFP	0.00741 ± 0.00061	28.0 ± 1.0
Serpent Nauchi-Kameyama	0.00716 ± 0.00005	24.0 ± 0.1
Serpent IFP	0.00732 ± 0.00019	21.3 ± 0.2
Serpent Perturbation	0.00724 ± 0.00008	22.5 ± 0.1
MCNP® 5	0.00760 ± 0.00010	22.6 ± 2.9

.

The MCNP^® and Serpent 2 models agree within one standard deviation of each other and Hartman 2013’s [13] calculation for the delayed neutron fraction. However, significant differences are observed in the predicted prompt neutron lifetime between the various methods with the closest agreement between the Serpent 2 perturbation method and Hartman 2013′s [13] ${\displaystyle \frac{1}{\nu }}$ approach. Additionally, the largest difference occurs between the Serpent 2 and MCNP^® 6 IFP calculations.

3. Conclusions

The results from the Serpent 2 model strongly agree with both the MCNP^® results and the measured values and are within one standard deviation of each other in all cases except for the Serpent 2 calculated total control rod reactivity worth, which slightly under-predicts the total rod worth when compared to the measured value. However, the important context for this work is to compare the benchmark of the Serpent 2 results to the MCNP^® results; the MCNP^® and Serpent 2 calculated values for total control rod reactivity worth are within each other’s relative error. The MCNP^® model’s average reactivity bias was calculated to be $ 0.05 ± 0.11. Similarly, the Serpent 2 model’s average reactivity bias was calculated to be $ 0.06 ± 0.08. Both models are found to be well within one standard deviation of each other for a critical system and the Serpent 2 model only differs from the MCNP^® model by approximately 0.004% when calculating average model reactivity bias from the critical control rod configurations listed in Table 1. Additionally, the control rod integral reactivity calculations for the MCNP^® and Serpent 2 models agree well with one another with all values being within one standard deviation of each other and the measured values.

The core excess calculated by the MCNP^® model was $ 4.68 ± 0.12 and $ 4.64 ± 0.09 by the Serpent 2 model, and the value measured on 31 October 2008 was $ 4.58. These calculated values are in strong agreement with one another and the measured value. When comparing the predicted fuel temperature coefficient of reactivity $\left(-{\alpha }_T\right)$, reactivity values calculated by MCNP^® and Serpent 2 as a function of fuel temperature are all within two standard deviations of each other. The MCNP^® model predicts a negative reactivity insertion of $ −0.0108 ± 0.0009 per K and Serpent 2 predicts $ −0.0104 ± 0.0010 per K. These results are in strong agreement with one another and the measured value of $ −0.01 per °C. It was also found that both models closely agree in the calculated power-per-element for a total steady state core power of 1 MW_th with the largest difference between the Serpent 2 and MCNP^® models for maximum power difference being 0.7 kW (0.07% of total core power and 5.6% of fuel element power) at grid location E7; the largest average difference occurred at grid location E7 with an average power difference of 0.4 kW (0.04% of total core power and 3.2% of fuel element power).

The MCNP^® and Serpent models agree within one standard deviation of each other and Hartman 2013′s [13] calculation for the delayed neutron fraction. However, significant differences are observed in the predicted prompt neutron lifetime between the various methods with the closest agreement between the Serpent perturbation method and Hartman 2013′s [13] ${\displaystyle \frac{1}{\nu }}$ approach. Additionally, the largest difference occurs between the Serpent 2 and MCNP^® 6 IFP calculations. This would suggest that care should be taken in the selection of a method for calculating the prompt neutron lifetime or the related mean neutron generation time as inputs to PRKE models for transient analysis. However, the Serpent 2 perturbation method agrees strongly with the MCNP^® methodology that was used to calculate the kinetic parameters for the SAR and, in general, the Serpent 2 model provides statistically similar answers to the MCNP^® models that have been used thus far in the licensing of the OSTR.