Inventories of Short-Lived Fission Gas Nuclides in Nuclear Reactors

Abstract

Taking inventories in reactor cores is critical for understanding their radioactive source terms and establishing the relationship between the activity concentration in the primary loop and the status of the reactor core’s fuel. However, there is a niche in which a simple but accurate relationship between reactor conditions and nuclide inventories can reliably predict the fission gas nuclide activities of the reactor core in the primary loop. In this study, a simple and efficient model called “Inventories of a Point Reactor for Fission Gas Nuclides” (IPRFGN) was proposed to calculate and interpret such inventories, in which a 10 MW high-temperature gas-cooled experimental reactor (HTR-10) was used as the test case. The present study findings were consistent with those of a general point–depletion burnup code such as the KORIGEN code. Here, the relative error was <1%. Based on the application of the IPRFGN model in HTR-10, the results indicate that the proposed IPRFGN model has provided the relationship between the inventories of fission gas nuclides in the core and the reactor conditions in all types of nuclear fission reactors. In the future, the IPRFGN model will be used for calculating fission gas nuclide inventories in various reactors.

1. Introduction

Fission gas nuclides, including the radioisotopes Kr and Xe, are gaseous products of the nuclear fission reaction of fissile material. Fission gas nuclides are important considerations in the design of radiation monitoring systems and the implementation of radiation protection procedures in nuclear reactors as significant source terms in the primary loop [1,2,3,4], as these are also reflections of the operation conditions of fuel elements in a reactor core [5,6]. For example, radioactivity measurements are required for accurate monitoring of the damage of tri-structural isotropic (TRISO)-coated particles in the fuel elements of pebble-bed high-temperature gas-cooled reactors (HTGRs), where the fuel pebbles move in the core [6,7]. Fission gas nuclides in the reactor core and primary coolant are also the main sources of radioactive nuclides released into reactor buildings through leakages and the environment via discharges from the stack or relief valve during depressurization accidents [8,9]. Hence, inventories are required to evaluate and ensure reactor radiation safety.

Although nuclides such as ¹²⁹I, ¹³¹I and ¹³⁷Cs exist, the main fission gas nuclides with a dominant contribution to the total activity in the primary coolant consist of ^85mKr, ⁸⁷Kr, ⁸⁸Kr, ⁸⁹Kr, ¹³³Xe, ^133mXe, ¹³⁵Xe, ^135mXe, ¹³⁷Xe, and ¹³⁸Xe regardless of the reactor type [10,11,12]. For instance, the activity fractions in various nuclear reactors such as M310, AP1000, the 10 MW high-temperature gas-cooled experimental reactor (HTR-10), and HTR-PM in normal operation conditions are approximately 81.90, 93.56, 69.77, and 84.06%, respectively [11], based on numerous online and offline radiation monitoring instruments that have been developed in the past to detect fuel elemental failures [13,14].

Inventories of fission gas nuclides are essential for quantifying the radioactive source terms and establishing the relationship between the activity concentration in the primary loop and the status of the fuel elements in the reactor core [15,16,17]. Reliable core inventories are also the basis for source term calculations, radiation protection, and accident analysis. Thus, accurately and efficiently calculating these inventories has been a relevant research area for the past several decades [18,19,20,21]. In addition, inventory calculations of radionuclides, including fission, fission metallic, and activation nuclides in various reactors are based on point depletion burnup equations, in which full-core calculations are divided into two steps: (1) determination of the point reactor equivalence model for a real reactor and (2) calculation of the solution for point depletion based on numerous inter-coupled ordinary differential equations (ODEs) [22,23,24,25].

Considering the neutron transport module in the core is necessary to determine the point reactor equivalence model for a real reactor, two approaches are applied for the coupling of neutron transport and depletion modules: the first approach involves solving the neutron transport equation using either deterministic or stochastic methods, in which the coupling is divided into deterministic neutron transport module depletion and stochastic neutron transport module depletion modules; the second approach involves simultaneously solving the neutron transport and point depletion equations using identified parameters, in which the equations can be classified into deep and shallow coupling modes [26].

Table 1. Point depletion burnup computer codes for nuclide inventory calculation.

Country	Code Name	Burnup Algorithms	References
Russia	AFPA	Transmutation trajectory analysis (TTA) Method	[33]
Sweden	INVENT		[34]
Spain	ACAB	Matrix exponential methods	[35]
U.K.	FISP6	TTA	-
	FISPIN	Exponential Euler difference method	[36]
	FISPACT		[37]
France	DRAWIN- PEPIN	TTA	[38]
	MECCYCO	A numerical method	[39]
Japan	DCHAIN	TTA	[40]
U.S.A.	CINDER		[41]
	ORIGEN	Matrix exponential methods	[42]
	ORIGEN-2		[43]
	ORIGEN-S		[44]
Germany	KORIGEN		[45]
	STACY-TNT		[46]
China	FLY		[47]
	DEPTH	TTA/matrix exponential methods	[48]
	NUIT		[49]
	GNIAC		[50]

summarizes the computer codes of the main point depletion burnup for nuclide inventory calculations based on related studies from various countries. This demonstrates that burnup algorithms can be divided into two categories: analytic and numerical methods [27], which have been verified and validated in several recent studies. For example, Li et al. (2021) [28] alleviated the testing oracle problem using metamorphic techniques for the NUIT code, while Ilas and Hiscox (2021) [29] validated ORIGEN-S integrated into SCALE 6.2.4 and ENDF/B-VII.1 data libraries to analyze the nuclide inventory in pressurized water reactors (PWRs). Alves et al. (2020) [25] calculated the inventories of fission products and source terms to simulate the Argonaut nuclear reactor inside a severe accident using ORIGEN-2. Zwermann et al. (2021) [30] also provided discussions on the propagation of nuclear data uncertainties to decay heat and nuclide density using DARWIN/PEPIN2, while Castagna and Gilad (2022) [31] investigated the radionuclide inventory in nuclear fuel under uncertainties in boron concentrations using high-fidelity models. As for HTGRs, Cui et al. [32] researched uncertainty propagation of fission product yields from uranium and plutonium using pebble-bed burnup calculations. These point depletion burnup computer codes require detailed reactor physics calculations, which are based on the reactor and fuel designs, burnup-dependent fuel composition, locations of specific fuel assemblies in the core, and operational data from the reactor, in which numerous input parameters are normally needed; hence, these often do not meet rapid core diagnosis requirements for radiation monitoring of the primary coolant. To date, only a few studies have focused on the factors that affect inventories of fission gases in the nuclear reactor core.

To understand the features of the production chain of short-lived fission gases in the core and establish the relationship between their inventories and reactor parameters, this study aims to investigate the factors affecting the inventories of fission gas nuclides in reactors using a simple, novel, and efficient model called the “Inventories of a Point Reactor for Fission Gas Nuclides” (IPRFGN) to calculate the inventories of short-lived fission gas nuclides based on an analytical approach (Figure 1). In this study, only two reactor operational parameters (the reduced reactor power in the final stage, or the reactor power, and the average burnup of fuel elements, or the burnup) were used, in which all complex inputs in the point depletion burnup computer codes were reduced to 15 characteristic parameters for various specific reactors.

To grasp the features of the production chain of short-lived fission gases in the core and establish the relationship between their inventories and reactor parameters will be significant for rapid core diagnosis in nuclear safety and nuclear emergency. This study elucidates, in detail, the factors affecting the inventories of fission gas nuclides in reactors. As shown in Figure 1, the present study proposes a simple, novel, and efficient model called the “Inventories of a Point Reactor for Fission Gas Nuclides” (IPRFGN) to calculate the inventories of short-lived fission gas nuclides for various types of nuclear reactors in an analytic form. Only two reactor operational parameters, which are the reduced reactor power in the final stage (denoted as reactor power) and the average burnup of fuel elements (denoted as burnup), are needed for the calculation of the inventories of short-lived fission gas nuclides for various types of nuclear reactors. All complex inputs in point depletion burnup computer codes are reduced into 15 characteristic parameters for several specific reactors.

To illustrate the model and calculate the inventories of various fission gas nuclides, HTR-10 is considered as an example. HTR-10 was built in Beijing, China, in 2000 and is the only pebble-bed type HTGR that can operate at full power in the world currently. The research on HTR-10 is of great importance for the development of HTGRs [7]. Further, the model is validated using the HTR-10 reactor as an example. The variations in typical fission gas nuclide inventories concerning reactor power and burnup are explained. The current model provides a clear and intuitive physical understanding of the relationship between the operating conditions of a reactor and the inventories of fission gas nuclides, which is useful for monitoring fuel elements in HTGRs and other nuclear fission reactors [13,14,53].

The remaining sections of the paper are organized as follows: The theoretical model and computational method, including the point depletion burnup equation and IPRFGN model, are presented in Section 2. Section 3 and Section 4 comprise the results and discussion based on the HTR-10 calculation.

2. Method and Model

As previously stated, inventory calculations of radionuclides, including fission, fission metallic, and activation nuclides in various reactors are based on point depletion burnup equations, which can be divided into two steps: (1) determination of the point reactor equivalence model for a real reactor and (2) calculation of the solution for point depletion based on numerous inter-coupled ordinary differential equations (ODEs). This section focuses on Step (2) and introduces general point depletion equations and the IPRFGN model as a simplification of the former due to the properties of fission gas nuclides. Furthermore, computer codes for corresponding methods, KORIGEN and IPRFGN, are presented.

2.1. Point Depletion Burnup Equation

A fundamental procedure in nuclide inventory calculation is numerically solving the point depletion burnup equation, which is approximated using a system of linear first-order differential equations with constant coefficients and has been applied to hundreds of complex burnup chains involving thousands of nuclides. The general point depletion burnup equation is as follows [44]:

$$\frac{d{N}_i}{dt}={\displaystyle \sum }_{j=1}^{n_j}{l}_{ij}{\lambda }_j{N}_j+\varphi {\displaystyle \sum }_{j=1}^N{f}_{ij}{\sigma }_j{N}_j-\left({\lambda }_i+\varphi {\sigma }_{a,i}\right)N_i,\hspace{1em}\hspace{1em}i=1,2,\dots ,N$$

(1)

where N_i represents the atom numbers of radionuclide i; N_j is the amount of precursor nuclide j or fissionable nuclides j; n_j is the total number of precursor nuclides i; N is the number of fissionable nuclides; λ_i is the decay constant of radionuclide i (s⁻¹); σ_a,i is the spectrally averaged neutron absorption cross-section (cm²); σ_j is the spectrally averaged cross-section that yields the nuclide i (cm²); l_ij is the fraction from radionuclide j decay into nuclide i; f_ij is the fraction from nuclides j to radionuclide i, particularly the fission yields from fissionable nuclides j to radionuclide I; and φ is the neutron flux averaged by space and energy (cm⁻² s⁻¹) [44,45].

Using Equation (1), the inventories of hundreds of radionuclides produced by nuclear fission and other successive reactions, such as decay and neutron absorption, can be calculated. In a simpler form, this equation can also be used to calculate the inventories of fission gas nuclides. The source term of fission gas nuclides, which can be assumed to be constant over a short time for fuel fission, is defined as S2. The value of S2 depends on the neutron flux and amounts of ²³⁵U (N_U5), ²³⁸U (N_U8), ²³⁹Pu (N_Pu9), and ²⁴¹Pu (N_Pu1) for uranium dioxide fuel elements. Furthermore, because their precursor atom decay constants are much lower than those of fission gas nuclides, the number of precursor atoms achieves equilibrium and remains constant. Thus, the first term on the right side of Equation (1) can serve as a constant value, which is defined as S1. The burnup equation for fission gas nuclide i can be approximated as follows:

$$\frac{d{N}_i}{dt}=-\left({\lambda }_i+\varphi {\sigma }_{a,i}\right)N_i+S\left(N_j,\varphi ,N_{U5},N_{U8},N_{Pu9},N_{Pu1}\right)$$

(2)

where S is a constant that is equal to S1 + S2.

The analytical solution can be easily derived by setting the initial atom $N_i \left(t=0\right)$ to 0 as follows:

$$I_i={\lambda }_i{N}_i=\frac{{\lambda }_{i}S}{\left({\lambda }_i+\varphi {\sigma }_{a,i}\right)}\left(1-e^{-\left({\lambda }_i+\varphi {\sigma }_{a,i}\right)t}\right)$$

(3)

where I_i is the activity inventory of nuclide i.

Owing to the large decay constant λ_i, fission gas nuclides can easily reach equilibrium, as shown in Equation (4):

$$I_i=\frac{{\lambda }_{i}S}{\left({\lambda }_i+\varphi {\sigma }_{a,i}\right)}$$

(4)

Based on Equation (4), the equilibrium activity of the fission gas nuclide i is dependent on the neutron flux, decay constant, neutron absorption cross-section, all precursor nuclide equilibrium activities, and fissionable nuclides amounts, particularly ²³⁵U, ²³⁸U, ²³⁹Pu, and ²⁴¹Pu.

2.2. IPRFGN Model

The IPRFGN model, which describes the various reactor cores as a point reactor characterized by several key parameters, was used to effectively calculate the inventories of fission gas nuclides in the point reactor. As shown in Figure 2, the calculation flow illustrates the processes, in which they are connected by dashed lines following the types of reactor, including the different enrichment levels of uranium and plutonium in the fuel elements. Here, the calculated inventories in similar enrichment levels and types of reactor depend on the given operational parameters such as the reactor power and burnup.

Fission gas nuclides normally have two general modes in terms of the number of decay chains: General Modes A and B. In this study, both modes were evaluated. The former is depicted in Figure 3a [45,54], in which the reaction branching ratios were obtained from KORIGEN nuclear data libraries, as these were the most recent data available from the Evaluated Nuclear Structure Data File (ENSDF) of the International Atomic Energy Agency (IAEA) [45,55,56].

Further, General Mode A was derived from the decay chain for ¹³³Xe and ^133mXe in Figure 3b, in which the 100% decay fraction was omitted for simplification. In Figure 3, Ai refers to the activity of nuclide i, where i can be equal to 1, 21, 22, 3, 01, or 02; and f_i,j refers to the decay fraction from nuclide i to nuclide j.

Equations (5)–(10) are the point depletion burnup equations for General Mode A, where ${{\displaystyle \sum }}_j{f}_{i,j}=1,j=1\dots n_j$. A_f,i is defined as the source of nuclide i obtained directly from fissionable fuel nuclides (atoms):

$$\frac{d{N}_{01}}{dt}=-\left({\lambda }_{01}+{\sigma }_{a,01}\varphi \right)N_{01}+\left(f_{02,01}{\lambda }_{02}{N}_{02}+f_{3,01}{\lambda }_3{N}_3\right)+A_{f,01}$$

(5)

$$\frac{d{N}_{02}}{dt}=-{\lambda }_{02}{N}_{02}+f_{3,02}{\lambda }_3{N}_3+A_{f,02}$$

(6)

$$\frac{d{N}_3}{dt}=-{\lambda }_3{N}_3+f_{21,3}{\lambda }_{21}{N}_{21}+f_{22,3}{\lambda }_{22}{N}_{22}+A_{f,3}$$

(7)

$$\frac{d{N}_{21}}{dt}=-{\lambda }_{21}{N}_{21}+f_{1,21}{\lambda }_1{N}_1+f_{22,21}{\lambda }_{22}{N}_{22}+A_{f,21}$$

(8)

$$\frac{d{N}_{22}}{dt}=-{\lambda }_{22}{N}_{22}+f_{1,22}{\lambda }_1{N}_1+A_{f,22}$$

(9)

$$\frac{d{N}_1}{dt}=-{\lambda }_1{N}_1+A_{f,1}$$

(10)

The following conditions were applied for the above equations:

• The (n, γ) neutron absorption reaction was only considered for ¹³⁵Xe and ¹³³Xe;
• ²³⁵U, ²³⁸U, ²³⁹Pu, and ²⁴¹Pu were considered for fissionable nuclides;
• All fission gas nuclides were assumed to be in equilibrium, where $\frac{d{N}_i}{dt}=0,i=1, 21, 22, 3, 01,\mathrm{or} 02$.

For the second characteristic, A_f,j can be written as follows:

$$A_{f,j}={\displaystyle \sum }_i{Y}_{i,j}{\sigma }_{f,i}\varphi N_i i=U5,U8,Pu9,Pu1$$

(11)

where σ_f,U5, σ_f,U8, σ_f,Pu9, and σ_f,Pu1 are the fission cross-sections of ²³⁵U, ²³⁸U, ²³⁹Pu, and ²⁴¹Pu, respectively (cm²); and Y_U5,j, Y_U8,j, Y_Pu9,j, and Y_Pu1,j are fission yields of nuclide j from fissionable nuclides ²³⁵U, ²³⁸U, ²³⁹Pu, and ²⁴¹Pu, respectively. All fission yields and cross sections were obtained from the nuclear data libraries of KORIGEN. Three-group pre-generated nuclear data were calculated by Karlsruhe Institute of Technology (KIT) using Evaluated Nuclear Data File (ENDF) in the IAEA database. The characteristic neutron flux spectrum was supplied for four types of nuclear reactors: HTGR, light water reactor (LWR), liquid metal fast breeder reactor (LMFBR), and molten salt breeder reactor (MSBR) [33,43]. Further inspection of Equation (11) reveals that A_f,j was proportional to the neutron flux φ.

The analytical results are listed below:

$$I_{02}={\lambda }_{02}{N}_{02}=f_{3,02}\left(A_{f,1}+A_{f,21}+A_{f,22}+A_{f,3}\right)+A_{f,02}$$

(12)

$$I_{01}={\lambda }_{01}{N}_{01}=\left(A_{f,1}+A_{f,21}+A_{f,22}+A_{f,3}+A_{f,02}+A_{f,01}\right)\frac{{\lambda }_{01}}{{\lambda }_{01}+{\sigma }_{a,01}\varphi }$$

(13)

where I_i is the inventory of nuclide i (Bq).

General Mode A includes four sub-modes with the same or simpler forms, and A₀, A₁, A₂, and A₃ were used as their representation, including three other sub-modes, as shown in Figure 4. Meanwhile, General Mode B was derived from the decay chain of ⁸⁸Kr, as shown in Figure 5b.

Hence, the equilibrium inventory of the fission gas nuclide i (I_i) can be obtained from the linear combination of the source of nuclide j in the entire decay chain of nuclide i (A_f,j) and can be expressed as Equation (14). Additional details are provided in

Table 2. Classification and solutions of fission gas nuclides for general modes.

General Mode	Nuclide	Solutions
A0	133Xe, 133mXe	$I_{01}=\left(A_{f,1}+A_{f,21}+A_{f,22}+A_{f,3}+A_{f,02}+A_{f,01}\right)$$\frac{{\lambda }_{01}}{{\lambda }_{01}+{\sigma }_{a,01}\varphi }$  $I_{02}=f_{3,02}\left(A_{f,1}+A_{f,21}+A_{f,22}+A_{f,3}\right)+A_{f,02}$
A1	135Xe, 135mXe	$I_{01}=\left(A_{f,1}+A_{f,2}+A_{f,3}+A_{f,02}+A_{f,01}\right)\frac{{\lambda }_{01}}{{\lambda }_{01}+{\sigma }_{a,01}\varphi }$  $I_{02}=f_{3,02}\left(A_{f,1}+A_{f,2}+A_{f,3}\right)+A_{f,02}$
A2	85mKr	$I_0=f_{2,0}\left(f_{1,2}{A}_{f,1}+A_{f,2}\right)+A_{f,0}$
A3	89Kr	$I_0=f_{1,0}{A}_{f,1}+A_{f,0}$
B	87Kr	$I_0=f_{11,0}{A}_{f,11}+f_{12,0}{A}_{f,12}+A_{f,0}$
	88Kr
	137Xe

. All justification of equations is discussed in the supporting information.

$$I_i=g\left(A_{f,j}\right)={\displaystyle \sum }_j{\alpha }_{j,i}{A}_{f,j}$$

(14)

where α_j,i is the linear coefficient of the source of nuclide i obtained directly from fissionable fuel nuclides j.

2.3. KORIGEN and IPRFGN

Based on the ORIGEN code created by Oak Ridge National Laboratory (ORNL), the KORIGEN code was developed to calculate isotope generation and depletion and was modified by the Institute for Neutron Physics and Reactor Technology (INR) at the KIT [45]. The KORIGEN code was used in this study to evaluate the nuclide inventories and depletion and burnup of nuclear fuels loaded into the reactor core, in which the neutron reaction rates of each burnup step were evaluated, and the radioactivity inventories of various reactors were evaluated regardless of the in-pile fuel or spent fuel.

The WrapKORIGEN code, developed by the Institute of Nuclear and New Energy Technology (INET) at Tsinghua University, refers to a series of Python scripts that were used in the pre- and post-processing of the KORIGEN-based inventory calculations for the full-core reactor; it was also used to examine the factors that can affect radionuclide inventories in this study.

Similarly, the IPRFGN code, which was similarly developed by the INET, focuses on the inventories of equilibrium and non-equilibrium conceptual reactors. The IPRFGN model was described above using an analytical approach.

3. Results

3.1. Conceptual Point Reactor for the Equilibrium Core of HTR-10

In this study, the conceptual point reactor was used as an idealized physical model to calculate the inventories of fission gas nuclides, in which a series of parameters were adopted to represent a specific reactor and determine core inventory calculations. Specifically, the neutron flux and atom numbers ²³⁵U, ²³⁸U, ²³⁹Pu, and ²⁴¹Pu were assigned as the characteristic parameters. However, the effect of ²⁴¹Pu can be neglected as pure uranium dioxide fuel elements were used in HTR-10. In the current conceptual point reactor, four parameters in the form of a quadratic function were adopted as follows:

$$\varphi =\left(a_{\varphi }{B}^2+b_{\varphi }B+c_{\varphi }\right)\cdot \frac{P_{final}}{74.6}$$

(15)

$$N_i=\left(a_i{B}^2+b_{i}B+c_i\right),\hspace{1em}\hspace{1em}i=U5, U8, \mathrm{and} Pu9$$

(16)

where a, b, and c are parameters of the conceptual point reactor, P_final is the reactor power (MW/tU), and B is the burnup (GWd/tU).

The various types of reactors and their corresponding reactor type with various loaded fuels were found to have divergencies. In

Table 3. Parameters of the conceptual point reactor for the equilibrium core of HTR-10.

	ϕ	NU5	NU8	NPu9
a	8.45 × 108	5.86 × 1021	−3.54 × 1021	−8.11 × 1020
b	2.49 × 1010	−3.15 × 1024	−6.00 × 1023	2.11 × 1023
c	2.11 × 1013	4.36 × 1026	2.10 × 1027	7.04 × 1024

, the conceptual point reactor for HTR-10 at equilibrium is presented. Accordingly, these were fitted with the result of the point depletion burnup computer code KORIGEN. Here, the inventories of short-lived fission gas nuclides for an individual reactor with specific loaded fuel were found to be dependent on two operational parameters: the reactor power and burnup in our conceptual point reactor.

The findings revealed that the obtained full-core reactor values were relatively close to the average conceptual point reactor. In real engineering design and operation, the conceptual point reactor can reliably and effectively substitute the full-core reactor. Hence, the results obtained in this study can similarly be substituted for full-core reactor values, in which the observed physical characteristics were noted in the equilibrium core of HTR-10. The KORIGEN code was used as a reference.

3.2. Neutron Flux

As the reduced reactor power in the final stage refers to the operational power of the reactor per ton of fissionable material (MW/tU) before the experiment duration, the operating history was not relevant. Meanwhile, the average burnup of fuel elements refers to the amount of energy extracted per mass of the initial loaded fuel. In addition, it is given by megawatt-days per metric ton of heavy metal loaded (MWd/MTHM); however, the unit adopted in this study was gigawatt-days per metric ton of uranium (GWd/tU).

In Figure 6, the neutron flux derived from the KORIGEN code for the equilibrium core of HTR-10 was shown to increase with an increase in the reactor power and burnup. The proportional relationship between the neutron flux and reactor power is due to the proportional relationship between the reactor power and the number of nuclear fission reactions per unit of time, as well as the proportional relationship between the neutron flux and the number of nuclear fission reactions per unit of time. Regarding the increase in neutron flux as burnup increases, the neutron flux rises to compensate for the consumption of total fissile material to maintain constant reactor power as burnup increases.

It also fitted with the quadratic function of burnup with a fixed reactor power, which can be defined as a function of the reactor power and burnup:

$$\varphi ={\varphi }_{ref}\frac{P_{final}}{P_{ref}}$$

(17)

where φ_ref is the reference neutron flux at the reference reactor power (P_ref, 74.6 MW/tU). φ_ref was fitted as a quadratic function using least-squares fitting:

$${\varphi }_{ref}\approx 8.45\times {10}^8{B}^2+2.49\times {10}^{10}B+2.11\times {10}^{13}$$

(18)

$$\varphi =\varphi \left(B,P_{final}\right)=\left(8.45\times {10}^8{B}^2+2.49\times {10}^{10}B+2.11\times {10}^{13}\right)\cdot \frac{P_{final}}{74.6}$$

(19)

In

Table 4. Variance between current results from Equation (19) and those from KORIGEN.

Pfinal (MW/tU)	B (GWd/tU)
	10	25	50	75	100	150
10	−0.35%	0.00%	−0.36%	−0.55%	−0.39%	−0.69%
22.22	−0.26%	0.12%	−0.40%	−0.59%	−0.33%	−0.20%
33.33	−0.26%	0.32%	−0.66%	−0.83%	−0.40%	−0.20%
40	−0.01%	0.33%	−0.50%	−0.68%	−0.39%	−0.36%
50	−0.01%	0.33%	−0.36%	−0.29%	−0.16%	−0.19%
60	−0.01%	0.33%	−0.25%	−0.02%	−0.01%	−0.08%
74	−0.20%	0.33%	−0.08%	−0.20%	0.20%	−0.06%
80	−0.01%	0.33%	−0.13%	−0.02%	0.18%	0.06%
90	−0.01%	0.33%	0.25%	0.13%	0.25%	0.30%
100	−0.35%	0.33%	−0.05%	−0.02%	0.30%	0.32%
110	−0.01%	0.64%	0.25%	0.10%	0.55%	0.49%
120	−0.01%	0.61%	0.25%	0.20%	0.57%	0.63%
130	−0.01%	0.59%	0.48%	0.29%	0.76%	0.62%
140	−0.01%	0.81%	0.47%	0.36%	0.76%	0.73%
150	−0.01%	0.78%	0.45%	0.60%	0.92%	0.83%

, the neutron flux values derived from Equation (19) were compared with those calculated using the KORIGEN code with an obtained burnup value and reactor power of 10–150 GWd/tU and 10–150 MW/tU, respectively. It was found that the values were relatively consistent, with relative errors of <1.0%. Hence, this demonstrates that the neutron flux is determined by fitting Equation (19) with the results from the KORIGEN code.

3.3. Fissionable Nuclides

The calculated results for the full-core inventory of the HTR-10 fueled by low-enriched uranium (LEU) indicated that most fission products were obtained from three main fissionable isotopes, namely, ²³⁵U, ²³⁸U, and ²³⁹Pu. The initial fraction of ²³⁵U in the uranium was 17%. Based on the calculations in this study, the number of atoms of ²³⁵U and ²³⁸U decrease from 4.05 × 10²⁶ and 2.10 × 10²⁷ to 9.46 × 10²⁵ and 1.93 × 10²⁷, respectively. In contrast, the number of ²³⁹Pu atoms increased from 9.08 × 10²⁴ to 2.05 × 10²⁵.

Mills et al. (2020) [57] proposed a method for estimating the fractional fission rates of major nuclides in light water and advanced gas-cooled reactors in terms of anti-neutrino emission, particularly when a reactor is at constant power. Here, a series of simplified equations with several key parameters were selected to describe the fractional fission rates of major nuclides by ignoring irrelevant parameters. This series was closely related to the neutron flux and the number of major fissionable atoms in the nuclear reactor. Hence, a similar simplification approach was used in the present study.

Based on the relationship between the fractional fission and the number of fissionable atoms, including the few effects of ²⁴¹Pu on the inventory calculation of HTR-10, the following equations are proposed to evaluate the number of major atoms (²³⁵U, ²³⁸U, and ²³⁹Pu) using the least-squares fitting method:

$$N_{U5}=5.86\times {10}^{21}{B}^2-3.15\times {10}^{24}B+4.36\times {10}^{26}$$

(20)

$$N_{U8}=-3.54\times {10}^{21}{B}^2-6.00\times {10}^{23}B+2.10\times {10}^{27}$$

(21)

$$N_{Pu9}=-8.11\times {10}^{20}{B}^2+2.11\times {10}^{23}B+7.04\times {10}^{24}$$

(22)

Figure 7 depicts that the number of atoms of ²³⁵U and ²³⁸U are significantly larger than those of ²³⁹Pu. However, the number of ²³⁹Pu atoms was only comparable with that of ²³⁵U in the same order of magnitude, given that the burnup value exceeded ~100 GWd/tU.

Figure 8 shows that the variations in the number of fissionable atoms were associated with an increase in the burnup value. These variations differed significantly for major nuclides, where relevant atom number variations take the number of fissionable atoms, with a burnup value of 10 GWD/tU used as a reference. These variations were obtained by subtracting the number of fissionable atoms in the initial fuel from those in the fuel with a higher burnup value from the number of fissionable atoms in the initial fuel. Changes in the number of ²³⁸U atoms were relatively small, with a maximum value of −7.80%, whereas the relative variations of the number of atoms of ²³⁵U and ²³⁹Pu were significantly larger with a maximum value of −76.63 and 125.90%, respectively. The linearity of the curve of ²³⁸U can also be attributed to the large value and small consumption of the number of atoms. Furthermore, the relative variation of the number of ²³⁹Pu was observed to increase with the increase in burnup, until it gradually decreased, which can be attributed to the competition processes between the neutron fission reaction of ²³⁹Pu and activation of ²³⁸U to ²³⁹Pu, which vary with the increase in burnup.

3.4. Inventories of Short-Lived Fission Gas Nuclides

Figure 9 shows the plot of the two inventories at fixed burnup values of 10 and 100 GWd/tU, respectively. All inventories of the fission gas nuclides were distributed within the range of 10¹⁵–10¹⁸ Bq/tU in a descending order as follows: ¹³³Xe, ¹³⁸Xe, ¹³⁷Xe, ⁸⁹Kr, ⁸⁸Kr, ⁸⁷Kr, ^85mKr, ^135mXe, and ^133mXe. Particularly, ¹³⁵Xe exhibited distinct behaviors. Figure 9 depicts that the inventories of fission gas nuclides excluding ¹³⁵Xe, i.e., ^85mKr, ⁸⁷Kr, ⁸⁸Kr, ⁸⁹Kr, ¹³³Xe, ^133mXe, ^135mXe, ¹³⁷Xe, and ¹³⁸Xe, were proportional to the reactor power in the final stage. Notably, the inventory of ¹³⁵Xe increased proportionally to the reactor power, with a gradual decrease toward the equilibrium state (~7.94 × 10¹⁶ Bq/tU).

Based on the general mode analysis and the determined relationship between the proportion of A_f,i and the neutron flux (φ), only the ¹³³Xe and ¹³⁵Xe inventories were relevant to the neutron flux. The equilibrium inventory for the fission gas nuclide i (I_i) was also found to be a linear combination of the source of nuclide j in the entire decay chain of nuclide i (A_f,j), in which the reactor power was proportional to the neutron flux at the same fuel burnup value. Hence, the inventories of fission gas nuclides excluding ¹³⁵Xe, i.e., ^85mKr, ⁸⁷Kr, ⁸⁸Kr, ⁸⁹Kr, ¹³³Xe, ^133mXe, ^135mXe, ¹³⁷Xe, and ¹³⁸Xe, were all proportional to the reactor power.

The inventories of ¹³³Xe and ¹³⁵Xe can be expressed as follows:

$$I_{01}=A_f\left(\varphi \right)\frac{{\lambda }_{01}}{{\lambda }_{01}+{\sigma }_{a,01}\varphi }$$

(23)

where A_f is the whole source from each nuclide in the corresponding decay chains, in which it was proportional to the neutron flux φ. Given that the value of σ_a,01φ was comparable to λ₀₁, such as those in ¹³⁵Xe, the inventory of ¹³⁵Xe increases as the neutron flux increases; however, it was found to progress more gradually compared to other inventories as λ₀₁ becomes more negligible as the neutron flux increases. A limit value also existed when φ approached infinity, as shown in Equation (24). For ¹³³Xe, φ did not approach infinity; thus, σ_a,01 was inferred to be inadequate, which makes σ_a,01φ incomparable with λ₀₁. Hence, this resulted in similar exhibited behaviors in the inventory of ¹³³Xe compared to those of other fission gas nuclides.

$$I_{01}=A_f\left(\varphi \right)\frac{{\lambda }_{01}}{{\sigma }_{a,01}\varphi }=A_f\left(Constant\right),\hspace{1em}\hspace{1em}{\sigma }_{a,01}\varphi \gg {\lambda }_{01}$$

(24)

As shown in Figure 10, for fixed reactor power, all inventories of fission gas nuclides decreased as the burnup value increased. Several isotopes, such as ¹³³Xe, ^133mXe, ¹³⁷Xe, and ¹³⁸Xe, were found to be more independent of the burnup in the lower burnup region. In contrast, all the inventories tended to have decreased proportional relationships, particularly in the higher burnup region.

Despite the simultaneous production of ²³⁹Pu in the low burnup region, the main inventories of fissionable nuclides, including ²³⁸U and ²³⁵U, gradually decreased. The neutron flux increased as the burnup value increased for the same reactor power. These findings indicate that the consumption of fissionable nuclides ²³⁸U and ²³⁵U can be accounted as the main contribution. Therefore, the inventory of fission gas nuclide i can be recalculated as follows:

$$I_i=g\left(A_{f,j}\right)={\displaystyle \sum }_j{\alpha }_{j,i}{A}_{f,j}={\displaystyle \sum }_k{\beta }_{k,i}\left(\varphi \right)N_k,\hspace{1em}\hspace{1em}j,k=U5, U8, Pu9$$

(25)

where β_k,i is a linear coefficient for radionuclide i based on the number of fissionable atoms k, as shown in Equation (26):

$${\beta }_{k,i}={\displaystyle \sum }_j\left({\alpha }_{j,i}{Y}_{k,j}{\sigma }_{f,k}\right)\varphi$$

(26)

For the same nuclide i (except ¹³⁵Xe), the variables in β_k,i were constant. The variation in I_i was derived from the divergences of φ, N_U5, N_U8, and N_Pu9, in which the sequence was: N_U8 >> N_U5 > N_Pu9 >> φ. Furthermore, with a similar increase in burnup value, ²³⁸U showed the least variation, followed by ²³⁵U and ²³⁹Pu. Hence, it is inferred that a larger value of β_U8,i corresponds to a more constant slope for the nuclide i (Figure 10). All nuclides in the decay chain defined as nuclide j will also be more dependent on the number of ²³⁸U atoms. I_i can be approximated using Equation (27) in the higher burnup region, as follows:

$$I_i\approx {\beta }_{U8,i}{N}_{U8}$$

(27)

As previously mentioned, N_U8 has sufficient linearity and thus explains the proportional decreases in the higher burnup region.

4. Discussion

4.1. Code-to-Code Comparison

Table 5. Code-to-code comparison with various reactor powers and burnups for ^85mKr.

Pfinal (MW/tU)	B (GWd/tU)
	10	25	50	75	100	150
10	0.14%	0.15%	−0.18%	0.10%	0.53%	−0.86%
22.22	0.18%	−0.02%	−0.02%	0.21%	0.38%	−0.47%
33.33	0.03%	0.14%	−0.19%	0.21%	0.38%	−0.71%
40	0.14%	0.20%	−0.12%	−0.02%	0.39%	−1.02%
50	0.04%	0.15%	−0.18%	−0.02%	0.53%	−0.70%
60	0.14%	0.11%	−0.22%	−0.02%	0.39%	−0.75%
74	0.04%	−0.07%	−0.31%	0.01%	0.41%	−0.84%
80	0.14%	0.07%	−0.26%	−0.02%	0.39%	−0.82%
90	0.08%	0.05%	−0.28%	−0.02%	0.47%	−0.84%
100	0.04%	0.03%	−0.30%	0.08%	0.36%	−0.92%
110	−0.36%	0.41%	−0.30%	−0.02%	0.45%	−0.87%
120	−0.36%	−0.07%	−0.59%	−0.02%	0.39%	−0.88%
130	0.41%	0.34%	−0.66%	−0.12%	0.45%	−0.89%
140	0.35%	−0.07%	0.08%	0.15%	0.49%	−0.90%
150	0.31%	0.29%	−0.03%	0.38%	0.18%	−0.91%

lists the relevant variations between the KORIGEN and IPRFGN codes for calculating ^85mKr, which were subsequently used to validate the proposed IPRFGN model. As these variations were obtained from the differences between the results of both models, the variations in other fission gas nuclides all corresponded to <1% under a reactor power of 0–150 MW/tU and a burnup value of 0–150 GWd/tU, which was similar to the results obtained in ^85mKr in Table 5.

The IPRFGN model is a widespread model based on a conceptual point reactor, which simplifies the real reactor into a point consisting of two operational parameters and 15 characteristic parameters. The KORIGEN solves the whole complex point depletion burnup equation with hundreds of parameters. The code-to-code comparison between the IPRFGN and KORIGEN in HTR-10 validated that the excessively significant parameters are captured. Thus, if the characteristic parameters are correct, the results of the IPRFGN model and the point depletion burnup equation will be similar. Furthermore, validation of other nuclear reactors will be studied in the future.

4.2. Inventories for a Non-Equilibrium Core of HTR-10

Inventories of short-lived fission gas nuclides can be readily calculated for equilibrium and non-equilibrium reactor cores as the characteristic parameters for similar reactor cores tend to be similar. It was observed that only a few operational reactor parameters, including the reactor power and burnup, were needed, as listed in

Table 6. Experiments in the primary loop of HTR-10.

Experimental Period	Operational Power (MW)	Average Burnup (GWd/tU)	Number of Fuel Elements
1 December 2004–21 December 2004	9.86	9.35	14,018
5 June 2015–14 June 2015	2.87	29.65	19,327
29 August 2019–31 August 2019	5.93	37.54	19,906

. The IPRFGN model results are presented in

Table 7. Inventories of fission gas nuclides (Bq) for non-equilibrium reactor.

Pfinal (MW/tU)	B (GWd/tU)
	2.87 MW	5.93 MW	9.86 MW
85mKr	9.90 × 1014	1.99 × 1015	3.68 × 1015
87Kr	2.06 × 1015	4.13 × 1015	7.66 × 1015
88Kr	3.02 × 1015	6.07 × 1015	1.12 × 1016
89Kr	3.31 × 1015	6.66 × 1015	1.24 × 1016
133Xe	5.70 × 1015	1.16 × 1016	2.06 × 1016
133mXe	1.64 × 1014	3.35 × 1014	5.93 × 1014
135Xe	3.09 × 1015	5.07 × 1015	7.00 × 1015
135mXe	7.07 × 1014	1.44 × 1015	2.54 × 1015
137Xe	5.09 × 1015	1.04 × 1016	1.83 × 1016
138Xe	5.36 × 1015	1.10 × 1016	1.93 × 1016
85mKr	9.90 × 1014	1.99 × 1015	3.68 × 1015
87Kr	2.06 × 1015	4.13 × 1015	7.66 × 1015
88Kr	3.02 × 1015	6.07 × 1015	1.12 × 1016
89Kr	3.31 × 1015	6.66 × 1015	1.24 × 1016
133Xe	5.70 × 1015	1.16 × 1016	2.06 × 1016

, which illustrates the non-equilibrium cores of various reactors. The proportion between inventories and reactor power indicates that HTR-10, as a non-equilibrium core, was present in the low burnup region. The inventories of fission gas nuclides in the non-equilibrium HTR-10 core during the experiment time were also determined to have an approximate value of 10¹⁴–10¹⁶ Bq.

5. Conclusions

The novel, simple, and efficient IPRFGN model can be used to interpret and calculate the inventories of short-lived fission gas nuclides. Regardless of the nuclear fission reactor type, the inventories of short-lived fission gas nuclides can be calculated using two steps: (1) determining the point reactor equivalence of a real reactor and (2) calculating solutions for point depletion based on numerous inter-coupled ordinary differential equations.

The IPRFGN model theoretically simplifies the neutron transport module and depletion module, including their coupling. Using the KORIGEN code, any type of fission reactor, such as the equilibrium HTR-10, can be represented by a few characteristic parameters, namely, the neutron flux and the number of fissionable atoms such as ²³⁵U, ²³⁸U, and ²³⁹Pu. Meanwhile, for an individual reactor with a specific loaded fuel, the inventories of short-lived fission gas nuclides are dependent on two crucial condition parameters, namely, the reactor power and the burnup. The IPRFGN model used in this study highlights the physical relationship between the inventories of short-lived fission gas nuclides and the operational parameters of a reactor, and it successfully provided a scientific basis for designing an effective and responsive online core diagnostic system and implementing radiation protections in the primary loop of nuclear reactors.

Based on the results of the equilibrium core of HTR-10, the following conclusions are made in this study: With the exception of ¹³⁵Xe, all inventories of fission gas nuclides are distributed between the range of 10¹⁵–10¹⁸Bq/tU and following descending order of ¹³³Xe, ¹³⁸Xe, ¹³⁷Xe, ⁸⁹Kr, ⁸⁸Kr, ⁸⁷Kr, ^85mKr, ^135mXe, and ^133mXe. The inventories of fission gas nuclides excluding ¹³⁵Xe, i.e., ^85mKr, ⁸⁷Kr, ⁸⁸Kr, ⁸⁹Kr, ¹³³Xe, ^133mXe, ^135mXe, ¹³⁷Xe, and ¹³⁸Xe, were proportional to the reactor power. However, the inventory of ¹³⁵Xe increased with an increase in reactor power and gradually reached equilibrium under a fixed burnup value. All the inventories of fission gas nuclides decreased as the burnup value increased. Several isotopes, such as ¹³³Xe, ^133mXe, ¹³⁷Xe, and ¹³⁸Xe, were independent of the burnup in the lower burnup region for a fixed reactor power.

Furthermore, the results for the equilibrium core of HTR-10 were relatively close to those of the values obtained from point depletion burnup programs, such as KORIGEN, with a relative error of <1.0%. Thus, this validates the proposed IPRFGN model used in this study. The inventories of fission gas nuclides in the non-equilibrium HTR-10 core throughout the experiment duration were approximately within the range of 10¹⁴–10¹⁶ Bq. As predicted by the IPRFGN model, their relationship with the reactor power was consistent.

Finally, as all nuclear fission reactors were found to be reduced to a point reactor, the relationships between the short-lived fission gas nuclide inventories and operational reaction parameters were similar, and the proposed IPRFGN model can be applied to both equilibrium and non-equilibrium cores of various fission reactors. In the future, the IPRFGN model will be widely used to calculate inventories of fission gas nuclides using a database of characteristic parameters from various reactors.