Hybrid method for numerical modelling of LWR coolant chemistry

Radiation Physics and Chemistry 127 (2016) 236–242

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Hybrid method for numerical modelling of LWR coolant chemistry Dorota Swiatla-Wojcik Institute of Applied Radiation Chemistry, Faculty of Chemistry, Lodz University of Technology, Zeromskiego 116, 90-924 Lodz, Poland

H I G H L I G H T S







A hybrid method for modelling radiation chemistry of the coolant is proposed. Non-homogeneous intra-track chemistry and secondary bulk reactions are combined. Principal reactions responsible for production of H2, O2 and H2O2 are indicated. Sensitivity to the rate of reaction H þH2O ¼ OHþH2 is discussed.

art ic l e i nf o

a b s t r a c t

Article history: Received 9 May 2016 Received in revised form 17 June 2016 Accepted 4 July 2016 Available online 5 July 2016

A comprehensive approach is proposed to model radiation chemistry of the cooling water under exposure to neutron and gamma radiation at 300 °C. It covers diffusion-kinetic processes in radiation tracks and secondary reactions in the bulk coolant. Steady-state concentrations of the radiolytic products have been assessed based on the simulated time dependent concentration profiles. The principal reactions contributing to the formation of H2, O2 and H2O2 were indicated. Simulation was carried out depending on the amount of extra hydrogen dissolved in the coolant to reduce concentration of corrosive agents. High sensitivity to the rate of reaction H þ H2O¼ OHþH2 is shown and discussed. & 2016 Elsevier Ltd. All rights reserved.

Keywords: Hydrogen chemistry Nuclear power engineering High temperature water radiolysis Numerical simulation

1. Introduction Stability and safety of light water nuclear power reactors (LWRs), including Pressurized Water Reactors (PWRs) and Boiling Water Reactors (BWRs), depend on our understanding of radiation-induced chemistry in high temperature water. Under the operating conditions the coolant is exposed to a mixed flux of fast neutrons and γ-rays at high temperature (T  300 °C) and high pressure (P4 100 bar) (Buxton, 2001; Baldacchino and Hickel, 2008; Takagi et al., 2011). Irradiation of liquid water initially results in the formation of short-living radicals (hydrated electrons, H, OH and HO2) and ionic species (H3O þ , OH  ). Mutual reactions of the transient species result in the formation of stable strong oxidants (H2O2, O2), and hydrogen (H2) (Buxton, 2001). Irradiation-assisted production of the oxy-hydrogen gas and formation of harsh chemical environments pose severe problems (Takagi et al., 2011; Yeh and Wang, 2008; Wilson et al., 2009). Austenitic steels and nickel alloys used in the currently operating LWRs have occurred more susceptible to intergranular stress corrosion cracking (IGSCC). The harmful effect of H2O2 and O2 is strengthened by low E-mail address: [email protected] http://dx.doi.org/10.1016/j.radphyschem.2016.07.005 0969-806X/& 2016 Elsevier Ltd. All rights reserved.

pH, ca. 5.8 at 300 °C compared to pH 7 at 25 °C, because the ionic product of water, Kw, increases with temperature. The corrosive environment may be additionally reinforced by the presence of trace impurities, which come from defective fuel, corrosion or erosion of fuel clad or corrosion products itself. Decomposition of trace impurities in the mixed field of ionising radiation and high temperature may result in the formation of water-soluble corrosive species contributing to degradation of performance of reactor components. Precise prediction of radiolysis effects in the nuclear reactor circuits and selection of conditions at which the radiolytic decomposition of the coolant is suppressed are essential to control IGSCC and maintain integrity of materials. Since direct analysis of the reactor water is possible at sampling points far away from circulation lines, simulation of radiation chemistry of the coolant is necessary to guide or replace difficult, dangerous, and expensive experiments, to deal with uncertainties or gaps in the data, and to validate the experimental work. Simulation is expected to be able to predict time profiles and steady-state concentrations of the chemical species under different operational conditions. It is also important to estimate the so-called critical hydrogen concentration (CHC), i.e. the amount of dissolved hydrogen gas needed to maintain a reducing environment and thereby suppress the net

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Fig. 1. Stages of water radiolysis.

radiolytic production of oxygen and hydrogen peroxide (Takagi et al., 2011). Radiation chemistry of water follows a sequence of physical events related with an energy loss by ionising particle (neutron, charged particle or photon) and the formation of a radiation track along the particle path (Fig. 1). At ca. 10  16 s the radiation track in liquid water contains energetic electrons, H2O þ ions, and electronically excited molecules. Partial recombination, dissociation of excited molecules, thermalisation and solvation of free electrons, and radical products result in the formation of the initial products within 10  12 s from the passage of an incident particle (Buxton, 2001). Initially, spatial distribution of the radiolytic species is highly non-homogeneous. The reactive species produced by low LET (Linear Energy Transfer) radiation are initially distributed in spherical isolated spurs. In the case of high LET, neutron or heavyion radiolysis, spurs overlap and cylindrical tracks of pseudo-twodimensional symmetry are formed. Diffusive expansion of the radiation track is associated with mutual reactions of the species and the formation of the primary products, as shown in Fig. 1 (Buxton, 2008). The time required for decay of spatial correlation depends on both LET and temperature (Swiatla-Wojcik and Buxton, 1995, 1998; Sanguanmith et al., 2011). For low LET radiation, such as γrays or fast electrons, it is commonly assumed that the nonhomogeneous stage of water radiolysis takes ca. 0.1 μs. Decay of spatial correlation ends the non-homogenous stage of radiolysis. By that time the primary products become homogeneously distributed in the bulk solvent. In the absence of other solutes, bulk reactions of the primary products result in the partial restoration of H2O molecules and formation of the stable molecular products (H2, H2O2, O2). The number of radiolytic species per 100 eV of energy deposited by ionising radiation at the end of the non-homogenous stage defines the radiation chemical yield or the G-value, commonly expressed in molecules per 100 eV or in mmol J  1, where the conversion factor is 1 molecule/100 eV ¼ 0.10364 mmol J  1. The radiation chemical yields of the primary products depend on temperature and LET. Along with the dose-rate pattern at each portion of the reactor loop and the kinetic model, the G-values of the primary species are necessary to simulate radiolysis of the coolant and model preventive water chemistry from the viewpoint of mitigation of IGSCC. Despite much research work undertaken to establish the temperature dependence for the G-values of the primary species for different LET characteristics (Elliot and Bartels, 2009) knowledge of G(OH) and G(H2O2) at 300 °C is rather uncertain. Another difficulty arises from the fact that in order to model the water radiolysis in reactor circuits one needs to know a set of relevant chemical

reactions and their rate constants at the coolant operating temperature. Particularly challenging is to recognise reactions that require high activation energies. Such reactions are insignificant under ambient conditions, but may become very important at reactor temperatures (Swiatla-Wojcik and Buxton, 2005). To address the above issues a hybrid computational method for numerical simulation of the coolant chemistry is proposed here. As described in Section 2 the hybrid method provides a comprehensive approach by combining diffusion-kinetic calculations to obtain the G-values and kinetic calculations of homogeneous chemistry to estimate steady-state concentrations of the stable molecular products in the coolant. The method can be employed for a wide range of conditions comprising temperature, composition of radiation, and pH. A computing capability of the hybrid method is presented in Section 3. In particular, the proposed computational procedure has been used to show sensitivity of the steady-state concentrations of O2, H2O2, and H2 to the recently proposed mechanism for the bimolecular recombination of two hydrated electrons (eaq  ) producing the H atom (Swiatla-Wojcik, 2015) and to the rate of the reaction between H and H2O being very controversial at high temperatures (Sanguanmith et al., 2011; Swiatla-Wojcik and Buxton, 2005, 2010; Sims, 2006; Alcorn, 2014). Rate of reactions leading to the formation of OH and H2 is a key issue for the radiation chemistry of the coolant.

2. Outline of the hybrid method In order to perform a numerical simulation of water radiolysis one has to solve a set of over 40 differential equations describing decay of the transient primary species (eaq  , H, OH, H2, H2O2, H3O þ , OH  , HO2) and formation of the secondary products in chemical reactions presented in Table 1 (Takagi et al., 2011; Elliot and Bartels, 2009; Elliot, 1994). To reduce uncertainties associated with the G-values, the hybrid method is a two-step procedure illustrated in Fig. 2. The first step is based on the deterministic diffusion-kinetic modelling of the non-homogeneous stage of water radiolysis formulated in the earlier studies (Swiatla-Wojcik and Buxton, 1995, 1998, 2000, 2005). As most of the secondary products are formed during the bulk homogeneous stage of radiolysis, it was shown that the reaction scheme describing the nonhomogeneous stage can be limited to the reactions bolded in Table 1, but the differential equations should comprise diffusional terms to account for a non-uniform spatial distribution of the reactants. Numerical integration of these equations up to the time limit corresponding to decay of spatial correlation provides the G-values or the primary yields. In the second step of the hybrid

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Table 1 Reactions contributing coolant radiolysis and rate constants at 300 °C, where not stated taken from (Elliot and Bartels, 2009; Elliot, 1994). For a reaction between like species, the value of k, not 2k is given. The reactions contributing the non-homogeneous stage of the fast neutron and gamma radiolysis are given in bold. The reactions that mainly contribute to the homogenous chemistry under constant exposure to the mixed radiation, are indicated by stars. No.

Reactiona

k (M  1 s  1)

No.

Reactiona

k (M  1 s  1)

1a 1b 2 3 4* 5* 6 7 8 9* 10* 11* 12* 13 14* 15* 16* 17* 18 19* 20 21 22* 23* 24F

eaq  þeaq  þ 2H2O-H2 þ2OH eaq  þeaq  þ H2O-eaq  þH þ OH  eaq  þH þ H2O-H2 þOH  eaq  þOH-OH  eaq  þH2O2-OH þOH  eaq  þO2-O2  eaq  þO2  þH2O-H2O2 þ2OH  eaq  þHO2-HO2  H þH-H2 H þOH-H2O H þH2O2-OH þH2O Hþ O2-HO2 Hþ HO2-OH þOH Hþ O2  -HO2  OHþ OH-H2O2 OHþ H2-H þ H2O OHþ H2O2-HO2 þ H2O OHþ HO2-O2 þ H2O OHþ O2  -O2 þ OH  HO2 þ HO2-H2O2 þ O2 HO2 þ O2  þH2O-H2O2 þ O2 þ OH  O2  þ O2  þ 2H2O-H2O2 þ O2 þ 2OH  H þH2O-H2 þ OH H2O2-0.5O2 þ H2O H2O-H þ þOH 

6.1  106b 2.0  1011b 5.0  1011 3.7  1011 2.7  1011 2.2  1011 1.6  1011 1.6  1011 1.0  1011 6.3  1010 2.2  109 6.1  1010 2.1  1011 2.1  1011 1.0  1010 7.8  108 4.2  108 3.2  1010 9.0  1010 4.1  107 4.0  108 3.5  107 1.0  104c 3.8  10  2d 6.5  10  2d

24B* 25F 25B* 26F* 26B 27F 27B* 28F* 28B* 29F 29B 30F 30B 31F 31B* 32F 32B 33 34 35 36 37 38 39F 39B

H þ þOH  -H2O H2O2-H þ þ HO2  H þ þHO2  -H2O2 H2O2 þ OH  -HO2  þ H2O HO2  þH2O-H2O2 þOH  H-eaq  þ H þ eaq  þH þ -H eaq  þH2O-H þ OH  H þOH  -eaq  þ H2O OH-H þ þ O  H þ þO  -OH OHþ OH  -O  þ H2O O  þ H2O-OHþ OH  HO2-H þ þO2  H þ þO2  -HO2 HO2 þOH  -O2  þH2O O2  þ H2O-HO2 þOH  O  þ H2-H þOH  H2O2 þ O  -HO2 þOH  OHþ HO2  -O2  þH2O OHþ O  -HO2  eaq  þ HO2  -O  þ OH  eaq  þ O  þH2O-2OH  O  þ O2-O3  O3  -O  þO2

1.1  1012 2.5  101d 5.7  1011 1.4  1011 1.8  108 1.7  105d 7.2  1011 2.0  103 4.2  1010 2.5  101d 5.7  1011 1.4  1011 1.4  108 1.6  105d 5.7  1011 1.4  1011 2.9  104 1.8  109 1.1  1010 1.7  1011 3.4  1010 6.9  1010 1.1  1011 3.2  1010 1.9  107d

a

To simplify the notation the radical dot is omitted. See Section 3.1. c See Section 3.2. d Units are s  1. b

method the primary yields are used to define both the initial concentrations of the primary species and rates at which they are produced when the coolant is exposed to the ionising radiation. The advantage of the hybrid method is that the G-values used in the second step satisfy both the material balance and the charge balance. It makes calculations consistent and more accurate. Details of the computational procedure for the non-homogenous primary stage and the homogeneous bulk chemistry are given in Sections 2.1 and 2.2, respectively.

2.1. Modelling non-homogenous track chemistry Modelling of the non-homogenous stage of water radiolysis is based on the extended diffusion-kinetic approach describing temporal and spatial development of isolated spherical spurs (Swiatla-Wojcik and Buxton, 1995) and continuous cylindrical tracks (Swiatla-Wojcik and Buxton, 1998), which are representative entities for low- and high-LET tracks, respectively. Simultaneous diffusion and mutual reactions of the transient

Fig. 2. Schematic illustration of the hybrid method.

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species are mathematically expressed by the following set of coupled differential equations:

production of the primary products at the rates which are defined by the primary yields and dose rate.

∂c i = Di∇2c i − ∂t

∂c i =− ∂t

∑ k ijcicj + ∑ ∑ k jkcjck + Pi ,

∂c n =− ∂t

∑ k njcn cj + ∑ ∑ k jkcjck ,

∑ k ijcicj + ∑ ∑ k jkcjck j

j

(1)

k

where ci, and Di denote concentrations and diffusion coefficients of the primary species, i¼eaq  , H, OH, H2, H2O2, H þ (H3O þ ), OH  , HO2. The two sums of reaction terms in Eq. (1), where kij and kjk denote the second order rate constants, describe the decay and restoration of the i-th species. The numerical algorithm, developed by Burns et al. (1972, 1984), divides the reaction space into a number of concentric zones and treats reactions in each zone and simultaneous mass flow between adjoining zones according to the Fick's law of diffusion. The reactants are placed into each zone according to the initial distribution that defines the integration boundary conditions. The extended diffusion-kinetic model (Swiatla-Wojcik and Buxton, 1995, 1998; 2000) describes the initial species distribution by the properly chosen set of parameters accounting for the early track events, occurring within ca. 10  12 ps and leading to the establishment of thermal equilibrium (see Fig. 1). In LWR coolant is exposed to a mixed flux of high-LET neutron radiation and low-LET gamma radiation. More than 90% of the energy transferred in neutron collisions produces a wide spectrum of recoil proton energies. For 2 MeV neutrons from the fission process, main contribution to the radiolysis comes from four recoil protons having the mean LET of 20, 27, 34.5, and 44 eV nm  1 (Elliot and Bartels, 2009). Since the maximum range of these secondary protons, ca. 10  6 m, is small compared to the fast neutron collision path, of the order of 10  2 m, the tracks of secondary protons from the same neutron can be analysed separately, like radiation spurs produced by photons. Following this simplification, mixed radiation is considered as consisting of low-LET, spherical, and high-LET, cylindrical, components developing independently. Integration of equation set (1) provides concentration of the primary species at the beginning of the homogeneous stage. The time required to stop integration of set (1) is not precisely known. In general the lifetime of spatial correlation depends on temperature, LET, and dose rate. Monte Carlo simulation predicted 0.64  10  7 s for the decay of low-LET track at 300 °C (Sanguanmith et al., 2011). For the purpose of the present study the upper integration limit of 1  10  7 s has been assumed for both gamma and neutron radiation. The computed primary yields are presented in Table 2.

j

j

j

i = primary products

k

j

n ≠ i (secondary products ) (2)

k

The production term Pi is considered only for the primary radiolytic products, i ¼eaq  , H, OH, H2, H2O2, H þ (H3O þ ), OH-, HO2 and not for the secondary ones n≠i. Eq. (3) relates Pi, expressed in M s  1, with the primary yield of the i-th species, Gi in molecules per 100 eV, dose rate in Gy s  1, and water density, ρ, in kg L  1.

Pi = 1.0364 × 10−7 × Gi × (dose rate) × ρ

(3)

In the case of mixed radiation net production rate, Pimix, is represented by the weighted sum of gamma (γ) and neutron (n) components:

P mix = 1.0364 × 10−7 × ρ i × ⌊f γGiγ × ( dose rate) + fnGni × (dose rate)n ⌋ γ

(4)

where factors fγ and fn describe composition of the radiation field in a given reactor section, (dose rate)γ and (dose rate)n are charγ acteristic dose rates, and Gi , Gin denote the primary yields in gamma and fast-neutron radiolysis, given in Table 2. The set of ordinary differential equations describing reactions of the primary and secondary products (see Table 1) is integrated numerically to provide steady state concentrations of the radiolytic products in a given part of the reactor circuit. If necessary reactions with added solutes or trace impurities can be easily included assuming that their rate constants are known at high temperatures. A convenient programming environment for implementation of the hybrid method is provided by the FACSIMILE package. Using FACSIMILE one can described complex chemical kinetics by means of a special high-level programming language. In the present formulation of the hybrid method an overlap of radiation tracks and spurs under constant exposure to ionising radiation is neglected. Although this problem requires further studies, an inaccuracy resulting from gaps and uncertainty in the reaction rate constants is supposed to be more significant.

3. Results and discussion

At this computational stage it is assumed that all the reactants are uniformly distributed throughout the medium and undergo homogeneous chemical kinetics. A set of coupled differential kinetic equations describes chemical reactions under constant

Simulation of the coolant chemistry depends on a large number of parameters and inputs. Apart from the technical information about flow rates, temperature profiles, in-core power distribution, radiation composition, dose rates, etc., the reaction rate constants between 250 and 350 °C are essential. Moreover, despite of the progress that has been made in the last decade there is still a critical need for constructing a reliable reaction scheme. The

Table 2 Primary yields, in molecules per 100 eV, computed assuming 1  10  7 s for the upper integration limit of equation set (1). For comparison the values in parentheses have been obtained from the polynomial fits to the measured G-values (Elliot and Bartels, 2009).

Table 3 Conditions assumed for simulation of the coolant chemistry.

2.2. Modelling bulk chemistry

Primary Yield

 eaq

H3O þ

H

OH

OH-

4.78 (5.74)

0.61 0.64 0.25 0.00 (  ) (0.64) (0.27) (  )

H2

Gamma

0.1 μs 3.20 3.80 Exp. (3.43) (  )

0.80 (1.56)

Neutrona

0.1 μs 1.42 Exp. (1.29)

0.73 3.66 0.52 1.19 (0.59) (2.87) (  ) (1.00)

a

1.95 ()

The G-values less than 5  10  3 are not shown in the table.

H2O2

HO2

0.43 0.01 (0.40) (0.03)

Neutral water Temperature

pH300C ¼5.8 300 °C

Mixed radiation: Gamma rays 33% 2 MeV neutrons 67% Dose rate: Gamma 2 MeV Neutrons

3.3 kGy s  1 6.6 kGy s  1

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Fig. 3. The concentration of the main radiolytic products as a function of exposure time calculated for the operating conditions listed in Table 3.

Fig. 4. The concentration of the radiolytic products after 100 s of exposure of liquid water to the ionizing radiation at 300 °C: (squares) – calculated for conditions listed in Table 3, (diamonds) – calculated assuming 100% gamma radiation and dose rate 10 kGy s  1. The observed trend is shown by arrows.

hybrid method provides the comprehensive approach to deal with these problems. The issues addressed below concern the formation of the stable molecular products O2, H2O2, and H2 under the conditions given in Table 3. Numerical simulations have been carried out assuming the reaction scheme from Table 1. The production rates of the primary species have been calculated based on Eq. (4) and the G-values given in Table 2. 3.1. Time-dependent profiles Numerical integration of differential kinetic Eq. (2) gives timedependent profiles describing how concentrations of the radiolytic products vary with time of exposure in the approach to equilibrium. Based on these profiles one can estimate the steady state concentrations. Fig. 3 shows that the equilibrium is reached at about 1  102 s. By this time the main products formed are H2, H2O2, O2, and HO2. As it can be seen in Fig. 4 the concentration of other radiolytic products is lower by at least two orders of magnitude. Compared to the earlier simulations for gamma radiolysis of pure water at 250 °C (Sunaryo et al., 1995; IAEA-TECDOC-996, 1998), the steady state concentration of the molecular products are higher. In order to show the effect of radiation composition, steady state concentrations of the radiolytic products have been calculated for gamma radiation and the dose rate 10 kGy s  1 (see

diamonds in Fig. 4). The change in cO2 and cH is small, but the steady state concentrations of the ionic species are significantly higher in gamma radiolysis. The increase is also seen for cOH, whereas cH2O2, cH2, cHO2 are diminished. Unlike the previous simulations (Sunaryo et al., 1995; IAEA-TECDOC-996, 1998) assuming equal production rate of eaq  and H þ (PH þ ¼Peaq), the hybrid method takes PH þ ¼POH  þ Peaq. Comparative simulations using the above approaches show significantly different values of steady state concentrations in neutron-gamma radiolysis. Namely, the omission of POH  results in much higher concentration of the ionic species and significantly lower concentrations of H2O2, H2, and HO2. Using the hybrid method one can gauge contribution of reactions to assess the principal ones. The numerical procedure has been adjusted to calculate time-dependent profiles for integrated contributions of the reactions from Table 1. The principal reactions of the bulk stage are indicated by stars. As it can be seen not all intra-track reactions are important in the bulk stage. This is the  case of reaction (1) between two eaq , significantly contributing to the hydrogen production in the non-homogenous stage of water radiolysis. At high temperatures both the mechanism and the rate of this reaction have been the subject of long discussion (Christensen and Sehested, 1986; Marin et al., 2007; Swiatla-Wojcik and Buxton, 1995; Elliot and Bartels, 2009; Swiatla-Wojcik, 2015).  Mechanism (1a) assumes that at 300 °C the encounter of two eaq leads to the formation of H2 at the rate by three orders of magnitude lower compared to the ambient conditions (Elliot and Bartels, 2009; Marin et al., 2007). Recently proposed mechanism (1b) postulates proton transfer when electrons occupy neighbouring cavities in the hydrogen-bond network and the formation  of the spatially correlated pair (H, eaq ) (Swiatla-Wojcik, 2015).  However, the subsequent recombination of eaq with H and the production of H2 is specific to the rigid hydrogen-bond network stiffen by the presence of supramolecular structures of continuously connected four-bonded molecules. Diminution of these structures at high temperatures promotes unconstrained diffusion of the radicals. Two independent simulations, respectively assuming (1a) and (1b), have shown that the calculated steady state concentrations of the stable molecular products O2, H2O2, and H2 are insensitive to the mechanism assumed for the bimolecular recombination of two hydrated electrons. However, although reaction (1) is insignificant for the bulk chemistry at 300 °C, it contributes to the non-homogenous stage. Modelling calculations carried out for gamma radiation revealed differences in the primary yields of the reducing species. The assumption of reaction  (1b) results in G(eaq ) lower by 3.8%, and G(H) and G(H2) increased by about 6.7% and 1.5%, respectively. Since the primary yields are used to calculate the production rates Pi , the calculated time profiles of the stable radiolytic products are expected to be slightly dependent on the assumptions made on reaction (1). IGSCC in austenitic stainless steels can be effectively eliminated by reducing the H2O2 level. The performed simulation shows that the corrosive hydrogen peroxide is mainly formed in reaction (14), but the contribution of reaction (14) depends on decay of the OH radical in reaction (15) and its restoration in reaction (22). Assuming that forward and backward steps require the same transition state, reactions (15) and (22) are often considered in the form of equilibria (Sunaryo et al., 1995; Swiatla-Wojcik and Buxton, 2005, 2010; Bartels, 2009; Elliot and Bartels, 2009). As reaction (15) is the key reaction controlling water radiolysis at reactor temperatures (Takagi, et al. 2011; Elliot and Bartels, 2009), kinetics of reaction (22) has to be known. The room temperature value of 10 72 M  1 s  1 obtained for k22 from a photolysis study of alkaline solutions (Hartig and Getoff, 1982) is by a few orders of magnitude higher compared to the thermodynamic estimates (Sunaryo et al., 1995; Swiatla-Wojcik and Buxton, 2005; 2010; Bartels, 2009) and

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2  103 M  1 s  1 (Elliot and Bartels, 2009) to (3.2 71.3)  104 M  1 s  1 (Swiatla-Wojcik and Buxton, 2005). Thus the uncertainty in k22 amounts to a factor of ten. As discussed in Section 3.1 reaction (22) significantly contributes the formation of the stable products O2, H2O2, and H2, which are of a particular interest from the point of view of reactor safety and integrity of materials. Fig. 5 shows the calculated steady state concentrations of the stable products assuming conditions from Table 3 and the value of k22 varying from 5  103 to 2.5  104 M  1 s  1 at 300 °C. Simulation with too low value of k22 may result in a significant underestimation of the amount of the stable products. The most sensitive to k22 is the hydrogen concentration, cH2. Taking the uncertainty in k22, Δk22 ¼5  103 M  1s  1, one obtains the uncertainty in cH2 ,

Fig. 5. Sensitivity of the steady state concentrations of O2, H2O2, and H2 to the rate constant k22 of the H atom abstraction from water calculated for conditions given in Table 3.

to the assessment based on the pulse radiolysis measurements of the dichloride anion decay in HCl solutions (Kazmierczak et al., 2015). Taking into account that the oxidation of water by the hydrogen atom in the gas phase requires an activation energy 75 kJ mol  1 (Baulch et al., 2005), k22 may increase about seven orders of magnitude as the temperature increases from 25 to 300 °C. Thus reaction (22) may significantly contribute to the production of H2 and OH under reactor conditions. 3.2. Sensitivity analysis A sensitivity analysis showing interdependence of various parameters and inputs is challenging. In particular, it is very important to examine how uncertainties resulting from guess estimates of the reaction rate constants employed in the kinetic calculations may affect the steady state concentrations of the stable radiolytic products depending on radiation composition, temperature, dose rates, etc. Here, the sensitivity analysis is focused on reaction (22). The kinetics of reaction (22) is still under discussion (Sunaryo et al., 1995; Swiatla-Wojcik and Buxton, 2005; 2010; Bartels, 2009; Sanguanmith, et al., 2011; Alcorn et al., 2014). At the operating temperatures the estimates of k22 ranges from

ΔcH2 =

∂cH2 ∂k22

× Δk22 ≈ 17 μM, compared to

ΔcH2O2 E9 μM for hydrogen peroxide, ΔcO2 E 4 μM for oxygen, and ΔcHO2 E0.3 μM for the hydroperoxyl radical. It has been found that sensitivity to k22 increases with a contribution of gamma radiolysis. A particular importance of reaction (22) under irradiation with gamma rays is consistent with the output of earlier simulations carried out for temperature 250 °C (Sunaryo et al., 1995). To mitigate the corrosion, H2 gas is injected into the coolant to convert O2 and H2O2 into water by radiolytic processes initiated by reaction (15). The scavenging of OH radical suppresses the formation of H2O2 in reaction (14), and the formation of H atom changes the oxidising conditions to the reducing ones as follows: +H

+H

+2H

(11)

(12)

(9)

O2 → HO2 → 2OH → 2H2 O

(5)

where the numbers correspond to the reactions given in Table 1. The hybrid method has been used to calculate stable concentration of the oxidants as a function of the dissolved hydrogen concentration. In Fig. 6 the ratio of the stable concentration in the presence of extra H2 to that calculated assuming that no H2 is added is presented as a function of the dissolved hydrogen concentration. The addition of hydrogen noticeably reduces the oxidant level. Compared to H2O2 the concentration of O2 is lowered more effectively. As expected the efficiency of process (5) depends on the rate of reaction (22) converting the H atom back into the hydroxyl radical. Simulation has been carried out assuming 1.5  104, 1  104, and 5  103 M  1 s  1 for k22. The vertical bars in Fig. 6 show high sensitivity of the calculated ratio to the rate constant of reaction (22).

Fig. 6. The effect of the dissolved hydrogen concentration on a decrease of the steady state concentrations of hydrogen peroxide (left) and oxygen (right) computed for conditions given in Table 3 and k22 ¼ 1  104 M  1 s  1. The vertical bars show sensitivity of the reduction level to the rate of the H atom abstraction from water in reaction (22). The upper and lower limits respectively correspond to 1.5  104 and 5  103 M  1 s  1 assumed for k22.

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4. Summary and conclusion The presented hybrid method combines diffusion-kinetic and kinetic calculations to deal with the non-homogenous and homogeneous stages of water radiolysis, respectively. This comprehensive approach has been used to simulate the radiation chemistry of the coolant under exposure to the mixture of neutron flux (67%) and gamma rays at 300 °C. Simulation has provided time-dependent concentration profiles for the radiolytic products under constant exposure to radiation. At the dose rates of 6.6 kGy s  1 assumed for 2 MeV neutron flux and 3.3 kGy s  1 for gamma rays, the equilibrium is reached at about 100 s and the main products formed by this time are H2, H2O2, O2, and HO2. The simulation carried out assuming equal production rate of eaq  and H þ (PH þ ¼Peaq) instead of PH þ ¼POH  þPeaq used in the hybrid method shows that the omission of the OH  production rate in neutron-gamma radiolysis modelling gives significantly lower steady state concentrations of H2O2, H2, and HO2. The hybrid method has been used to assign the principal reactions contributing to the formation of H2, O2 and H2O2. Calculations show that the steady-state concentrations of the stable products are insensitive to the mechanism and rate of the bimolecular recombination of two eaq  , but are highly sensitive to the reaction rate constant of the H atom abstraction from water. Importance of the latter reaction increases with a contribution of gamma radiolysis. The radiolytic reactions are fast compared to the coolant residence time determined by the flow rates at different parts of the primary circuit. At each region an equilibrium is always established. The steady state concentrations depend on dose rate and radiation composition, treated as input parameters. The radiolysis modelling at different regions provides the species distribution inside the primary circuit which combined with the flow rates can be used to calculate the electrochemical corrosion potential (ECP). Further development and application of the hybrid method towards evaluation of CHC and simulation of the effect of additives or trace impurities are necessary. The presence of even small amounts of impurities such as metal cations, halide anions, etc., reacting with the radiolytic species may change both the corrosive environment and the hydrogen concentration with a consequent effect on CHC.

Acknowledgement Financial support under Research task No. 7 “Study of hydrogen generation processes in nuclear reactors under regular operation conditions and in emergency cases, with suggested actions aimed at upgrade of nuclear safety” financed by the National Research and Development Centre of Poland in the framework of the strategic research project entitled “Technologies Supporting Development of Safe Nuclear Power Engineering” is greatly acknowledged.

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