A Numerical Simulation of Radiation Chemistry for Controlling the Oxidising Environment in Water-Cooled Nuclear Power Reactors

: Maintaining the integrity of materials of light-water nuclear power reactors requires the development of effective methods to control and minimise the corrosive environment associated with the radiolysis of a coolant. In this study, the behaviour of the oxidising environment is simulated using a hybrid method. The hybrid method has advantages in that the production of radiolytic species under exposure of the coolant to ionising radiation is simulated while providing material and charge balances. Steady-state concentrations of stable and transient oxidising agents are calculated as a function of radiation composition and dose rate by numerical integration of the system of kinetic equations describing radiation chemistry of neutral water, the alkaline solution, and the hydrogenated systems at 300 ◦ C. The importance of the reactions and equilibria constituting the radiolysis scheme of the coolant is assessed. The inﬂuence of the presence of a base and the injected H 2 on the yield of key reactions responsible for the formation of the main oxidants H 2 O 2 and O 2 are discussed. Simulation indicated the synergic effect of H 2 gas and base added to the coolant on diminishment of the steady-state concentration of oxidants.


Introduction
Light-water reactors (LWRs), categorised into boiling-water reactors (BWRs) and pressurised-water reactors (PWRs), are the most widespread type of nuclear power reactors [1]. In LWRs, water plays a role of a neutron moderator and a coolant. The cooling water operates at high pressure and temperature conditions (7-7.5 MPa, 285 • C in BWRs and ca. 15.5 MPa, 315 • C in PWRs) and is exposed to intense fluxes of gamma rays and fast neutrons [2] (pp. 960-1023). Exposition to ionising radiation results in the formation of transient radical species: hydrated electrons (e aq •− ), hydroxyl radicals ( · OH), hydrogen radical atoms (H · ), hydroperoxyl radicals (HO 2 · ), ions: H 3 O + , OH − , O 2 − , HO 2 − , O − , and stable molecular products: molecular hydrogen (H 2 ), hydrogen peroxide (H 2 O 2 ), molecular oxygen (O 2 ) [3] (pp. 331-364). The stable oxidants (H 2 O 2 and O 2 ) create a corrosive environment contributing significantly to stress corrosion cracking of the structural materials [1,2,[4][5][6]. The harmful effect of oxidants is strengthened by the low acidity (low pH) of water at the operating temperatures. For example, because the ionic product of water (K w ) abruptly increases with temperature, the pH of liquid neutral water is 5.8 at 300 • C, compared with pH 7 at 25 • C. To minimise the concentration of dissolved oxidants and increase pH, the chemicals such as H 2 , hydrazine (N 2 H 4 ), ammonia (NH 3 ), lithium hydroxide (LiOH) are added to the coolant [1]. A quantitative understanding of how these additives change a corrosive environment in the nuclear core is still insufficient but essential to control stress corrosion cracking and maintain the integrity of materials in reactor circuits. In particular, it is important to estimate critical hydrogen concentration (CHC), i.e., the amount of dissolved hydrogen gas needed to suppress the net production of O 2 and H 2 O 2 in the reactor circuit [2] (pp. 960-1023).
Direct measurements of the chemical environment at the reactor core are not feasible because of high pressure, high temperature, and intense radiation field. Analysis of the reactor water is possible only at sampling points far away from circulation lines. Therefore, computer simulation is indispensable to model the radiation chemistry of the coolant under different operational conditions and assess the response of the oxidising environment to the additives (H 2 , LiOH, N 2 H 4 , NH 3 ).
As knowledge of high-temperature water radiolysis develops the simulations have been widely applied to estimate the chemistry of the coolant, both in the core and in associated pipelines [7][8][9][10][11][12][13][14]. Usually, simulation of the coolant chemistry is based on the numerical solution of a set of ordinary differential equations describing reactions of the primary and secondary chemical species, which are formed during the exposition of cooling water to ionising radiation. The reliability of such a simulation largely depends on the accuracy of the input data including rate constants for the zero-, first-, and second-order reactions. The former rate constants are directly related to the radiation chemical yields of the primary species formed in ca. 10 −7 s after the ionising event [3] and expressed in the number of species per 100 eV of energy deposited in the medium.
During the past few decades, the input data for the simulation of the high-temperature radiolysis of water have been amassed by several research groups based on the measurements or extrapolation of the data available for lower temperatures [13,15,16]. The variations between outputs of simulations using the datasets from different laboratories were examined by Katsumura et al. [13]. These authors found variations particularly resulting from uncertainties in the primary yields of · OH, H 2 , and H 2 O 2 for neutron radiolysis, and confirmed the significance of reaction (42) between H · and H 2 O (Table 1) indicated by the earlier simulations of the coolant chemistry [8,10]. Later simulation using the hybrid method showed that the uncertainty in the rate constant of 5 × 10 3 M −1 s −1 may result in the unsureness of the estimated steady-state concentration of H 2 , H 2 O 2 , and O 2 of 17, 9, and 4 µM, respectively [14]. Table 1. Production rates P i(g) and P i(n) , of the primary species obtained assuming the dose rate value of 1 kGy·s −1 and using the G values for gamma and fast neutron radiolysis of water at 300 • C from ref. [14]. Last row describes loss of water molecules resulting from irradiation.
Since 2005, the role of reaction (42) in the radiolysis of water at high temperatures has been under intensive scientific debate [17][18][19][20][21][22]. One of the difficulties in recognising the importance of water oxidation by H · in the liquid phase results from high activation energy [19] and a negligibly slow rate at ambient conditions [18,19,23]. The latest pulse radiolysis measurements in an aqueous iodide solution showed an increase in the rate constants, from ca. 4 × 10 3 to 6 × 10 4 M −1 s −1 , in a temperature range of 270-350 • C [22]. These measurements confirmed earlier evaluations using thermodynamics [19] and muonium chemistry [21].
In this study, we exploited the hybrid method for numerical simulation of LWR coolant chemistry [14] and the measured high-temperature rate of water oxidation by H · [22] to identify the principal reactions responsible for the production of oxidants in the pressurised liquid water exposed to gamma (γ) and fast neutron radiation at 300 • C. Steady-state concentrations of oxidants calculated assuming averaged values of reaction rate constants reported from different laboratories are presented for mixed radiation as a function of gamma to fast neutron dose rate ratio ranging from 0 to 1. The effectiveness of additives such as H 2 and LiOH in suppressing the production of stable oxidants is simulated and discussed.

Materials and Methods
Numerical simulations were performed using the hybrid method, described in detail elsewhere [14] and schematically illustrated in Scheme 1. The concept of the hybrid method is to combine diffusion-kinetic calculation for the radiolytic species initially clustered in radiation tracks and kinetic calculation of reactants homogenously distributed in the bulk solvent. Numerical integration of a set of diffusion-kinetic equations up to 0.1 µs provided the primary yields (or G values) of the primary species (e aq •− , • OH, H • , H + , OH − , H 2 , H 2 O 2 , HO 2 • ). The primary yields, still often expressed in molecules per 100 eV of absorbed energy, were then recalculated to production rates P i according to the required dose rate. Simulations were performed for the dose rates of 1 and 10 kGy·s −1 , which are similar to the in-core dose rates of gamma and fast neutron radiation. Scheme 1. Schematic illustration of the hybrid method [14]: c i and D i are molar concentrations and diffusion coefficients of the primary species; P i (in M·s −1 ) = 1.0364·10 −7 ·G i (in molecules per Table 1 presents production rates P i(g) and P i(n) , obtained for gamma and fast neutron radiolysis of water at 300 • C. For mixed radiation, production rates were calculated using the equation The advantage of employing the hybrid method is that the production of primary radiolytic species satisfies both the material balance and the charge balance. At the concentration of additives [OH − ] = 2·10 −4 mol·kg −1 and [H 2 ] ≤ 1.5·10 −4 mol·kg −1 , considered in this study, the production rates for alkaline and hydrogenated solutions can be assumed the same as for neutral water at 300 • C [24].
The production rates specified in Table 1 were used as zeroth-order terms in a system of differential kinetic equations, which was defined with respect to 13 species (i.e., e aq •− , involved in the 1st and 2nd order reactions listed in Table 2. The reaction set and the associated high-temperature rate constants were compiled based on the data reported from different laboratories. Table 2. Reaction set for the radiolysis of high-temperature water and rate constants k at 300 • C (units for 2nd and 1st order reactions are M −1 ·s −1 and s −1 , respectively). For reactions between similar species, the value of k, not 2k, is given.

No.
Reaction In order to obtain steady-state concentrations, the system of differential equations was numerically integrated up to the time limit t max = 100 s using the FACSIMILE 4 (ver. 4.2.50, MCPA Software, Harwell, UK) package. Except for H 2 O, H + , and OH -, the initial concentration of the reactants was set to zero. The molar concentration of H 2 O was calculated using the density of water at 300 • C, 0.72 kg·dm −3 . The initial concentration for H + and OH − , were obtained from the relation pK w = pH + pOH, where pK w , expressed in molar units, equals 11.64 at 300 • C [16].

Results
The results of the kinetic simulations presented here were obtained for neutral water and for an alkaline solution containing 2·10 −4 mol·kg −1 LiOH using the reaction scheme and the rate constants from Table 2.
Concentration profiles of the radiolytic species shown in Figure 1 indicate that the steady state is reached after ca. 10 s. Within 10 and 100 s, the relative change in species concentration is less than 0.08%.
The main species responsible for the formation of oxidising environment are oxygen and hydrogen peroxide. The steady-state concentrations of • OH and HO 2 • are lower by about two orders of magnitude.   [16], whereas γ rays are representative for low-LET radiation (LET~0.2 eV·nm −1 ), producing higher yields of radical than molecular radiolytic products. The number of generated oxidants depends on the dose rate. A particularly significant increase with increasing dose rate was obtained for • OH and H 2 O 2 .
The hybrid method allows one to assess the change in pH under exposure to ionising radiation. The simulation showed a slight decrease in pH, which is more pronounced for fast neutron radiation-namely, the 100 s exposure to fast neutrons leads to a decline in pH by 0.0023 at the radiation dose of 1 kGy·s −1 and by 0.0039 at 10 kGy·s −1 , whereas exposure to γ rays leads to a decline in pH by 0.0013 at 1 kGy·s −1 and by 0.0028 at 10 kGy·s −1 .

Alkaline Aqueous Solution
In comparison with neutral water, radiolysis of alkaline solution results in lower steady-state concentrations of H 2 O 2 , O 2 , HO 2 • , H 2 , H • , and O 2 •− but in the higher concentrations of e aq •− , • OH, and HO 2 -. Figure 3 presents how the addition of base alters the steady-state concentrations of O 2 , H 2 O 2 , • OH, and HO 2 • in the mixed gamma-neutron radiolysis. Simulations were performed for fraction of γ rays varying from f g = 0 (high LET neutron radiolysis) to f g = 1 (low-LET gamma radiolysis). In the alkaline solution, the steady-state concentration of stable oxidants is significantly reduced. A change in O 2 concentration is more pronounced at the dose rate 1 kGy·s −1 and develops with increasing f g . Simulation shows the decrease by 60% and 85%, respectively, for fast neutron (f g = 0) and gamma (f g = 1) radiolysis at 1 kGy·s −1 , compared to 32% (f g = 0) and 74% (f g = 1) at 10 kGy·s −1 . In contrast to O 2 , the decrease in the concentration of H 2 O 2 is not very sensitive to either radiation composition or dose rate. At a smaller dose rate, the amount of H 2 O 2 drops by 81%. At 10 kGy·s −1 , decreases by 73% and 77% were obtained for f g = 0 and f g = 1, respectively.
Steady-state concentrations of HO 2 • and • OH are less sensitive to the addition of base. At 1 kGy·s −1 , a decrease in [HO 2 • ] but an increase in [ • OH] by 25% and 50% were obtained for f g = 0 and f g = 1, respectively. Taking into account that the concentration • OH is by several orders of magnitude lower, the overall number of oxidants in the alkaline solution is greatly diminished.

Effectiveness of Hydrogen Addition in Suppressing Production of Stable Oxidants
Molecular hydrogen is one of the reducing products of water radiolysis, but the amount of radiolytic H 2 is not enough to suppress the formation of stable oxidants. Therefore, to change the oxidising conditions into the reducing ones, hydrogen gas was injected into the coolant. Hydrogen injection promotes reaction (43), converting · OH into the reducing H · radical atom, which initiates the transformation of O 2 to H 2 O in a sequence of reactions, as illustrated in Scheme 2.  The steady concentration of O 2 calculated vs. the increasing amount of H 2 is presented in Figure 5. Compared with H 2 O 2 , the concentration of O 2 responds significantly more to the amount of added H 2 . The reduction in O 2 in neutral water is not very sensitive to LET and dose rate. In the case of alkaline solution, the synergic effect of H 2 and OH − is more pronounced for gamma radiolysis.
In Table 3, the effectiveness of hydrogen injection is expressed by a ratio of the steadystate concentration of H 2 O 2 and O 2 in the absence of extra hydrogen and in the presence of 1.5·10 −4 mol·kg −1 H 2 . Data are shown as a function of dose rate (1 and 10 kGy·s −1 ), LET (fast neutron and gamma radiation), and pH (neutral and alkaline water at 300 • C). An increase in pH enhances the reducing effect of H 2 . The influence of pH is slightly more pronounced at the higher dose rate and lower LET.

Discussion
The reaction schemes assumed for simulation by different groups vary at some points. To account for the different values of reported rate constants, the arithmetic mean was assumed as indicated in Table 2, and the following reactions were considered as parallel reactions: (1) and (57); (9) and (10); (14) and (15)
The most significant are equilibrium reactions, but the assessment of their importance should take into account a net contribution of each equilibrium, expressed as a difference between the integrated contributions of forwarding and backward reactions.  The importance of other reactions is revealed in Scheme 3. The contributions, integrated up to 100 s, were classified into the following three categories: Compared with neutral water, the significance of reactions initiated by e aq •− , and the alkaline forms of • OH and HO 2 • , i.e., O •− and HO 2 − , is noticeably increased in the alkaline solution, whereas the increase in R6 and R12 seen for neutral hydrogenated system indicates more efficient scavenging of • OH and H 2 O 2 by H • in this system. The radiation chemistry of the alkaline hydrogenated solution is controlled by six reactions: (3), (4), (5), (7), (48), and (57). This is a consequence of significant right shifts of equilibria (40) ↔ (41) and (38) ↔ (39) (Figure 6). In irradiated alkaline hydrogenated solutions, the reducing conditions are maintained by simultaneous production of e aq •− and H • .

Scheme 3.
Graphical presentation of importance of reactions from Table 2. The integrated contributions R were classified into categories I, II, and III, defined in the text above the scheme. Colours correspond to neutral water (orange), alkaline solution (green), and hydrogenated (3·10 −5 mol·kg −1 H 2 ) systems: neutral (violet) and alkaline (yellow). Based on the simulation carried out for 1 kGy·s −1 .

Key Reactions Contributing to Production of Oxidants
The simulations carried out in this study confirm that H 2 O 2 and O 2 are the main radiolytic products responsible for the formation of the oxidising environment in the coolant. The contributions of key reactions leading to the formation and decay of these oxidants in fast neutron (n) and gamma (g) radiolysis of neutral water and the alkaline solution are presented in Figures 7 and 8. The action of added hydrogen is shown for [H 2 ] = 3·10 −5 mol·kg −1 . Contributions of all reactions were integrated up to 100 s.
Exposition of the coolant to ionising radiation results in the continuous production of H 2 O 2 in zeroth-order reactions. In Figure 7, integrated production rates are represented by P(g) and P(n). Since the production rate of H 2 O 2 by fast neutrons is higher (Table 1), P(n) exceeds P(g). At the small concentrations of base and H 2 , considered here, the production of H 2 O 2 is independent of the composition of the irradiated system. Apart from the production, the formation of H 2 O 2 is controlled by reactions (3), (14), (35), while reactions (7), (17), (34) determine the decay of H 2 O 2 . Since the contribution of reaction (35) prevails, the net contribution R35-R34 is shown in the upper panel. This net contribution is only relevant for the gamma radiolysis of alkaline solution. Production P(n) is the most important source of H 2 O 2 in fast neutron radiolysis of neutral water, whereas P(g), R3(g), and R14(g) are comparable. Contribution R14(g) is highly increased by H 2 injection to neutral water but minimised in the presence of a base.
The enormous increase in R3(g) observed for the alkaline solution and neutral hydrogenated system is balanced by the decay of H 2 O 2 in reaction with the radicals, e aq •− and · OH. The synergic effect of H 2 and OH − is seen in the diminishment of R3(g) obtained for the alkaline hydrogenated solution.
Contributions of key reactions directly responsible for the formation and decay of O 2 are displayed in Figure 8.
In neutral and neutrally hydrogenated systems, O 2 decays in reaction (13), producing HO 2 • , which reforms O 2 in reaction (19) with • OH. The restoration of O 2 is less productive in the hydrogenated system, thus diminishing the concentration of O 2 ( Figure 5). In the alkaline solution, O 2 is produced in reaction (19) but effectively scavenged by e aq •− in reaction (8). In the alkaline hydrogenated system, R19(g) and R19(n) are of the order of 10 −6 and 10 −5 mol·kg −1 , respectively, and the formation of O 2 is suppressed. Figure 7. Integrated contributions of key reactions responsible for the formation and decay of H 2 O 2 calculated for fast neutron (n) and gamma (g) radiolysis of neutral water, alkaline solution, and hydrogenated (3·10 −5 mol·kg −1 H 2 ) neutral and alkaline systems. Integrated production rates are represented by P(g) and P(n). Based on the simulation for 1 kGy·s −1 .

Conclusions
Simulation of water chemistry at 300 • C and under exposure to ionising radiation was performed using the hybrid method [14] and the measured rate constant of H • + H 2 O → • OH + H 2 reaction [22]. The steady-state concentration of oxidants (O 2 , H 2 O 2 , HO 2 • , • OH) were calculated for neutral water (pH 573K = 5.8) and an alkaline solution (pH 573K = 8.0), as a function of added hydrogen (0 ≤ [H 2 ] ≤ 1.5·10 −4 mol·kg −1 ), dose rate (1 and 10 kGy·s −1 ), and the fraction of γ rays (0 ≤ f g ≤ 1) in the mixed fast neutron-gamma radiation. The concentration of main oxidants (O 2 , H 2 O 2 ) increases with the dose rate but decreases with the increasing fraction of γ rays. At higher pH the total amount of O 2 , H 2 O 2 is significantly diminished. The simulation indicates that the addition of H 2 to the alkaline solution effectively suppresses the formation of oxidising environment. This synergic effect of H 2 and OH − may lower CHC.
The importance of reactions contributing to the coolant radiolysis (Table 2) by fast neutrons and γ rays were discussed for neutral, deaerated water and the alkaline solution, and both of these systems contained 3·10 −5 mol·kg −1 H 2 .
The synergic effect of hydrogen gas and base added to the coolant, demonstrated in this study, provides an important guideline for the development of more efficient methods for controlling the oxidising environment and minimising corrosion problems in PWR engineering.