Theoretical analysis of long-lived radioactive waste in pressurized water reactor

Ze-Xin Fang ? Meng Yu ? Ying-Ge Huang ID? Jin-Bei Chen ? Jun Su ID ?Long Zhu ID

Abstract Background The accelerator-driven subcritical transmutation system (ADS) is an advanced technology for the harmless disposal of nuclear waste. A theoretical analysis of the ingredients and content of nuclear waste,particularly long-lived waste in a pressurized water reactor(PWR),will provide important information for future spent fuel disposal.

Keywords Radiotoxicity · PWR · Five Gaussians · Longlived nuclides · Fission fragments yields

1 Introduction

With the development of the nuclear industry, the amount of radioactive waste in storage worldwide has rapidly increased. An efficient prediction of the composition of spent fuel in nuclear reactors plays an important role in the design of facilities for radioactive waste management and has been a topic of significant interest.In[1],formulas and basic data were proposed for calculating the fission product radioactivity for a thermal neutron reactor. Similarly,codes for the fission products in the primary loop of a pressurized water reactor (PWR) were developed [2, 3].Simulations and calculations of the neutron flux densities have also been extensively studied [4–10].

In recent years, significant progress has been made in the mechanism of nuclear fission [11–18], which provides essential information for estimating the fission products in a nuclear reactor.The fission process can be described as a potential energy surface guiding the evolution of the nuclear shape [19, 20]. According to the scission model,the shell effects can be reflected in the fission yield and can be distinguished according to different fission modes,which is related to the deformation of the nucleus at the scission point[20,21].The improved scission-point model describes the charge distributions well [22]. The Gaussian fitting approach has been widely and successfully used in investigations into various fission products. By approximating different fission modes to Gaussian distributions,the dependence of the different fission modes on the mass and charge distributions can be studied. In general, the experimental data of a neutron-induced fission yield are measured with a fixed incident neutron energy within a certain energy range. Therefore, to study the continuous behavior of the fission yield with incident neutron energy,it is necessary to establish a continuous relationship between the yield and excitation energy. One of the most frequently adopted phenomenological approaches is to approximate the fission yield through a superposition of several Gaussian distributions [23]. However, most curve fitting results have been found through studies using a fixed neutron energy or specific compound nucleus.In this study,we theoretically analyse the evolution of fission products in a PWR using the multi-Gaussian function in combination with the most probable charge model and the concept of a di-nuclear system.

The calculations are based on the parameters and information of a typical domestic PWR. The neutron spectrum plays an important role in studies on the fission product yields in a reactor.Wigeland et al.[24]divided the neutron spectrum into two energy ranges:thermal and fast,depending on the incident neutron energy. The epithermal energy range is also defined to better describe the spectrum[25]. The semi-empirical method can be useful for determination of the neutron spectrum [26].

Figure 1 shows the neutron spectrums in different nuclear reactors. It can be seen that a general incident neutron energy range in a reactor varies from 0.001 eV to 10 MeV.The formula used in this study was established to calculate the fission yields within this energy range. As mentioned above, the fission process in a nuclear reactor produces substantial fission fragments,which influence the reactivity of the reactor. However, fissile nuclides in a nuclear reactor continuously produce large amounts of long-lived nuclides,some of which,including239Pu,241Pu,and233U, can be extracted and reused in fission reactions[27, 28]. However, some fission fragments, such as90Rb,107Rh,and135Xe are highly radioactive.The transmutation of minor actinides also contributes to long-term radiotoxicity [29]. The ADS system is flexible for lowering such waste [30], and the storage of spent nuclear fuel should be considered crucial [31]. Therefore, it is necessary to investigate the properties of the fission fragment evolution in a PWR in a theoretical manner.

The remainder of this paper is organized as follows. In Sect.2,the details of the theoretical method are described.The results and discussion are presented in Sect.3.In Sect.4, we provide some concluding remarks regarding this study.

2 Model

2.1 Multi-Gaussian functions

The Gaussian model was first proposed by Wahl [32],and this approach, based on the five Gaussians, has been widely used and continuously improved in estimations of fission fragment distributions. The expression for the five Gaussian superposition can be written as [33]:

where Yirepresents the proportion of each Gaussian component.In addition,σiand Δiare the Gaussian parameters,2nt is considered as the total number of neutrons emitted during the fission process, and AF is the mass of the compound nucleus.

2.2 Potential energy surface

Fig. 2 (Color online) Potential energy surface for the reaction 235U(n,f). The dashed and dotted lines indicate the fragment combinations with the minimum potential energy and the configuration in the symmetry fission, respectively

Relatively low potential energies result in high fission yields of the corresponding fragments [38–40]. Owing to shell closures of Z = 50 and N = 82, the valley with the minimum value Vmin can be seen, which results in relatively high fission yields of approximately Z = 50 and N =82. By contrast, the potential energy Vmid at the central position influences the fission yield at approximately Z =46 and N = 72.

2.3 Nuclear charge distribution

To obtain the isotopic fission yield, it is also assumed according to the most probable charge model [41] that the fission yield in the isobaric chains with mass number A follows a Gaussian dispersion:

Considering the charge conservation and symmetry of the mass distribution, k and b are determined as follows:

where AL and AH represent the mass numbers of light and heavy fragments, respectively. In an isobaric chain,

2.4 Prediction of components in spent fuel

The inventory of components in a PWR can be obtained through the following formula [43, 44]:

where Niis the nuclide number density of isotope i. In addition, σf，l, σc，j, and σa，irepresent the microscopic effective fission cross section, gamma capture cross section, and absorption cross section, respectively. Moreover,Φ denotes the neutron flux in the reactor,yi，lis the yield of fragment i produced by the fissile nuclide l,λiis the decay constant of nuclide i, and Kp，kis the decay branching ratio of parent nuclide k. It can be seen that the production process of nuclide i contains several parts. For heavy nuclides,the formation from the decay process and neutron capture are considered. For fission fragments, the fission yield was also considered. The consumption process includes decay and neutron absorption.

The effective microscopic reaction cross section is defined as [43]

In the above equation, σx，ris defined as the effective microscopic cross section for the reaction r of nuclide x.In addition, En is the neutron energy in the reactor, and χ is the neutron distribution probability density, which is simplified as space-independent. The integral bound at 10 MeV and 0.001 eV is determined based on the energy range of the neutron spectrum in the PWR. We can obtain the following equation by a change in variable u=ln（E0/En）, which provides a new integration boundary, which is approximated as 23.03 0.

where E0=10 MeV, u is often referred to as lethargy, and φ=Enχ（En） is referred to as the normalized neutron spectrum.

3 Results and discussion

3.1 Determination of Gaussian parameters

The fitting results of233，235，238U(n,f)and239，240，241Pu(n,f)are shown with experimental data from [45–55] in Figs. 3 and 4.

Fig.3 (Color online)Fitting results for 233U(n,f), 235U(n,f),and 238U(n,f).The lines represent the curve-fitting results.Black dots with error bars denote the experimental data [52–55]

Fig. 4 (Color online) Fitting results for 239Pu(n,f), 240Pu(n,f) and 241Pu(n,f). The lines represent the curve-fitting results. Black dots with error bars denote the experimental data [45–51]

For uranium,Δiand σiexhibit a linear relationship with the mass number of a compound nuclei. The Gaussian parameters are expressed as a function of the mass number and excitation energy. The parameters in Eq. (1) can be expressed as follows:where par1, par2, par3, and par4 are listed in Table 1. In addition, Vmid and Vmin denote the values of potential energy with a symmetry configuration and combinations with proton and neutron shell closures, respectively, as shown in Fig. 2. Moreover, E*is the excitation energy of the fission system.

The value of Yidepends on the excitation energy. The parameters σiand Δiwere chosen to be fixed by changing the excitation energy. The values of the Gaussian parameters for239Pu,240Pu, and241Pu are listed in Table 2.

3.2 Comparisons with experimental data

Owing to the scarce independent yield data of the fission products, we fit the data to the cumulative yields to investigate the energy dependence. In Figs. 5 and 6, the calculated isotope yields from the fitted mass distributions of235U(nth,f) and239Pu(nth,f) were compared with the experimental data [56–59]. Numerical comparisons of the experimental independent and simulated yields for235U(nth,f) and239Pu(nth,f) are shown in Tables 3 and 4.As shown in Tables 3 and 4, we present the errors in the experimental fragments and compare the simulation results with the experimental data.Within the permissible range of errors,the thermal neutron-induced fission yields predicted by our simulations are in good agreement with the experimental data,which proves the validity of our methods and models.

To further verify the feasibility of our method in the establishment of fission product charge distribution, we compared the calculated charge distribution and experimental data in the reactions235U(nth,f)and239Pu(nth,f),as shown in Fig. 7). The experimental data were obtained from [59]. The calculated peak is slightly higher than the experimental data. However, both the calculated yield and experimental data reached the maximum at approximately Z = 54, which might suggest a transition of shell closure from Z = 50 to Z = 54.

Table 2 Fixed Gaussian parameters(σ4,σ5,Δ4,and Δ5 are assumed to satisfy the conditions σ4=σ2,σ5=σ1,Δ1+Δ5=0,and Δ2 +Δ4=0)

3.3 Neutron spectrum

Based on the Daya Bay nuclear power plant, a neutron spectrum was established by modeling an actual reactor core using MCNP4C [60]. The neutron spectrum in the nuclear reactor generally ranges from 0.001 eV to 10 MeV and is separated into three parts: thermal neutron energy,epithermal (intermediate) neutron energy, and fast neutron energy. Thermal neutron energy ranges from 0.001 to 0.1 eV,where most fission reactions take place in a PWR.The fast neutron energy ranges from 105eV to 10 MeV. Most of these neutrons are emitted before and after fission. The energy of epithermal neutrons is between 0.1 and 105eV.In thermal neutron reactors, neutrons emitted during the fission process have an average energy of 2 MeV. The neutrons lose energy by elastic or inelastic collisions with nuclei in the moderator medium until they become thermal neutrons.Most fission reactions in thermal nuclear reactors are induced through thermal neutrons.

The neutron spectrum is often referred to as

where φ is the neutron spectrum, χ（En） is the neutron distribution probability density at an energy of En.

A piecewise function was used to describe the neutron spectrum. In the thermal neutron region, χ（En） is often approximated using a Maxwellian–Boltzmann distribution.A method of superposing five to seven partial Maxwellian distributions to represent the neutron spectrum within the thermal and epithermal range was proposed in [25]. We found that a single Maxwellian distribution is sufficient todescribe the neutron spectrum within the thermal range.The multi-Maxwellian distribution mainly takes effect in the transition part between the thermal and epithermal ranges. The formula used is as follows:

Table 1 Values of par1, par2,par3, and par4 for different parameters

Fig. 5 (Color online) Normalized isotope yields of fission fragments in 235U(nth,f) are compared with the experimental data. The hollow points are the calculated results. The red dashed lines are the guidelines. The black solid points represent the experimental data.Data for light fragments were taken from [56]. Data for fragments with Z = 54 and Z = 55 were obtained from [57] and [58]

Fig.6 Normalized isotope yields of fission fragments in 239Pu(nth,f)are compared with the experimental data. The hollow points are the calculated results. The red dashed lines are the guidelines. The black solid points represent the experimental data. Experimental data were obtained from [59]

where αM is related to the temperature of the moderating medium [25]:

Here, Tm is the temperature of the moderating medium.

The experimental thermal neutron energy distribution in the case of a water-moderated reactor was compared in[61]. Figure 8 shows a comparison of the experimental thermal neutron spectrum in water moderated reactors at 291.15 and 371.15 K with the calculation results in this study. It can be seen that the calculated results can reproduce the experimental data quite well.

where C is a constant determined by the continuous condition at the boundary of the thermal and epithermal energy ranges.

In the fast neutron energy range of the PWR, χ（En） is approximated by the thermal neutron-induced fission spectrum of235U. The experimental data and curve fitting results of the235U fission spectrum are presented in[62, 63]. In [63], the prompt neutron spectrum from thermal neutron-induced fission in235U using the recoil proton method was recently measured. In [62], a photographic plate method and time-of-flight method are employed.Two different formulas are used in curve fitting, all of which fit well with the experimental data. In this study, the fission spectrum of235U is considered as a linear combination of the two formulas and can be written as

where ω denotes the weight, and EM, a, and b are constants. These were all determined in [64] using the leastsquares method.In addition,C1is determined based on the continuous condition at the boundary of the epithermal and fast energy ranges. The normalized neutron spectrum within all energy ranges was multiplied by a normalized constant Cnorm.

In this study, the boundaries between the three energy ranges were 10-0.6and 105.7eV.In addition,Tm is taken as 563.15 K.In Fig. 9,the normalized neutron spectrum in the PWR was compared with the calculated results obtained in[60]. In our mathematical description of the neutron spectrum,the resonance in the epithermal energy range was not considered.

3.4 Effective cross section in PWR

In Table 5, we list the effective fission and neutron capture cross-sections of several heavy nuclides calculated using Eq. (11). The cross-section data are taken from the ENDF library, and the neutron spectrum proposed above(Eqs. (16), (18), and (19)) are used to calculate the effective cross sections.

The calculated results were compared with the effective cross sections from [43]. The cross sections from theENDF reflect the resonance absorption of neutrons around the epithermal energy range. However, for some nuclei such as238U, the resonance phenomenon of the reaction cross sections leads to a deviation during the integration process in Eq. (12).However,in[43],the differential cross sections are continuous without showing the resonance phenomenon, which makes the integration results more precise. Because238U(n, γ) plays an important role in the uranium-plutonium cycle, we use the fission-to-capture ratio in [43] to calibrate the capture cross section of238U.

Table 3 Comparison of experimental independent yield data and simulated yield data for 235U(nth,f)

Table 3 continued

Table 4 Comparison of experimental independent yield data and simulated yield data for 239Pu(nth,f)

Fig. 7 (Color online) Normalized charge distribution of the fission fragments in 235U(nth,f) and 239Pu(nth,f) are compared with the experimental data. Red dots are the calculated results, and red lines are the guidelines. The black solid points are the experimental data taken from [59]

Fig.8 (Color online)Comparison of the thermal neutron spectrum in the case of a water moderated reactor at 291.15 and 371.15 K.Hollow points are the experimental data [61]. Lines represent the theoretical results calculated in this study

3.5 Operational parameters of nuclear power plant

The characteristics of the Daya Bay nuclear power plant were adopted [65], and some characteristics are listed in Table 6. It is assumed that the reactor operates at a full power of 330 days per year. The energy production is assumed to be constant.In this study,the initial enrichment is assumed to be 3%, and the fuel cycle performance with235U enrichments of above 5% was studied in [66]. The moderator temperature was assumed to be 290°C. The neutron flux was determined using the following equation:where P is the thermal power of the reactor, ∑f denotes the macroscopic fission cross section,Φ is the neutron flux,and Ef represents the energy released in one fission event of235U.In addition,V is the volume of the reactor core.In this study, it is considered that the energy released in one fission event of235U is 200 MeV.In general,decreasing the fission cross section leads to an increase in the neutron flux as the fuel consumption increases.

Fig. 9 The neutron spectrum established in this study is compared with the results simulated by MCNP4C in [60]. The solid line represents the neutron spectrum in this study. Dots refer to the calculations [60]

3.6 Long-lived heavy nuclides in PWR

In this section, the inventory of long-lived heavy nuclides as a function of fuel consumption is studied. The fuel consumption was calculated using the following equation:

Table 5 Effective fission(σf)and neutron capture(σc)cross sections for several heavy nuclides

where BU is the uranium burnup, P represents the thermal power as a function of time,T indicates the total operation time of the reactor. According to [67], a reactor in Daya Bay nuclear plant goes through a shutdown and refueling process every 1 and a half years. Each refueling process lasted for approximately 40 days. In addition, the reactor operates normally at full power on other days. Therefore,for actual fuel utilized for a 1-year period and fuel burned incessantly under full power for approximately 330 days,it is assumed in this study that the burnup in both cases is equivalent. As a result, for a time scale of 1 year, the numerator of Eq. (21) can be expressed as the maximum and constant thermal power multiplied by 330 days. In addition, MU indicates the total mass of uranium in the fuel.

Figure 10 illustrates the evolution of inventories of uranium isotopes with burnup. It is noted that for the isotopes of uranium,the inventory of238U remains essentially constant. Here, (n, γ) is the primary consumption path of238U owing to the small effective microscopic fission cross sections.With a gradual decrease in the slope,the value of235U starts to stabilize at deep burnup after 180 GW·d/t.Compared to the inventory of235U,234U, and236U are insignificant under a low fuel consumption before reaching 50 GW·d/t.However,the inventory of236U exceeds that of235U after the burnup at approximately 50 GW·d/t. The inventory of234U is higher than that of235U at a deeper burnup. The inventory of233U is maintained at an extremely low level after reaching a steady state.

Partitioned from the PWR, plutonium can be reutilized in the fuel transition to the thorium fuel cycle in a thermal molten salt reactor[68].The variation in plutonium isotope inventories with burnup is shown in Fig. 11. It is observed that the inventory of239Pu is higher than that of several isotopes.The significant generation of239Pu at low burnup is the result of a large amount of neutron capture by238U and the relatively short half-lives of β-decay of239U and239Np. In addition, the240Pu and241Pu levels were maintained at essentially the same level after stabilization. The inventory of242Pu increases with the burnup and approaches that of239Pu. In Ref. [28], the calculated results in a thermal molten salt reactor using Th-U mixed fuel reflect the same relative inventory of242Pu. The main source of242Pu is the orbital electron capture of242Am.Its relatively short half-life results in a significant inventory of242Pu at a deeper burnup.

In Fig. 12, the inventories of neptunium isotopes are presented. It can be seen that both237Np and238Np stabilized after burnup at approximately 80 GW·d/t. The inventories of the americium and curium isotopes are presented in Figs. 13 and 14. The inventories of the light isotopes are relatively high for both americium and curium isotopes. The heavier isotopes are essentially produced by the neutron capture reaction of lighter isotopes.In addition,some isotopes tend to have the same value at a deeper burnup.The inventory of243Am was close to that of242Am.The same phenomenon was observed for pairs of (243Cm,244Cm) and (245Cm,246Cm). This can be explained by comparing the microscopic fission cross sections shown in Table 7.For nuclei such as244Cm,246Cm,and243Am,their fission cross sections are much smaller than those of other isotopes, which leads to less consumption and gradual accumulation.

Figure 15 presents the variation of inventories of uranium and plutonium isotopes with the burnup. It can be seen that the calculated results of the relative inventories and the variation trend in our model are in surprisingly good agreement with the experimental data from Daya Bay. Further investigation is therefore justified, although the calculated absolute inventory of each isotope is higher than the experimental value because the enrichment of uranium used in our simulations was higher than the fuel used in [69], whereas the assumed neutron flux of the reactor is kept at a relatively high level.

The total macroscopic fission cross sections and the principal contribution of fission cross sections of fissile nuclides are presented in Fig. 16. It was observed that thetotal fission cross section continued to decrease during the reactor operation. At a lower burnup, it decreases with a slower rate of change, which is mainly caused by a rapid consumption of235U,alleviated by the growth of239Pu and241Pu. With the stabilization of the239Pu and241Pu inventory, the sustained decrease of235U leads to a decrease in the total macroscopic fission cross sections.

Table 6 Operating parameters of Daya Bay nuclear power plant

Fig. 10 (Color online) Calculated inventories of uranium isotopes in PWR as a function of burnup

Fig. 11 (Color online) Calculated inventories of plutonium isotopes in a PWR as a function of burnup

Fig. 12 (Color online) Calculated inventories of neptunium isotopes in a PWR as a function of burnup

Typically, PWR nuclear power plants have a discharge burnup of over 45 GW·d/t. Table 8 shows the predictions of the inventory of heavy long-lived nuclides at 50 GW·d/t and 196 GW·d/t under full-power operation.

3.7 Fission fragments in a PWR

As shown in Table 9, we simulated the nuclide inventory on three decay chains.According to the data provided in [70], the relative inventories of long-lived nuclides are reasonable. Figure 17 presents the evolution of several long-lived nuclides in a PWR under 100% power. As can be observed in Fig. 17, long-lived nuclides increase linearly during the evolution. This accumulation is due to the long half-lives.A few long-lived nuclides can be consumed within a fuel cycle.

On the one hand, Table 9 indicates that the short-lived nuclides are maintained at an extremely low level. For example,the inventory of135Sn is approximately 10-13kg/t.On the other hand,the long-lived nuclide135Cs is greater than the others, and during a 600-day period, increases linearly. For the isobars of135Cs, it can be seen that their concentration does not increase significantly because of their short half-life; they increase slowly and start to stabilize at a deeper burnup. Specifically, the inventories of135Xe and135I are all 10-4kg/t at 20 GW·d/t.

Fig. 13 (Color online) Calculated inventories of americium isotopes in a PWR as a function of burnup

Fig.14 (Color online)Calculated inventories of curium isotopes in a PWR as a function of burnup

Table 7 Effective fission(σf)and neutron capture(σc)cross sections for Cm and Am isotopes

4 Summary

Fig. 15 (Color online) Comparison of heavy nuclide inventories between (A) experimental data [69] and (B) calculated results in a PWR

Fig. 16 Calculated total macroscopic fission cross sections and the principle components of fission cross sections

As an important aspect of the study on the fission mode and fission fragment inventories in nuclear reactors, fragment yields of fissile nuclides have been investigated for decades. The five Gaussian functions and the most probable charge model were combined to investigate the fission yields in a PWR.The potential energy surface based on the di-nuclear system concept was applied to evaluate the physical properties of the fission process. In addition, the neutron incident energy effects on the fission product distribution are considered.

The neutron spectrum in a PWR under stabilization was established using a piecewise function. The inventories of long-lived heavy nuclei and fission fragments in the PWR were predicted by solving a set of differential equations coupled to multiple variables. In this study, mathematical expressions for the neutron energy spectrum in a PWR were established, and effective microscopic cross sections were calculated using data from the ENDF library. The operating parameters of the Daya Bay nuclear power station were used to simulate the inventories of heavy nuclei and fission fragments. For the heavy nuclei, the evolutionof uranium,plutonium,neptunium,americium,and curium isotopes were investigated. The inventories of234U and236U exceed that of235U after reaching a discharge burnup.Among the plutonium isotopes, the inventory of242Pu increases and gradually reaches the same level as239Pu under a deep burnup.Because of the single decay chain for americium and curium isotopes and their large difference in effective fission cross sections, the inventories of the isotopes decrease with an increase in mass number. By contrast, isotope pairs such as243Am and242Am tend to have similar inventories after stabilization.Upon discharge burnup, except for238U, the inventories of239Pu,240Pu,241Pu,242Pu,237Np,235U, and236U are predominant in a PWR. For fission fragments,the evolution of several longlived fission fragments is predicted, and inventories of isobars of135Xe were calculated. We also compared the calculated inventories of uranium and plutonium isotopes with experimental data from Daya Bay. Surprisingly, theevolution trends and relative values were in good agreement with the data. In the future, our group will optimize the calculation of the reaction cross sections and consider more conversions between nuclides, allowing the method to be improved and better predict the inventories of longlived isotopes.

Table 8 Prediction of heavy long-lived nuclides in a PWR

Table 9 Prediction of fission products in a PWR

Fig.17 (Color online)Evolution of long-lived nuclides during a fuel cycle

AcknowledgementsThe authors thank Dr. Li-Le Liu and Chao-Sheng Zhou for their useful discussions.

Author contributionsAll authors contributed to the study conception and design. Material preparation, data collection, and analysis were performed by Ze-Xin Fang, Meng Yu, and Long Zhu. The first draft of the manuscript was written by Ze-Xin Fang and Meng Yu.All authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.