Linear energy transfer of fission fragments of 235U and nucleation of gas bubbles in aqueous solutions of uranyl nitrate

https://doi.org/10.1016/j.anucene.2020.107379Get rights and content

Highlights

  • Mechanics of bubble formation in aqueous fissile solutions examined.

  • Correlations derived for radius of fission track bubbles in uranyl nitrate solution.

  • Data presented quantifying the number, size and location of fission track bubbles.

  • Data and correlations provide foundations for a mechanistic model of radiolytic gas.

Abstract

Fission fragments emitted in a fissile solution create tiny gas bubbles, the size of which is determined by the linear energy transfer (LET) of the particles. The LET of fission fragments of 235U in aqueous solutions of uranyl nitrate has been determined, and using methods adapted from the literature, the size of gas bubbles generated along the tracks of these particles has been estimated, revealing important variations with respect to particle LET and solution properties. Empirical correlations are presented for the maximum radius of radiolytic gas bubbles in unsaturated solutions of uranyl nitrate as a function of solution temperature and concentration. These can be used to predict the critical concentration of dissolved hydrogen necessary for the appearance of gas voids during nuclear criticality transients. The findings are intended for use in a future model of nuclear criticality transients in aqueous fissile solutions for the purposes of nuclear criticality safety assessment.

Introduction

The term criticality accident describes an unplanned and uncontrolled criticality excursion in a fissile material. A review of criticality accidents (McLaughlin et al., 2000) carried out at the turn of the century showed that a very large proportion of these accidents occur during operations involving the handling of fissile liquids. Some well known examples include the 1958 accident at the Y-12 National Security Complex in Oakridge, Tennessee (Patton et al., 1958) and the 1999 accident at the JCO fuel fabrication plant in Tokaimura, Japan (IAEA, 1999). Fortunately, such accidents are becoming increasingly rare thanks to advances in the field of nuclear criticality safety.

Aqueous fissile solutions are created in many processes throughout the nuclear industry and pose a particular set of hazards with respect to nuclear criticality safety. Fissile liquids are common in nuclear fuel fabrication and reprocessing, where solid fuels are dissolved in aqueous solutions containing acid for the purpose of homogeneous chemical processing. Fissile liquids also occur in Aqueous Homogeneous Reactors (AHR), a type of reactor in which the fuel solution and moderator comprise an aqueous solution of a uranium-containing salt, typically uranyl nitrate or uranyl sulphate. The system may be critical, such as the MIPR concept (Souto et al., 2005, Cooling et al., 2014), or a subcritical accelerator-driven system, like SHINE (Cunningham, 2015). These reactors are expected to become an important future source of radionuclides for medical applications (IAEA, 2009, National Academies of Sciences, 2018).

As with any operation involving the manipulation of hazardous materials, the processing and handling of aqueous fissile solutions is subject to rigorous risk assessment processes. These processes require a means to determine the potential consequences of accidental scenarios, and where accurate models are not available, risk assessors are forced to adopt conservative assumptions. There is therefore a need for accurate models to inform risk assessment processes and avoid the need for unnecessarily conservative safety margins. The aim of the work presented in this study is to contribute to the improvement of kinetic models for simulating nuclear criticality transients in fissile liquids, by providing the basis of a mechanistic model of radiolytic gas formation. The formation of radiolytic gas is an important factor determining the behaviour of fissile liquids during nuclear criticality excursions.

When the nucleus of a fissile atom undergoes nuclear fission in aqueous solution, the fully ionised fission fragments are emitted at high velocity from the site of fission. Except for a very small number of ternary and quaternary fissions (Gönnenwein, 2004), each fission event results in the creation of two charged fission fragments. Unlike in heterogeneous reactors, where fission occurs inside solid fuel elements surrounded by cladding, fission fragments created in fissile liquids are emitted directly into the aqueous solution. They travel in opposite directions to each other, leaving tracks of radiolysis products in their wake. The quantities and proportions of radiolysis products created depends on the linear energy transfer (LET) of the fission fragments (Lane et al., 1958).

The radiolysis products created by the fission fragments lead to the formation of radiolytic gas bubbles. The formation of these bubbles causes negative reactivity feedback by increasing the voidage of the solution and reducing its density. Bubbles of radiolytic gas have been observed to appear only after a threshold fission energy has been reached (Spiegler et al., 1962, Blue et al., 1964). This threshold energy corresponds to the energy required for the concentration of dissolved gas in the solution to reach the “critical concentration” at which the nucleation of radiolytic gas on the microscopic fission track bubbles becomes possible (Souto et al., 2005). Accurate models of this process are essential for predicting the time-dependent behaviour of fissile liquids during nuclear criticality excursions. They are also required for accurate models of AHR operation.

Many of the current models for simulating the kinetics of nuclear criticality transients in fissile solutions, such as CRITEX (Mather and Shaw, 1986), TRACE (Liem and Naito, 2015) and FETCH (Pain et al., 2001), require calibration of their radiolytic gas models against experimental data. The calibration is specific to a particular fuel solution composition and transient, limiting the predictive capability of these models. Meanwhile, other models (Souto et al., 2005, Cooling et al., 2014) make simplifying assumptions about the physics of the nucleation process, which prevents the capture of important effects related to radiolytic gas. The purpose of the work presented in this paper is to provide information necessary for a more fundamental, mechanistic model of radiolytic gas formation. In particular, the objective of this study is to provide a means of rapidly estimating the size distribution of fission track bubbles for a given solution composition and set of system conditions. This information can be used to determine the critical concentration of radiolytic gas in the fuel solution.

The model used in this study to estimate the size of fission track bubbles as a function of LET is adapted from a model originally published in 1958 (Ghormley, 1958) and updated in 1963 (Norman and Spiegler, 1963) and 1973 (Deitrich and Connolly, 1973). At the time this model was originally developed, only approximate estimates were available for the LET of fission fragments in aqueous fissile solutions. The novelty in the current work consists of calculating precise LET profiles for the fission fragments of 235U in solutions of uranyl nitrate and applying the model of bubble formation to determine the size distribution of bubbles produced as a function of the solution properties and system conditions. These results are then used to examine the influence of various parameters on the critical concentration for the nucleation of bubbles of radiolytic gas during a nuclear criticality excursion in a fissile liquid. Correlations are also presented allowing for the rapid estimation of the critical concentration. The results will provide the basis for a future study, in which the mechanics of gas bubble formation will be examined by means of a numerical simulation.

Much work was done in the fifties, sixties and seventies to examine the mechanics of fission tracks in aqueous fissile solutions and their role as nucleation sites for bubbles of radiolytic gas and steam.

In 1958, Ghormley of Oak Ridge National Laboratory exposed aqueous solutions of uranyl sulphate to fission recoil particles and measured the maximum superheat that could be applied before the appearance of stable bubbles (Ghormley, 1958). The purpose was to test the hypothesis that water vapour created due to the energy transfer from the fission recoil particles to the water would create tiny bubbles that could act as nucleation sites for boiling. It was noted that a significant superheat was required before visible bubbles appeared, from which Ghormley concluded that bubbles measuring approximately 1.4 μm in diameter were generated along the fission tracks and that visible bubbles would only be observed if the liquid superheat was sufficient for water to vaporise at the interface of bubbles of this size. It was calculated that approximately 28 MeV would be required to vaporise enough water to create the bubble and that this amount of energy would be deposited by a fission recoil particle over approximately 4 μm of track.

Ghormley’s work was based on the thermal spike theory of bubble nucleation in fissile liquids. In 1963 it was extended by Norman and Spiegler who, using an energy balance for the creation of vapour bubbles in fissile aqueous solutions, predicted the size of bubbles deposited along the fission track as a function of LET (Norman and Spiegler, 1963). Norman and Spiegler notably extended the theory, previously concerned only with steam bubble generation in superheated solutions, to the formation of hydrogen gas bubbles in subcooled aqueous solutions.

Spiegler describes their theory of gas bubble formation in subcooled aqueous solutions in a 1962 technical report (Spiegler et al., 1962). The theory can be summarised as follows:

  • a jet of vapour is formed along the track of a fission fragment as energy is transferred from the fission fragment to the water surrounding it;

  • the jet of water vapour breaks up into sections which become distinct vapour bubbles;

  • if the solution is subcooled the water vapour condenses on a timescale of the order 10-8 s;

  • a bubble of hydrogen is left which, in a solution where the dissolved gas concentration is relatively low, will collapse on a timescale of the order ~10-6 s.

The hydrogen bubble that remains will collapse unless the concentration of dissolved gas in solution is sufficient to establish a net flow of gas into the bubble. This is the critical concentration at which fission track bubbles will grow, leading to the appearance of voids due to radiolytic gas. The critical concentration of hydrogen in the liquid phase depends on the partial pressure of hydrogen gas in the bubble and the solubility of hydrogen in the fuel solution. It is determined fromCcrit=HH2/solPH2,b=HH2/solPl+2σRbyH2where Ccrit is the critical concentration of hydrogen gas in the fuel solution, HH2/sol is the Henry’s law constant for the solubility of hydrogen in the solution, PH2,b is the partial pressure of hydrogen inside the bubble, Pl is the liquid pressure, σ is the fuel solution surface tension, Rb is the bubble radius and yH2 is the mole fraction of hydrogen gas inside the bubble.

Up until this point, the discussion has been limited to hydrogen gas. However, nitrogen gas is also produced as a direct product of radiolysis of uranyl nitrate solutions and will also influence the size of the gas bubbles deposited in the tracks of fission fragments. Bidwell et al. (1956) showed the amount of nitrogen gas produced to be two orders of magnitude less than the quantity of hydrogen produced. Any effect on the size of bubbles produced is therefore expected to be negligible and will not be considered here. Oxygen gas is also produced during radiolysis of aqueous fissile solutions. However, it is not a direct radiolysis product but a product of ancillary reactions which take place later (Lane et al., 1958). Therefore, oxygen gas production will not affect the initial size of radiolysis bubbles but it will affect the critical concentration at which these bubbles begin to grow. Critical concentrations quoted in this work are calculated based on the assumption that the solution contains no dissolved oxygen. However, the effect of oxygen on the critical concentration will be incorporated into models to be presented in future studies.

The work of Norman and Spiegler was examined in a 1973 study (Deitrich and Connolly, 1973) by Dietrich and Connolly. Their experiments supported the hypothesis that particle LET is the factor determining the size of bubbles generated along fission tracks. They noted, however, that the values predicted by the theory of Norman and Spiegler overestimated the superheat required to nucleate steam bubbles in their own experiments, and they proposed a modification to the characteristic track length over which bubbles are formed (see Section 2.1).

The theory described above relies on the assumption that hydrogen gas will nucleate on the bubbles created by fission fragments at a lower concentration than that required for homogeneous nucleation. It is well known that homogeneous nucleation in certain gas/liquid systems tends to occur at lower concentrations than current theory predicts, and it is therefore useful to consider whether homogeneous nucleation might occur in the case of a fissile liquid at concentrations lower than the critical concentration discussed above.

An extensive review by Lubetkin (2003) of numerous theories to explain the discrepancy between the predictions of the classical theory and the relative ease with which homogeneous nucleation seems to occur, concluded that dissolved gas molecules act as surfactants, reducing the liquid surface tension, which in turn, reduces the concentration required to bring about homogeneous nucleation. Lubetkin notes that the surfactant effect is smallest for helium and also very small for hydrogen. Whereas carbon dioxide has been shown to undergo homogeneous nucleation at concentrations as low as 5.4 times the saturation concentration, hydrogen gas was found to require a concentration of at least 80 times saturation (Lubetkin, 2003).

According to the data of Norman and Spiegler (Spiegler et al., 1962, Norman and Spiegler, 1963), the critical concentration required for heterogeneous nucleation of hydrogen bubbles on fission tracks is approximately 28 times the saturation concentration of the solution. This is well below the concentration required for homogeneous nucleation to occur. Homogeneous nucleation can therefore be ruled out as a factor in the formation of gas voids in fissile solutions, lending further credibility to the thermal spike model.

The size of bubbles created on the tracks of fission fragments depends on the LET of the fragments as this determines the quantities of water vapour and hydrogen gas created. This was confirmed experimentally by Deitrich and Connolly (1973).

The LET of a fission fragment varies along the fission track and is a complex function of particle velocity, the stopping power of the medium with respect to that particle and the charge of the ionised particle (Tavernier, 2010). Unlike smaller charged particles which tend to retain their charge until they are almost stopped, fission fragments, which have an exceptionally large initial charge, deionise continuously along the track (Chadderton, 1988).

While there is a relatively large body of experimental data available in the literature for protons, electrons and alpha particles, there is relatively less for larger charged particles. In 1970, this motivated Northcliffe and Schilling (1970) to look at ways of predicting stopping powers for larger particles by extrapolation of the experimental data available at the time. While more recent stopping power tables have been compiled for some ions, those of Northcliffe and Schilling remain unmatched in the open literature for the range of charged particles and media covered.

A 2003 review by Helmut and Schinner (2003) of available data and models for calculating the stopping powers of heavy ions found that the Northcliffe and Schilling tables provided good accuracy at reproducing the general trends in stopping power, while failing to reproduce the Z1- and Z2-oscillations in stopping power observed as a function of particle (Z1-) and target medium (Z2-) atomic numbers. This is not surprising, since Northcliffe and Schilling extrapolated their values by assuming that the stopping power was a smooth function of the atomic numbers of the particles and target media. For this reason it was chosen to model the fission fragment LET using the software package SRIM (Ziegler et al., 2010). SRIM is capable of predicting the Z1- and Z2- oscillations noted by Paul and Schinner. For comparison, the LET profiles of the 6 highest-yielding fission fragments of 235U were calculated using both SRIM and the tables of Northcliffe and Schilling. The comparisons can be found in Appendix A and confirm a relatively close agreement with some notable differences between the two methods.

The LET of a heavy charged particle, such as a fission fragment, is quite different from that of smaller charged particles. With electrons, protons and alpha particles, the LET often starts relatively low, remaining relatively constant for most of the length of the particle track, before increasing sharply to a peak, then dropping to zero. This peak, known as the Bragg peak, occurs because, even as the particle is losing energy towards the end of the track, its reduced velocity increases its ability to interact with passing matter, leading to an increase in the rate of energy transfer from the particle to the medium.

Fission fragments may be emitted with energy on the same order of magnitude as an alpha particle, however, fission fragments have much greater mass, and therefore much lower velocities. Since it is velocity which determines the ability of the particle to interact with, and transfer energy to, the matter surrounding it, the LET of fission fragments starts close to its maximum, and Bragg peak’s in the LET profiles are either absent or very small (see Section 3). A notable difference between the LET profiles calculated using Northcliffe and Schilling compared to SRIM was that plateaus or peaks due to the Bragg peak effect were significantly less pronounced in the LET profiles predicted by SRIM.

Section snippets

Methodology and data

The following section presents the methodology used to predict values for fission fragment LET and the resulting distribution of bubble sizes created along the fission track. The methods are applied as indicated in the literature, subject to some modifications which are described below.

Results

In this section, results are presented for the LET of fission fragments in aqueous fissile solutions and the size of hydrogen gas bubbles deposited along the tracks of those fission fragments. The size of vapour bubbles deposited along the fission tracks has already been thoroughly examined in the literature, as discussed in previous sections, and no further examination will be presented here. This section is concerned with the creation of hydrogen gas bubbles, along fission fragment tracks,

Conclusions

The LET profiles of fission fragments in aqueous solutions of uranyl nitrate have been calculated and used to predict the distribution of bubble sizes generated along the tracks of these fragments as they travel through the solution. The critical concentration at which hydrogen gas appears as voids in a fissile solution is determined by the size and availability of nucleation sites where dissolved gas may come out of solution. The bubbles created in the wake of fission fragments can serve as

Data statement

In accordance with EPSRC funding requirements all supporting data used to create figures in this paper may be accessed at the following URL: http://doi.org/10.5281/zenodo.2617156.

CRediT authorship contribution statement

George E. Winter: Conceptualization, Methodology, Software, Data curation, Formal analysis, Writing - Original draft preparation, Writing - Review & Editing, Visualization, Investigation. Christopher M. Cooling: Conceptualization, Supervision, Writing - Review & Editing. Matthew D. Eaton: Conceptualization, Supervision, Writing - Review & Editing, Funding Acquisition.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

Mr. G.E. Winter would like to acknowledge the support of EPSRC and AWE plc through the industrial CASE programme. Dr. C.M. Cooling and Dr. M.D. Eaton would like to thank EPSRC for their support through the following grants: Adaptive Hierarchical Radiation Transport Methods to Meet Future Challenges in Reactor Physics (EPSRC Grant No.: EP/J002011/1) and Nuclear Reactor Kinetics Modelling and Simulation Tools for Small Modular Reactor (SMR) Start-Up Dynamics and Nuclear Criticality Safety

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      Citation Excerpt :

      The critical concentration therefore depends on the size of the gas bubbles deposited along the fission tracks. A method for determining the bubble size distribution was presented in a previous study (Winter et al., 2020) and will be applied in the present study to examine the processes leading to the formation of radiolytic gas bubbles in a fissile solution during saturation with radiolytic gas. Models of radiolytic gas production in fissile liquids typically fall into one of two categories: empirical models which are calibrated against experimental data (Barbry et al., n.d.) and models that avoid the need for experimental calibration by making simplifying assumptions which prevent them from capturing all the transient effects due to radiolytic gas (Souto et al., 2005; C.M. Cooling, 2014).

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