the kinetics of the reactions of tellurium with stainless steel surfaces and silver aerosols

Nuclear Engineering and Design 180 (1998) 1–27

The kinetics of the reactions of tellurium with stainless steelsurfaces and silver aerosols

C. Gonzalez a,*, A. Alonso b

a Laboratory of Statistics, ETSII, Madrid Polytechnical Uni6ersity, c/Jose Gutierrez Abascal 2, 28006 Madrid, Spainb Nuclear Regulatory Council of Spain, c/Justo Dorado 11, 28040 Madrid, Spain

Received 27 September 1997; received in revised form 24 October 1997; accepted 31 October 1997

Abstract

Tellurium is a very reactive fission product having great importance because of its own contribution to the healthhazard if it is released to the environment and the fact that it is the precursor of decay chains involving iodine. Thispaper describes a new model to analyze and quantify the chemical behavior of the tellurium fission product in theprimary system of pressurized water reactors under severe accident conditions. The great uncertainty on telluriumchemical behavior and the lack of specific experimental evidences suggest the need of a theoretical approach to theproblem. The model quantifies the chemical interaction of tellurium vapors with silver aerosols and with the stainlesssteel of the pipes and structural surfaces. It is also aimed to determine the effect of oxide layers on the reaction rates.All these phenomena alter the transport through the primary system and modify the source term. Hence, theimportance of an adequate description of these chemical processes. The model has been incorporated into the RAFT1.1 transport code and has been verified and validated by using the results and boundary conditions of MARVIKENAerosol Transport Test 4. The results show their good agreement with the experimental data. © 1998 Elsevier ScienceS.A. All rights reserved.

1. Introduction

Modelling the chemical behavior of telluriumand its compounds in the reactor coolant system(RCS) and reactor pressure vessel under lightwater reactors (LWR) severe accident conditions isnecessary due to two main reasons: first, telluriummay contribute to the health hazard if released tothe environment; second, tellurium is the precursorof some decay chains involving iodine.

In general, the interactions of fission product(FP) vapors among themselves, with aerosols andwith structural materials in the RCS, may providemany mechanisms for modifying (increasing, re-tarding or completely inhibiting) the release ofFPs into the containment. The new compoundsgenerated under these conditions alter the volatil-ity and transport properties of FPs, so a detailedstudy of the chemistry in the primary system isneeded. The main factors affecting these processesare temperature, pressure, H2/H2O ratio, gas com-position, radiation field, impurities, radionucliderelease rates and state of structural surfaces.

* Corresponding author. Tel.: +34 1 3363149; fax: +34 13363005.

0029-5493/98/$19.00 © 1998 Elsevier Science S.A. All rights reserved.

PII S0029 -5493 (97 )00293 -8

C. Gonzalez, A. Alonso / Nuclear Engineering and Design 180 (1998) 1–272

There are many uncertainties associated withthe processes dominating FP behavior in the pri-mary system. These uncertainties are even greaterfor tellurium, because it is a very reactive element.In fact, it is generally recognized that telluriumchemical behavior is not well understood. It reactswith other FPs (i.e. cesium), with the carrier gas(H2/H2O), with structural materials such as tin,silver, stainless steel (SS), and so forth.

There exists theories describing tellurium (andother FPs) chemistry under severe accident condi-tions. They are well founded from the thermody-namical point of view. They calculate theequilibrium concentrations of the predicted chem-ical species in the system (Wichner and Spence,1985). Nevertheless, this approach has some limi-tations, for instance, the right identification of thecompounds is needed, as well as the thermody-namic data for all of them under the conditions ofinterest. Furthermore, the equilibrium assumptioncan be unacceptable in some cases (Dickinson,1987); as an example, let us consider the interac-tion between FP vapors and solid structural com-ponents, it is influenced by kinetics as well as bythermodynamic considerations.

In the case of tellurium, the most importantinteractions are those taking place between tel-lurium and aerosols (mainly silver) and structuralsurfaces, so the kinetic effects must be accountedfor in a predictive model for tellurium chemicalbehavior. The new model presented in this articleprovides the tools for solving the problem underconsideration. It is a model that combines bothapproaches: the equilibrium hypothesis is as-sumed in the gaseous phase, for the reactionsamong vapors, while the kinetic approach is usedfor the chemical reactions between tellurium andaerosols and structural surfaces.

For the silver–tellurium system and tempera-tures above the melting point of the product or ofthe substrate, the overall reaction rate is linearand it is controlled by the diffusion of telluriumvapor from the bulk of the gas (gaseous phase) tothe surface of the aerosols. At temperatures belowthe melting point, the diffusion of the silver ionsthrough the product layer (Ag2Te), which will befree of pores, will control the reaction whichobeys a parabolic rate. These assumptions are

based on experimental results and in the study ofother fission product–aerosol systems. The exper-imental results also indicate that the kinetics ofthe reaction itself is very rapid so there are not‘real’ (not diffusion) kinetics limitations.

For the SS–tellurium system and temperaturesabove the melting point of the product (FeTe0.9),the overall reaction rate is linear and it is con-trolled by the diffusion of tellurium vapor fromthe bulk of the gas (gaseous phase) to the struc-tural surface. At temperatures below the meltingpoint, the diffusion of the iron ions through theproduct, FeTe0.9, which will be free of pores, willbe the rate determining step. This step obeys theparabolic law.

For the oxidized stainless steel–tellurium sys-tem, the reaction product will vary depending onthe temperature, FeTe2 or FeTe1.451. The overallreaction rate is controlled by the diffusion of ironatoms through the layer of oxide (Fe3O4) belowthe melting point. Otherwise it will be controlledby the diffusion of tellurium from the bulk of gasto the surface.

Again these assumptions are based on experi-mental results. As for the silver–tellurium system,the experimental results also indicates that thereaction rate itself is very fast, so there are not‘real’ kinetics limitations.

The main point of this article is to accomplishthe problem of the interaction of tellurium withsilver aerosols and SS in the primary circuit ofpressurized water reactors (PWRs) at tempera-tures lower than the melting point of the productformed.

Initial work has been devoted to identify thetheories which could be applied for analyzing andfor quantifying the interaction of tellurium withsurfaces (aerosols and pipes). The model reportedhere has drawn upon Wagner’s theory (Wagner,1936). The initial purpose of this theory was toexplain the oxidation of metals, however, theideas outlined can be applied to develop similarmodels for other materials and products. In fact,the chemical reaction of tellurium with silver andwith the iron of the SS can be described as anoxidation process.

The theory presented in this paper leads to amodel for tellurium chemical behavior based on

C. Gonzalez, A. Alonso / Nuclear Engineering and Design 180 (1998) 1–27 3

the physical/chemical properties of the productmaterials, on experimental evidences, on the anal-ysis of accident sequences and on the previousknowlegde gleaned from earlier studies on tel-lurium behavior during the Three Mile Island-2accident (Vinjamuri et al., 1984). Section 2 of thisarticle describes the severe accident conditionswhich are the initial and boundary conditions ofthe problem. Section 3 describes the kinetic modelfor chemical reactions of tellurium with silveraerosols. In Section 4, the kinetic model for thechemical reaction of tellurium with SS is devel-oped. An alternative to take into account thepresence of oxide layers on the surfaces is alsoderived. Section 5 contains a summary of theimplementation of the new model into the RAFT1.1 code (reactor aerosol formation and trans-port) (Im et al., 1987). Section 6 illustrates theapplication of the model to the analysis of theMarviken Aerosol Transport Test 4 (MXE-204,1985) and contains a discussion of the results.Section 7 summarizes the main results and sug-gests areas for further research.

2. Accident conditions

The accident sequences with the greatest contri-bution to risk (USNRC, 1989) in PWRs are:TMLB’, AB, S2D, V and SGTR, followingWASH-1400 notation (USNRC, 1975). The con-ditions existing in the primary system for eachsequence will be the boundary conditions for tel-lurium chemical behavior.

These boundary conditions are: (a) the partialpressure of tellurium, which quantifies the amountof tellurium available for chemical reactions; (b)the presence of reducing conditions, which willinfluence on the speciation of tellurium; (c) theresidence times, in both the upper plenum and hotleg, which will determine the existence of somesaturation and diffusion phenomena that modifythe chemical reaction rates, and (d) the gas andsurface temperatures, which may modify the inter-action mechanisms.

The analysis of the conditions existing in theRCS for the sequences with the greatest contribu-tion to risk provides the range of variation of the

variables which determine the boundary condi-tions (Alonso et al., 1988; Harman and Clough,1991; Hobbins et al., 1991; Alonso et al., 1992),these are

10–6000 PaTellurium partial pressure:0.5:1–Z1H2/H2O mole ratio:

Residence Time:0.1–1500 sUpper Plenum:0.1–70 sHot leg:

Gas temperatures:500–3000 KCore:550–1700 KUpper Plenum:500–2000 KHot leg:

Surface temperatures:500–3000 KCore:

Upper Plenum: 550–1400 KHot leg: 500–1500 K

20–170 minDuration of release:

The model we are developing will be of nopractical use unless it accurately describes thechemical behavior of tellurium under the condi-tions given above.

3. The silver–tellurium system

Silver is the main component of control rods(85%) in PWR reactors. Some analyses of severeaccident sequences indicate that 71.3 wt.% ofaerosols released is metallic, mainly from silver(Cubicciotti and Sehgal, 1984). The interactionbetween silver aerosols and tellurium vapors re-duces the vapor pressure of tellurium in the RCSand modifies its deposition mechanisms. As re-ported previously (Gonzalez and Alonso,1991a,b,c), there is some experimental evidencethat confirms the importance of tellurium reactionwith silver aerosols. The reaction has been ob-served in the Marviken Aerosol Transport Test(ATT) experiments (MXE-204, 1985), however,no quantitative studies were performed to analyzeits extent. Some experiments carried out at SandiaNational Laboratory indicated that the reactionbetween tellurium and a silver plate obeyed alinear rate of 1 cm s−1 (Sallach et al., 1984).


Measurements of aerosol composition in theSFD 1–4 experiments indicate that volatile FP’s(especially cesium) make up 25–50% of the aero-sol with the remainder of the material being con-trol materials (especially tin) (Petti et al., 1994).The experiments indicate that the amounts ofcadmium, tin and zirconium in the aerosol areeach about a factor of ten greater than theamount of silver (Petti et al., 1991) and all thesematerials form tellurides. Therefore, the impor-tance of these tellurides on the transport of tel-lurium in the RCS should be also addressed andsubjected to a detailed theoretical study as pro-posed for the silver tellurides.

There is not, as far as the authors know, spe-cific separate-effect tests for the analysis of tel-lurium interaction with silver aerosols, such as thethermogravimetric studies performed at Winfrithto study the silver–iodine, cadmium–iodine andmanganese–iodine systems (Henshaw et al.,1991). Nevertheless, although the characteristicsof the product formed in the silver–telluriumsystem are quite different, some conclusions ob-tained from the experiments mentioned above areof general application in metal–FP systems.

The main compound of the silver–telluriumsystem is the Ag2Te (silver telluride) obtainedfrom the reactions

2Ag(c)+1/2xTe2(g)+ (1−x)Te(g)�Ag2Te(c),

and

2Ag(c)+H2Te(g)�Ag2Te(c)+H2(g).

The study of its structural and electrical charac-teristics is essential for understanding the mecha-nisms leading to its formation from the elementalconstituents. As described in the sequel, at tem-peratures lower than the melting point of theproduct, the main mechanism will be the diffusionof silver ions through the Ag2Te product layer toreact with tellurium.

The phase diagram for the silver–tellurium sys-tem (Hansen, 1958) shows a polymorphic modifi-cation taking place at 405 K in presence of excessof tellurium and at 418 K in the pure compoundand in the presence of excess of silver. This wouldindicate a certain range of homogeneity. The tran-sition temperature represents a point at which the

silver atoms in the structure lose the rigidity char-acteristic of the solid state while the telluriumatoms retain a rigid structure. The structure of thelow temperature modification is monoclinic/or-thorhombic and is called b-phase. The structureof medium temperature (a-phase, stable from405–418 to 1073 K) is face centered cubic formedby tellurium (Sakuma and Saitoh, 1985). Thestructure of high temperature, a %-phase, is bodycentered cubic. Considering the range of tempera-tures reached in the vessel and primary systemunder severe accident conditions, phases a and a %will be the most important.

The Ag2Te has characteristics of superionicconductor and its crystal structure is far frombeing perfect with the mobile silver ions (Ag+)distributed randomly over the available sites inthe crystal lattice (Kobayashi, 1989), that in thecase of the face centered cubic lattice are fouroctahedral sites and eight tetrahedral sites. Studiesbased on molecular dynamics indicate that theeight Ag+ ions stay at the tetrahedral sites formost of the time and move towards their neigh-boring tetrahedral sites through the vicinity of theoctahedral sites along zigzag paths (Kobayashi etal., 1988). The disordered arrangement of thecations can be imagined to arise from Frenkeldisorder by reducing the energy difference be-tween lattice positions and interstitial sites to sucha degree that a considerable fraction of ions occu-pies interstitial sites (Grientschnig and Sitte,1991).

Silver belongs to group IB of the periodic tableand tellurium to group VIA, so the silver–tel-lurium interaction is similar to an oxidation pro-cess, an exchange of electrons which represents apassage to a more stable state and which canoccur spontaneously, without external provisionof energy.

The interaction will be considered to be con-trolled by tellurium diffusion in the gaseous phasewhen the temperature exceeds the melting point ofAg2Te (1232 K). At temperatures lower than 1232K, the reaction mechanism is different and ourexplanation is the following: given that the molevolume of the telluride formed (Ag2Te, Vm=40.4cm3 mol−1) is bigger than that of the reactingmetal (Ag, Vm=10.3 cm3 mol−1), a protective


layer, free from pores, may be formed (followingthe theory of Pilling and Bedworth) (Evans,1960). This is usually associated with the outwardmovement of metal which leaves vacancies atplaces previously occupied by the cations. Underthese conditions, Wagner (1953) has shown thatthe reaction rate constant may be expressed interms of the mobilities of the reacting componentsand the appropriate concentration or activity gra-dients which are established across the growingproduct layer.

Some experiments carried out also by Wagner(1953) have indicated that in the reactions be-tween silver and sulfur, it is silver and not sulfurwhich has migrated through the sulphide layergiven that in a-Ag2S silver ions have a greatmobility. Similar behavior can be expected forAg2Te, as the ionic radii of Te2− (2.21 A) is evengreater than that of S2− (1.84 A) and both ofthem are greater than that of Ag+ (1.26 A). Thishypothesis has been also justified by studies basedon molecular dynamics (Kobayashi et al., 1988)and experiments with X-rays (Sakuma andSaitoh, 1985).

This indicates that the reaction does not stopafter the surface layer has been formed but, inabsence of pores, the reaction can continue onlyas a result of Ag+ diffusion process ocurringwithin this surface layer. Obviously, this is a veryslow process if it is compared with the passage ofgas through porous material or with the diffusionin gaseous phase, even when the tellurium supplyto the aerosol surface is maintained (high tel-lurium vapor pressure). Interestingly, it can beconcluded that at temperatures below the meltingpoint of Ag2Te (1232 K), the diffusion of silverions through the product layer (Ag2Te) is the ratedetermining step of the reaction, whenever therewas enough amount of tellurium for the reaction.

Under these conditions, the reaction rate obeysthe parabolic rate law, which follows directlyfrom Fick’s first law of diffusion. If the flux ofdiffusing substance passing per unit time throughan unit area of a plane at right angles to thedirection of diffusion is measured as the increaseof thickness of the product layer per unit time(dx/dt), this will be inversely proportional to thethickness of the layer (x), because the concentra-

tion gradient in the layer will be proportional to1/x, i.e.

dxdt

=K %x

.

The integration of this expression is

x2=2K %t,

where K %, Tamman rate constant (cm2 s−1), isproportional to the diffusion coefficient. The pre-vious expression is valid if a quasi-stationary stateis assumed: i.e. the concentrations of the metallicsubstrate, Ag, and gaseous reactant, Te2, at thephase boundaries of the layer are independent oftime during the reaction.

In the literature, it is not unusual to express theparabolic rate law in terms of the rational rateconstant, K (g equivalent cm−1 s). The paraboliclaw can be then written as

dndt

=Kqx

, (1)

where n, is the amount of product formed (Ag2Te)(g equivalent); q, the cross section of the system(cm2), and x, the thickness of the product layerformed (cm). Therefore, the meaning of the ratio-nal rate constant is the number of chemical equiv-alents of silver telluride formed per s and cm2

when the product layer is 1 cm thick. Both con-stants, the Tammann rate constant K %, and therational rate constant K, are related by the expres-sion

K=K %V

, (2)

where V is the equivalent volume of productformed (cm3 equivalent−1).

Mathematical treatments apply equally forcases where cations move outwards and anionsinwards so long as the product film is continuingto thicken uniformly and remains unbroken.Wagner developed a theory to model the processin which the metal ions and electrons, in this casesilver ions and electrons, migrated through theproduct layer and their movement was the rate-controlling factor. He deduced an expression forthe rational rate constant based on the mobilityor diffusivity of the moving particles and on the


appropiate concentration or activity gradientswhich are established across the growing productlayer (Wagner, 1953)

K=1

�Z2�F2

& m2¦

m 2%

(t1+ t2)t3s dm2, (3)

where K is the rational rate constant; t1, t2 and t3,the transference numbers of cations, anions andelectrons respectively; s, the conductivity of theproduct layer (V−1 cm−1); F, the Faraday con-stant (96 500 J V−1 equivalent); m2 the chemicalpotential of the anions in the inner (primed) ofouter (double primed) side of product layer (ergg−1 atoms), and Z2 the valence of the anion(equivalent mol−1).

When compounds have predominant electronicconduction, it is easier to measure the self-diffu-sion coefficient than the transport numbers; underthese conditions, the rational rate constant isgiven by one of the two following equalities

K= �Z2�C2& a2¦

a 2%

[(Z1/�Z2�)D1*+D2* ] d ln a2,

K=Z1C1& a1%

a 1¦[D1*+ (�Z2�Z1)D2* ] d ln a1, (4)

where Z1 and Z2 are the chemical valences ofsilver and tellurium (1+ and 2−, respectively); C1

and C2, the concentration of silver and tellurium(g atoms cm−3 of Ag2Te); a1, a2, the thermody-namic activities of silver and tellurium in thetelluride; the primed and double primed quantitiesreferring to the conditions at the inner and outersurfaces of the growing layer, respectively, andD1* and D2* (cm2 s−1), the respective self-diffusioncoefficients defined as the diffusion coefficient inthe absence of a chemical concentration gradientand experimentally measured with radioactivetracers (Landler and Komarek, 1966).

Simplifications can be made to previous expres-sions by considering the electronic and structuralproperties of the Ag2Te. The Ag2Te is a superi-onic conductor with t3=1. Besides, the mobilityof silver ions is very much greater than that oftellurium ions, which means t1Z t2 and D1*ZD2*(Grientschnig and Sitte, 1991). The number ofcations Ag+ is much lower than the number ofpositions available for their movement, therefore,

cation mobility can be considered virtually con-stant in the narrow homogeneity range of theAg2Te. In consequence, both the conductivity andthe self-diffusion coefficient for silver can be con-sidered constant and independent of the composi-tion (Okazaki, 1977), so the new expressions are

K=1

�Z2�F2 t1s& m2¦

m 2%

dm2=t1s

�Z2�F2 (m2¦−m2% ), (5)

K= �Z2�C2

Z1

�Z2� D1*& a2¦

a 2%

d ln a2=C2Z1D1* lna2¦a2%

,

(6)

K=Z1C1D1*& a1%

a 1¦d ln a1=C1Z1D1* ln

a1%

a1¦. (7)

Eqs. (5) and (6) are more convenient becausethe final value does not depend on the choice ofthe standard state to which the activities are re-ferred, and the errors associated with the Gibbs–Duhem integration are avoided.

The activities, assuming ideal conditions, areproportional to the square root of the partialpressures of tellurium. PTe2¦ is the partial pressureof tellurium in the outer side of product, i.e. thepressure of Te2 on the aerosol surface which willbe determined by the accident conditions and bythe mass transport coefficient from the bulk gasto the aerosol surface. PTe2% is the partial pressureof tellurium in equilibrium over Ag2Te+Ag ob-tained from the literature (Knacke et al., 1991),therefore,

K=t1s

�Z2�F2

RT2

ln�PTe2¦

PTe2%

�, (8)

K=C2Z1D1*

2ln�PTe2¦

PTe2%

�. (9)

For the silver–tellurium system, the value C2,for a telluride composition of 37.16 wt.% tel-lurium and an average density of 8.5 g cm−3, isequal to 0.0248 g atoms of tellurium cm−3 oftelluride, Ag2Te; T is the absolute temperature(K). Therefore, the only unknown parameters areD1* and the product t1�s, ionic conductivity of theAg2Te or sAg.

An implicit assumption in Wagner’s theory isthat the product growing on the metal substratehas properties identical to those of the homoge-


neous bulk product of the same chemical compo-sition. Wagner showed this to be true for thegrowth of Ag2S on silver (Himmel et al., 1953).Wagner’s treatment also assumes that ions andelectrons migrate independently of one anotherand that the Nerst–Einstein equation relating theionic mobilities and the respective self-diffusioncoefficients is satisfied. Two different expresionsare available

D1*=kTsAg

e2Z21nAg

=mAgkT

eZ1

, (10)

where mAg is the ionic mobility or the velocityreached under the action of an unit force; e, theelectronic charge (u.e.a); Z1, the valence of silverions; k, the Boltzman constant (8.615�10−5 eVK−1); nAg, the Ag ion density and D1*, the self-diffusion coefficient or conductivity diffusion co-efficient of the ions which governs the ionic flux inthe hypothetical situation of ideal mixing andabsence of electric field (Grientschnig and Sitte,1991). Experimental results obtained using atracer technique have shown that the Einsteinrelation does not hold in silver chalcogenides andthat the deviation from the Einstein relation canbe expressed quantitatively by introducing theHavens ratio (Okazaki, 1977; Kobayashi, 1989),the expression of which is

f=eD1*Z1

mAgkT, (11)

where f depends on the defects in the lattice whichinfluence ion movements, and on the crystal struc-ture.

Thus, the new self-diffusion or tracer coefficientof the tracer ions is

D1**= fD1*= fkTsAg

e2Z21nAg

= fmAgkT

eZ1

. (12)

There exists in the literature values for D1** andsAg. Some of them are experimental, as those ofMiyatani and Okazaki (Okazaki, 1977), and someof them have been obtained theoretically by usingmethods of molecular dynamics (Kobayashi et al.,1988). The value of the Havens ratio can beobtained from the relation between the ionic con-ductivity and the self-diffusion coefficient.

Given that two series of experimental resultsexist and agree between them and that the tem-perature range covered is greater than that corre-sponding to the theoretical ones, it was decided touse the experimental values.

In the experimental results, at temperatureslower than 1075 K, both the self-diffusion coeffi-cient and the ionic conductivity follow an Arrhe-nius type behavior

D1**=D0 exp(−o/kT), (13)

where D0 value is 2.0523�10−4 cm2 s−1 and o isthe activation energy for an ionic diffusion (0.14eV). The conductivity is

sAg=s0 exp(−os/kT), (14)

where s0 is a constant (18.086 V−1 cm−1) and os

the activation energy associated to conductivity(0.13 eV).

A discontinuity can be observed at 1075 Kwhich corresponds to the structure change in theAg2Te from face centered cubic (a-phase) to bodycentered cubic (a %-phase) (Tachibana et al., 1989).At temperatures higher than 1075 K, the values tobe used are D1**=6�10−5 cm2 s−1 and sAg=6V−1 cm−1.

As Wagner’s theory assumes that the Nerst–Einstein relation is satisfied, the values for theself-diffusion coefficient to be introduced in theformulation must be those obtained directly fromtheir relationship with the conductivity, so attemperatures lower than 1075 K

D1*=D1**

f

=2.0523�10−4�1

0.73�exp

� −0.148.616�10−5�T

�,

(15)

and at temperatures greater than 1075 K

D1*=D1**

f=

6�10−5

0.73=8.22�10−5. (16)

By introducing the values of the self-diffusioncoefficient in Eq. (9), the rational rate constantwill be

T51075 K:


K�equi

cms�

=3.4873�10−6�exp� −0.14

8.616�10−5�T��ln

PTe2¦PTe2%

.

(17)

1075BTB1232 K:

K�equi

cms�

=1.0193�10−6�lnPTe2¦PTe2%

. (18)

The calculation of the Tamman rate constantfrom Eq. (2) and the parabolic rate constant, Kp

2

which refers to the tellurium which has been takenup by the specimen from the gas phase is ex-plained in Appendix A. The choice of one of thethree different options, rational rate constant,Tamman rate constant or parabolic rate constant,will depend on the final variable to be calculated.

Equivalent expressions can be obtained if therational rate constant is calculated from the ionicconductivity by using Eq. (8)

T51075 K:

K�equi

cms�

=4.0371�10−9�T�exp� −0.13

8.616�10−5�T��ln

PTe2¦PTe2%

.

× (19)

1075BTB1232 K:

K�equi

cms�

=1.3393�10−9�T�lnPTe2¦PTe2%

. (20)

From these equations, the Tamman rate con-stant and the parabolic rate constant can be ob-tained in the same way as explained in AppendixA.

4. The stainless steel–tellurium system

As previously discussed, tellurium behavior inthe RCS of PWRs is greatly affected by theenvironment (oxidizing or reducing), by the typeof chemical species and by the interaction of thesespecies with aerosols and structural materials.

There are three series of experiments focussedon analyzing tellurium–SS interaction under theconditions prevailing in the primary circuit of aPWR during a severe accident: the studies carriedout at the Sandia National Laboratories (Sallachet al., 1984); the experiments performed at Win-frith (Bowsher et al., 1983), and the experimentsof the DEVAP program (Le Marois and Megnin,1994). Sallach et al. (1984) explained the tel-lurium–SS reactions in terms of a deposition ve-locity or linear rate constant which valuedepended on the initial conditions of the surface(oxidized, unoxidized). The reaction productsidentified were Fe2.25Te2 on SS and FeTe2 whenthere was present an oxide layer of magnetite(Fe3O4). The experiments carried out at Winfrith(Bowsher et al., 1983) showed evidence of chemi-cal interaction between tellurium and 304 SS atthe grain boundaries of the metal above 573 K,with preferential formation of nickel andchromium tellurides. In the DEVAP program (LeMarois and Megnin, 1994), the boundary condi-tions were those of the most likely accident se-quences at low pressure and the geometricconditions were those of an steam generator tube.Post-test analyses showed that the tellurium de-posits depend on both the nature and state of thealloy and on the temperature.

Special attention is being given nowadays totellurium–SS interaction due to the corrosionproblems encountered in the fuel cladding of FastBreeder Reactors. There are numerous experi-ments devoted to characterize this phenomenonor to obtain the values of the parameters whichare needed for calculating the reaction rates. Ex-periments have been performed by Saito andcoworkers (Saito et al., 1987; Furuya et al., 1988);Gotzmann et al. (1973), Lobb and Robins (1976)and coworkers; Maiya and Busch (1975), andmore recently by Magara et al. (1991) who haveanalyzed the high temperature corrosion of iron-chromium alloys (1.17, 5.65 and 11.96 at.% Cr) bytellurium. They have measured the reaction ratesand the activation energies for the reactions atexperimental conditions quite similar to thoseidentified to occur in the primary circuit ofPWRs. They observed the reaction rates obeyingthe parabolic rate law. By comparing the reaction


activation energies with the activation energies forthe diffusion of iron in b, d, d % and o irontellurides, they found that the rate-determiningstep of the reaction depends on temperature andchromium concentration.

4.1. Extrapolation to se6ere accident conditions

From the previous review (of the literaturementioned above), some tentative conclusions canbe established. The great affinity between tel-lurium and SS is clearly demonstrated under abroad range of conditions:� Temperatures ranging from 573 to 1500 K.� Tellurium partial pressure ranging from 1 to

600 Pa.� Residence time ranging from hundreds of sec-

onds to hundreds of hours.The temperature covers all the expected range

conditions for the surface temperatures in upperplenum and hot leg. The range for the partialpressure of tellurium is wide, although in theTMLB% sequence the tellurium partial pressuremay reach �6000 Pa. The residence time is themost difficult parameter to evaluate, given thatmost of the experiments have been performedfollowing an annealing process, which it is not thescenario of a severe accident. Moreover, this verybroad range does not cover sufficiently the condi-tions of interest; there is not enough data forshort and medium residence times, thus someextrapolation is needed.

In most of experiments the presence of a doublelayer has been observed; the outer layer is mainlycomposed of iron, nickel and tellurium while theinner layer is composed of chromium, telluriumand oxygen, if present. It is likely that tellurium issupplied to the inner layer by transport throughmicrofissures and by dissociation of the outertelluride layer at the interface between the innerand outer reaction layers. The parabolic law isdetermined by the outward diffusion of iron andnickel in the inner reaction layer.

There is also agreement on the great influenceof oxygen, because the formation of chromiumoxides is thermodynamically more favorable thanthe formation of chromium tellurides and wouldprevent the diffusion of iron (the main component

of the SS) and the SS–tellurium interaction woulddecrease. The order of thermodynamic stability isfound to be chromium tellurides\nickel tel-lurides\ iron tellurides.

The presence of an oxide layer produces passi-vation effects. Under PWR normal operation con-ditions, the water pH is neutral or slightlyalkaline. The Pourbaix diagram (Maroni et al.,1990) indicates for these conditions that the mostlikely oxides are Fe3O4 and FeCr2O4. Fe2O3 isstable only at a steam/hydrogen ratio greater than1�105 and, therefore, it should not be expected toform in an overheated core (Hobbins et al., 1987).Experiments carried out at Sandia (Sallach et al.,1984) have considered a Fe3O4 layer on the SS toanalyze the passivation effects originated by theoxides; in this particular case, tellurides withhigher tellurium content are formed in theproduct layer.

From the analysis of the sequences with thehighest contribution to risk in a PWR, sometentative conclusions can be established.

It is in sequences of high or intermediate pres-sure where the longer residence times and thehigher tellurium partial pressure make the interac-tion of tellurium with surfaces to be important. Itis also thought that the passivation effects origi-nated by the presence of the product layer and ofan oxide layer should be of special importance.

By gathering both sources of information (ex-perimental and theoretical from the study ofsevere accident scenarios) a picture on the way theinteraction tellurium–SS may occur can be out-lined.

On 316 SS unoxidized alloy, which is the mostcommonly used in pipes and structures of PWRs,iron telluride FeTe0.9 (b-phase) is initially formedafter the reaction of tellurium with iron given thatiron is the main constituent of this alloy (�70wt.% of iron). The compound has tetragonalstructure and the diffusion of iron in FeTe0.9

controls the reaction at temperatures lower thanthe melting point. As the reaction goes on, ironfrom the alloy migrates through the product layerto react with tellurium; the iron activity in thealloy decreases while that of chromium increases,favoring the formation of chromium tellurides inan inner layer (mainly Cr2Te3). It is likely that


tellurium is supplied to the inner layer by trans-port thought microfissures and by dissociation ofthe iron telluride layer at the interface between theinner and outer reaction layers. This idea is inagreement with the experimental results and it isconfirmed by thermochemical calculations, giventhat the chromium telluride formed is more stablethan the b-phase iron telluride (Saha et al., 1985;Baba et al., 1988; Azad and Sreedharan, 1989;Pulham and Richards, 1990).

The presence of chromium telluride will preventthe iron diffusion from the alloy to the outer layerand iron tellurides with higher tellurium contentwill be formed, for instance d-phase FeTe1.427 andd %-phase FeTe1.451 with hexagonal structure, ando-phase FeTe2 with orthorombic structure. Sup-porting evidence has also been provided by themore recent work of Magara and his coworkers(Magara et al., 1991). It is worth mentioning thatd and d % iron tellurides are soluble in thechromium tellurides given that all of them havethe same structure.

If this situation continues, iron tellurides willfinally vanish and the only layer remaining will bethat of chromium tellurides. This result has beenexperimentally observed by Lobb and Robins(1976), although the alloy which they used had ahigher chromium content than the 304 or 316 SSalloys of structures and pipes of a PWR.

All these processess are represented in Fig. 1;they would be completed if the contact time be-tween tellurium and SS was long enough and ifthe tellurium partial pressure was sufficient duringthis period. In fact, steps (a–g) in Fig. 1 take along time, so if the restrictions imposed by theaccident conditions are taken into account, onlysteps (a) and (b) will take place. Fig. 1a meansthat tellurium vapor reaches the structural surfaceand Fig. 1b means that FeTe0.9 is formed and theiron ions must diffuse through this product layerto react with tellurium.

On an 316 SS oxidized alloy there is initially apassivation effect due to the oxide layer (Fe3O4)existing on the surface which prevents the diffu-sion of iron, so iron tellurides with higher tel-lurium content will be formed. The reactionproduct will vary depending on the availability ofiron, and the reaction rate will be controlled by

the diffusion of the iron through the oxide layerat temperatures lower than the melting point.From 273 to 709 K, the product formed will beFeTe2 and from 709 to 1085 K it will be FeTe1.451.Fig. 2 represents the initial and final states of theprocess. Step (a) means that tellurium vaporreaches the structural surface which is coveredwith Fe3O4. Step (b) means that FeTe2 andFeTe1.451 are formed and the iron ions must dif-fuse through the oxide layer and product layer toreact with tellurium. In this case, the passagethrough the oxide layer is the slowest.

4.2. Theoretical approach to SS– telluriuminteraction

Under some of the conditions prevailing in asevere accident scenario, tellurium interactionwith 316 or 304 SS is controlled by the irondiffusion through the product layer, or the oxidelayer; in these cases Wagner’s theory should beapplied to obtain an analytical expression for theparabolic rate constant. Two different studieshave been made; the first assumes that the SS isunoxidized, while the second assumes that anoxide layer exists on the surface.

4.2.1. Unoxidized stainless-steelTaking into account the maximum residence

time of tellurium in the upper plenum and hot legin a PWR under severe accident conditions, themean composition of the SS (16–18 wt.% of Cr,12–14 wt.% of Ni and 68–72 wt.% of Fe) and themaximum partial pressure of tellurium, it is as-sumed that the product formed on the SS is theiron telluride in b-phase (FeTe0.9) and that therate-controlling step is the diffusion of ironthrough this product, at temperatures lower thanthe melting point of the product.

This assumption is also based on the knowledgeof the iron-tellurium phase diagram (Baba et al.,1988) and taking into consideration the structureof the crystal and the type of lattice defects whichare present. The existence of the solid tetragonalb-phase is confirmed for temperatures lower than1117 K, where it decomposes peritectoidally intoFe and rhombohedral high temperature b %(#FeTe0.9); it has a homogeneity range from 45.9 to48.5 at.% Te, at 988 K.


Fig. 1. Interaction of tellurium with SS. (a) Initial state; (b) formation of FeTe0.9, diffusion of iron, iron activity decreases in thealloy; (c) chromium activity increases, tellurium penetration through microfissures, decomposition of FeTe0.9; (d) formation ofCr2Te3, iron diffusion hindered; (e) formation of FeTe1.427 and FeTe1.451; (f) solubility of FeTe1.427 and FeTe1.451 in Cr2Te3; and (g)final state.

If other iron tellurides such as d (FeTe1.427) ord %(FeTe1.451) phases were considered, the maxi-mum temperature of solid phases should be 1085and 922 K for the o-phase (FeTe2). Above thesetemperatures, the mass transport in gaseous phaseshould be the rate determining step.

The order of thermodynamic stability for the

different phases is bBd=d %Bo, which confirmsthe experimental evidence of the progressive for-mation of the tellurides with greater telluriumcontent.

The b-FeTe0.9 has a density of 6.77 g cm−3 at293 K and the cell content for the nominal com-position FeTe0.9 is 2.19 atoms of iron and 1.98


atoms of tellurium. The structure can be definedas tetragonal anti-PbO type, a transition betweenthe B1O and C38 structure types (Gronvold et al.,1954). The tellurium atoms form an approxi-mately cubic close packing, the iron atoms occupytetrahedral sites on the second layer and the ironatoms in excess occupy the octahedral sites ran-domly. The average jump frequency of the ironatoms in the b-phase is two orders of magnitudegreater than those in the d % and o phases, and it isclosely related to the jump distance, which islower in the b-phase than in other phases (Ma-gara et al., 1990a,b).

The b-FeTe0.9 is a non-stoichiometric com-pound, an n-type conductor with metal atomexcess. The mobility of electronic defects is ordersof magnitude greater than that of ionic defects, sodiffusion will be controlled by ionic defects. Thistype of defect is usual in oxides, sulfides andselenides, all of them with similar behavior totellurides (Hauffe, 1965).

Wagner’s theory is used to calculate the ratio-nal rate constant given by Eqs. (3) and (4). Themeaning of the variables is the same as the oneexplained for the silver–tellurium interaction, butin this case the subindex 1 refers to iron, and theproduct formed to FeTe0.9. Neither Eq. (3) northe second equality of Eq. (4) can be used becausethere are no data in the literature for either theionic conductivity of b-FeTe0.9 (t1�s) or the ironactivities, a1, in b-FeTe0.9. Only one value of a1 isavailable at 973 K (Ipser and Komarek, 1974).

Experimental results indicate that the irontransference number, t1, and the self-diffusion co-efficient of iron in b-FeTe0.9, D1*, are much higherthan those corresponding to tellurium, t2 and D2*,respectively (Magara et al., 1990a). By assuming

ideal conditions, the thermodynamic activity oftellurium in FeTe0.9 is considered to be propor-tional to the square root of the equilibrium partialpressure of tellurium in the gaseous phase; there-fore the simplified expression will be

K= �Z2�C2& PTe2¦

PTe2%

[(Z1/�Z2�)D1* ] d ln PTe1/22 . (21)

Although the average composition of the b-FeTe0.9 layer depends slightly on temperature, nosignificant error is introduced by assuming C2 tobe constant over the range from 600 to 1117 K.The value to be used for C2 is 0.036 g atoms oftellurium per cubic cm of FeTe0.9. It is based onan average telluride composition of 67.28 wt.%tellurium (FeTe0.9) and an average density of 6.77g cm−3. The integration limits correspond to theboundaries of the b-phase field at the temperaturein question.

Thus, with �Z2�=2 and the ratio Z1/�Z2� ex-pressed as a function of the atomic ratio N2/N1,which will vary from the inner side to the outerside of the product layer (Z1N1=Z2N2), the ratio-nal rate constant becomes

K=0.036�& PTe2¦

PTe2%

N2

N1

D1* d ln PTe2. (22)

From 673 to 1073 K, in the iron-rich side of theb-phase field, N2/N1=0.811, while in the tel-lurium-rich side N2/N1=0.942 (Prasad et al.,1988). On the outer side of product layer, PTe2¦ isdetermined by the accident conditions and thetransport of Te2 towards the surface; on the innerside of product layer (Fe+FeTe0.85), a slightcomplication arises because of the fact that theavailable data for PTe2% refer only to the tempera-ture range from 866 to 999 K (Prasad et al.,1988).

At this point some assumptions must obviouslybe made regarding the magnitude of D1* and itsvariation versus temperature and versus telluriumpartial pressure in order to evaluate the rationalrate constant from Eq. (22). Two different ap-proaches can be followed:1. To consider the self-diffusion coefficient of

iron in b-FeTe0.9 to be independent of concen-tration in the range of homogeneity of theb-phase at a given temperature. This means

Fig. 2. Interaction of tellurium with oxidized SS. (a) Initialstate; and (b) diffusion of iron through Fe3O4 controls thereaction, formation of FeTe1.451 and FeTe2.


that the mobility of iron ions is not affected bythe different composition in the range of homo-geneity and it is therefore independent of thetellurium partial pressure. This seems to be agood assumption for n-type semiconductors sinceit has been deduced theoretically and observedexperimentally (Wagner, 1936; Hauffe, 1965).2. To consider the self-diffusion coefficient ofiron in b-FeTe0.9 to be dependent on concentra-tion in the range of homogeneity of the b-phase.The self-diffusion coefficient will vary dependingon the tellurium partial pressure in equilibriumwith the exact composition in the range of homo-geneity of the b-phase.

There are no data in the literature to choose thebest option, therefore it has been decided to fol-low both approaches and to develop the associ-ated formulation so that no possibility isneglected or forgotten.

HYPOTHESIS 1: Self-diffusion coefficient ofiron in b-FeTe0.9 independent of concentration.Under this assumption, the expression for therational rate constant is

K=0.036�N2

N1

�D1*�lnPTe2¦PTe2%

. (23)

Recently measured values for the self-diffusioncoefficient of iron in b-phase for the compositionFeTe0.893 can be found in the literature (Magaraet al., 1991, 1992, M. Magara, 1992, Private com-munication) (the homogeneity range of the b-phase goes from FeTe0.811 to FeTe0.942). Theself-diffusion coefficient follows an Arrhenius lawwith the expression

D1*=8.6�10−3�exp�−91 500

RT�

. (24)

The self-diffusion coefficient is valid for thehomogeneity range of the b-phase and it will beused to calculate the rational rate constant whichbecomes

K�equi

cms�

=2.79�10−4�exp�−11 004.9

T��ln

PTe2¦PTe2%

. (25)

The calculation of the Tamman and parabolicrate constants is explained in Appendix B.

HYPOTHESIS 2: Self-diffusion coefficient ofiron in b-FeTe0.9 dependent on concentration. Inthis case, it is necessary to study carefully theself-diffusion coefficient, its activation energiesand the structure of the formed iron telluride. D1*will vary depending on N2/N1, that is, dependingon the exact composition in the homogeneityrange. On the other hand, N2/N1 will vary de-pending on the equilibrium partial pressure oftellurium, so D1* will depend on the partial pres-sure of tellurium which will determine the metalexcess or defect (Shewmon, 1963). It can be ex-pressed in general terms as

D1*=cte�Nd�Wt (26)

where Nd is, the defect fraction in the lattice andWt, the jump frequency. The constant, cte, in-cludes lattice parameters. The general expressionsfor Nd and Wt are

Nd= (PTe2)1/n exp�−DH0

RT�

,

Wt=n exp�−DHm

RT�

. (27)

where DH0 is the molar activation enthalpy or theformation energy of a mol of point defects inb-phase; y the frequency, and DHm, the activationenthalpy of diffusion of iron in b-phase.

The Nd value will be determined from the devi-ation from the stoichiometry and for a givencomposition it is independent of the temperature.The variation of D1* with temperature at anygiven composition (constant defect fraction) isobtained from DHm since:d ln D1*

d1T

;N d

=d ln Wt

d1T

=−DHm

R. (28)

However, as expressed in Eq. (26), the self-dif-fusion coefficient is proportional to the defectconcentration which changes with the composi-tion of the atmosphere (the equilibrium partialpressure of tellurium). The b-FeTe0.9 is an n-typesemiconductor with metal excess. If the equi-


librium partial pressure of tellurium increases, theexcess of iron will decrease and the self-diffusioncoefficient of iron in the b-phase will decrease asthe composition reaches the tellurium rich side ofthe homogeneity range. This behavior is charac-teristic of all the n-type semiconductors. Frommeasurements of the equilibrium partial pressureof tellurium at the different compositions of theb-phase in its range of homogeneity (Ipser andKomarek, 1974), it can be deduced that the con-centration of point defects in b-iron telluride isproportional to PTe2

−1/2 at a given temperature,

NdT=cte*(PTe2)−1/2. (29)

This proportionality can be explained theoreti-cally if it is assumed that interstitials iron ions aredoubly charged (Fe+ +) and the electrons in ex-cess are associated to the interstitial iron ions togive complexes of the form

[�Fe+ +� ].

The b-FeTe0.9 can be obtained as

FeTe� [�Fe+ +� ]+12

Te2,

where

[�Fe+ +� ]

is the defect concentration, Nd. From the equi-librium constant, the value of Nd is obtained

[�Fe+ +� ]= (PTe2)−1/2 exp�−DH0

RT�

.

This result can be substituted in Eq. (26) to give

D1*=cte�(PTe2)−1/2 exp�−DH0

RT�

exp�−DH0

RT�

.

(30)

so, at constant temperature, the self-diffusioncoefficient of iron will decrease linearly withPTe2

−1/2.The activation energy of diffusion of iron in

b-phase at constant composition, DHm, is 91.5 kJmol−1 (Magara et al., 1991, 1992, M. Magara,1992, Private communication). The formation en-thalpy of the point defects DH0 can be obtainedfrom data on the equilibrium partial pressure oftellurium over the b-phase which exists in theliterature (Ipser et al., 1974).

At constant pressure, the expression for Nd is

Ndp=cte�exp

�−DH0

RT�

, (31)

through which one can calculate DH0 from thedefect fraction at different temperatures. By usingthe data from Ipser et al. (1974) at PTe2=188.3Pa, the value of DH0 is obtained to be 72.8 kJmol−1. This value agrees with that obtained in arecent study carried out by Magara et al. (1992)(DH0=72.5 kJ mol−1), who also observed thedefect fraction dependence on PTe2

−1/2.These values are substituted in Eq. (30) to give

D1*=cte3�(PTe2)−1/2�exp�−164 300

8.31451�T�

. (32)

The variable DH= −164.3 kJ mol−1 (DH=DH0+DHm) is defined as the activation energy ofenthalpy for diffusion at constant tellurium par-tial pressure and physically indicates that at con-stant tellurium partial pressure, the compositionof the telluride changes with temperature (Shew-mon, 1963). The value of cte3 can be obtainedfrom Eq. (24), and from tellurium partial pressurein equilibrium over FeTe0.892 at 923 K (0.683 Pa).These data can be introduced in the expressions ofD1* at constant temperature, at constant telluriumvapor pressure and at constant composition togive cte3=93.695 (cm2 s−1). The final expressionfor the self-diffusion coefficient of iron in b-phaseis

D1*=93.695�(PTe2)−1/2�exp�−164 300

8.31451�T�

. (33)

Previous expression will be sustituted in Eq.(22) to calculate the rational rate constant at agiven temperature as

K

=0.036�exp�−164 300

8.31451�T�

×�93.695�& PTe2¦

PTe2%

N2

N1

(PTe2)−1/2 1PTe2

dPTe2.

(34)

The integration limits correspond to the telluriumpartial pressure at both sides of the product layerat a given temperature.


To apply Eq. (34) it is necessary to know, foreach temperature, the tellurium partial pressure inequilibrium with the N2/N1 ratio over all theb-phase field. Unfortunately, a complete series ofdata can not be found in the literature, but it canbe concluded that no significant error is made ifthe value of N2/N1 is considered to be constantand equal to 0.9 in all the range of the b-phase.In the literature there only exists data for calculat-ing the rational rate constant at 923 K, thus itseems to be a good solution to consider N2/N1

constant, so the analytical calculations of theintegral can be performed, resulting in

K

=0.036�93.695�exp�−164 300

8.31451�Tn

×�0.9�2�� 1

PTe2%−

1

PTe2¦n

. (35)

Under accident conditions PTe2¦ZPTe2% , so

K�equi

cms�

=6.071436�(PTe2% )−1/2�exp�−19760.6

T�

=0.036�0.9�2�D1* (innerside) (36)

Two important conclusions can be obtaineddirectly from the previous equation: (a) The ratio-nal rate constant for the formation of b-FeTe0.9

does not depend on the external tellurium partialpressure, which clearly agrees with the resultsobtained for other types of n-semiconductor com-pounds (with metal excess) (Wagner, 1936, 1951;Jost, 1960; Hauffe, 1965; Shewmon, 1963). (b)The rational rate constant depends on the self-dif-fusion coefficient of iron in the b-FeTe0.9 mea-sured in the iron rich side of the compound.

The Tamman rate constant, K %, and theparabolic rate constant, Kp

2, are calculated as ex-plained in Appendix B.

4.2.2. Oxidized stainless steelThe most common feature is that under normal

operation conditions of a PWR, an oxide layer isformed on the SS. Recent studies (Saito et al.,1985) indicate that two different layers can be

formed, the outer, in contact with the primarysystem atmosphere, formed by Fe3O4, and theinner, in contact with the alloy, formed by acompound of the type FeCr2O4. The presence ofthis oxide layer will modify the chemical interac-tion of tellurium with the SS, and the telluridesformed will be those of higher tellurium content,the phases d-FeTe1.427, d %-FeTe1.451 and o-FeTe2.

There exists data in the literature for the self-diffusion coefficient of iron in Fe3O4 at tempera-tures ranging from 1023 to 1273 K (Himmel et al.,1953), for the self-diffusion coefficient of iron ind %-FeTe1.515 when 798 KBTB1023 K (Magaraet al., 1990a) and for the activation energy for thediffusion of iron in d-FeTe1.347 (Magara et al.,1991). There also exists data indicating that theactivation energy for the diffusion of iron ino-FeTe2 is probably very high, because the diffu-sion coefficient is too small to be measured evenat 873 K, near the peritectic point.

From the available data, it can be deduced thatthe diffusion of iron in Fe3O4 is slower than thediffusion through the tellurides formed, so thediffusion through the oxide will be the limitingstep for the tellurium–SS reaction when a mag-netite layer is present on the surface.

From the iron–tellurium phase diagram, it canbe deduced that phases b and o coexist in a rangeof values from room temperature to 792 K. Thetellurium partial pressure in equilibrium with thismixture is reported in the literature (Baba et al.,1988) from 656 to 759 K.

Phases b and d % coexist between 792 and 838 K.The tellurium pressure in equilibrium with bothphases can be obtained in the temperature range803–818 K (Baba et al., 1988). Phases b and d

coexist between 838 and 909 K, but there is nodata for the tellurium pressure over both phases.From 909 to 1073 K, the b-phase coexists withthe g-phase, which is an intermediate phase in thisrange of temperatures between phases b and d ;however, neither its composition nor its structureare known, this being the reason why it will beassumed that the b-phase also coexists with the d

and d % phases.All this information will be used for calculating

the rational rate constant for the chemical interac-tion between tellurium and oxidized SS, along


with the assumptions that the diffusion of iron ismuch higher than that of tellurium and the activi-ties of tellurium can be substituted by the corre-sponding tellurium partial pressure, so Eq. (21)will be used.

It is worth mentioning two additional hypothe-ses:

1. The product formed will vary depending onsurface temperature. Below 792 K, the productwill be the o-phase. Therefore, assuming the nom-inal composition of FeTe2 (82.04 wt.% tellurium)and an average density of 8.092 g cm−3 (Gron-vold et al., 1954), C2 will be 0.052 g atoms oftellurium per cubic cm of FeTe2. D1* will beassumed to be the corresponding to the diffusionof iron in magnetite; PTe2¦ will be given by theaccident conditions, and PTe2% will be the dissocia-tion pressure of the o-phase; in this case theequilibrium pressure between the b and o phaseswill be used by extrapolation to the temperaturesof interest. The resulting expression for the ratio-nal rate constant will be

K�equi

cms�

=0.104�exp�−229 900

R�T��ln

PTe2¦PTe2%

.

(37)

2. From 792 to 1085 K the products are d andd % iron–tellurides. For calculation purposes onlythe d %-phase is considered. If the diffusion of ironin magnetite is the rate-determining step (D1* isself-diffusion coefficient of iron in magnetite) andC2 is �0.0462 g atoms of tellurium per cubic cmof FeTe1.451, the equation for the rational rateconstant reduces to

K�equi

cms�

=0.348�exp�−229 900

R�T��ln

PTe2¦PTe2%

.

(38)

C2 is based on a nominal composition of FeTe1.451

(76.83 wt.% tellurium) and an average density of7.674 g cm−3 (Gronvold et al., 1954). The inte-gration limit PTe2¦ is given by the accident condi-tions and PTe2% corresponds to the dissociationpressure of d %-phase, in this case the equilibriumpressure between the b and d % phases is used byextrapolation to the temperatures of interest. Thevalue used for C2 is based on an average irontelluride composition FeTe1.451 and an averagedensity of 7.674 g cm−3.

3. From 1085 K on, the product is above themelting temperature and the mass transfer oftellurium in gaseous phase will control the reac-tion.

5. Implementation of the model into the RAFT1.1 code

The model has been built with a modular struc-ture allowing for an easier implementation intotransport codes. It was decided to include it intothe RAFT 1.1 code (Im et al., 1987).

A unique feature of the RAFT is the inclusionof the homogeneous nucleation mechanism, whichprovides the capability to predict the particle sizespectrum of the aerosols without making anyassumptions concerning initial size and numberdensity. The aerosol surface temperature is differ-ent to the gas temperature. RAFT 1.1 does notconsider the chemical reaction between vaporsand aerosols, it only treats condensation. There-fore the inclusion of the new model to address thereaction of tellurium with silver aerosols has beenmade as follows: at temperatures above the melt-ing point of the product formed, tellurium vaporreacts with silver aerosols following a linear ratewhich is calculated using the mass transfer coeffi-cients and equations included in the RAFT 1.1code. At temperatures below the melting point,the theoretical expressions for the reaction rate,which will be controlled by the diffusion of silverthrough the product layer, have been developedand implemented in the RAFT 1.1 code. Once theAg2Te aerosols are formed, their deposition mech-anisms are the same than for the other aerosolsand they are the original models included inRAFT 1.1.

As silver–tellurium interaction progresses, alayer of Ag2Te covers the surface of the aerosolwhich is transported through the primary circuit.The coupling is made at each time step and foreach element of spatial coordinate, dx. Thus thetellurium consumption in a dx is introduced in themass balance equation as a sink, i.e. in the nextdx the available tellurium for reaction will havebeen reduced. Fig. 3 shows the process. The labelsdx1, dx2 and dx3 mean the spatial coordinates of


Fig. 3. Coupling of the model for the tellurium–silver interac-tion.

each dx, so that in the next time step, the productthickness is the sum of previous ones. Fig. 4shows this process. At time t0, and in the spatialcoordinate dx1, tellurium reacts with SS and alayer of product is formed; the thickness of thislayer of product is dy10. At the same time, in thespatial coordinate dx2, tellurium also reacts withSS and the product layer formed has dy20 thick-ness. As these layers of product are not removedfrom the primary system, in the next time, t1, theywill exist, so if tellurium reacts again with the SSin the spatial coordinate dx1, it will form aproduct layer of dy11 thickness, but in this spatialcoordinate, the total thickness of the productlayer will be dy10+dy11. Similarly, the thicknessof product in the spatial coordinate dx2 will bedy20+dy21 after the time t0+ t1.

There exists user options for choosing betweenthe different possibilities of the model. For in-stance, the calculation of the rational rate con-stant for silver–tellurium interaction can be madeeither using the data for self-diffusion coefficientor the data for the ionic conductivity. For tel-lurium–SS interaction, the rational rate constantcan be calculated by three different ways:1. To consider that the self-diffusion coefficient is

independent of the variation of product com-position FeTe0.9 in the range of homogeneityof the phase. Under this circunstance, the log-arithmic dependence of the rational rate con-

the primary circuit as they are treated by theRAFT code. The thickness of the product isaccumulated at each dx, in such a way that at theend of the circuit it will be possible to know itsvalue for aerosols of a determined size. In the nexttime step, at the beginning of the circuit, thevalues of the product thickness are set to zero. Itis assumed that the interaction of tellurium withsilver aerosols does not modify the size distribu-tion of aerosols; the heat transfer processes arenot considered.

Regarding tellurium–SS interaction, the RAFT1.1 code also considers the condensation of tel-lurium vapor on structural surfaces and thechemisorption. The overall chemisorption rate isobtained by considering diffusion from the bulkof the gas to surface and chemisorption to act inseries. RAFT 1.1 uses 10 cm s−1 as chemisorptionrate, that leads (as the authors of RAFT mention)the process being limited by the diffusion of tel-lurium from the bulk of the gas to the surfacerather than being controlled by surface kinetics,given that tellurium appears to be very reactive.

The new proposed model considers that at tem-peratures higher than the melting point of theproduct, the diffusion of tellurium from the bulkof the gas to the structural surface controls thereaction and the reaction rate is calculated usingthe mass transfer coefficients and equations in-cluded in the RAFT 1.1 code. Below the meltingtemperature the equations obtained in Section 4are applied.

For tellurium–SS interaction the coupling ismade in such a way that the thickness of theproduct is accumulated in each time step, dt, for

Fig. 4. Coupling of the model for the tellurium–SS interac-tion. (a) Time= t0; and (b) time= t1 (t1\ t0).


stant on the tellurium partial pressure in theatmosphere will be observed.2. To consider that the self-diffusion coefficientdepends on the variation of product compositionFeTe0.9 in the range of homogeneity. In this case,the rational rate constant will be practically inde-pendent of the tellurium partial pressure in theatmosphere.3. The rational rate constant is calculated consid-ering that there exists an oxide layer on the SS.The oxide produces passivation effects and theproduct formed will vary depending on the tem-perature.

6. Application to MARVIKEN ATT 4

There are not, as far as the authors know,separate effect experiments available to be used inthe validation of the specific developed models.Therefore, the objective has been to obtain resultsto verify that the model behaves in the foreseenway after its inclusion into the RAFT 1.1 code.

The verification consisted of the analysis of theMARVIKEN ATT 4 experiment with themodified/improved version of RAFT 1.1. Thepurpose of the Marviken Aerosol Transport Tests(ATT) was ‘to create a data base on the behaviorof vapors and aerosols produced from overheatedcore materials in large-scale facilities representingtypical water reactor primary systems and pres-sure vessels for risk dominant scenarios’ (MXE-301, 1985). The data obtained can be used in thevalidation of theoretical models assessing thetransport of fission products and aerosols in de-graded core accidents. Some modifications weremade to the original plant to satisfy the ATTrequirements, as a result, a large-scale model of aLWR primary cooling circuit was available, in-cluding a reactor vessel, simulated internal struc-tures, pressurizer, relief tank and associatedpiping.

Five experiments, within the ATT series, havebeen performed at this facility. Four of them (1,2a, 2b, 7) were designed to study the transport ofcesium, iodine and tellurium, while one of them(4) was intended to study the transport of cesium,iodine and tellurium along with silver and man-

ganese. From the experimental matrix it is de-duced that test 4 is the most complete becausehigh density aerosols were included. This test hasbeen previously analyzed by other users of RAFT(MXIP, 1985), which allows for a comparison ofresults. It has been decided to use this experimentfor the verification exercise.

The boundary conditions, gas and surface tem-peratures, pressure, tellurium partial pressure andso forth, correspond to the MARVIKEN ATT 4experiment (MXE-204, 1985). More detailed in-formation on the characteristics of the facility andof the experiment can be found in the referencesmentioned previously. In this experiment, thetransport of FPs such as cesium, iodine, telluriumalong with structural materials as silver and man-ganese was analyzed under boundary conditionsand geometry similar to that of PWRs. This ex-periment allows for the use of both models: themodel for the interaction of tellurium with sur-faces and the model for the interaction with silveraerosols.

The duration of the experiment was 4740 s.During this time, 59.3 kg of cesium, 10.9 kg oftellurium, 210 kg of silver and 15.2 kg of man-ganese were injected. The rates of steam, water,N2, Ar and H2 were, respectively, 36, 81.0, 15.3and 0.3 g s−1. However, some problems arisedand only 92% of Ag, 86% of Mn, 82% of Cs, 40%of I and 80% of Te which was injected wasrecovered, which introduced some uncertainties inthe mass balance. However, as the main results,those comparing the original RAFT 1.1 and theRAFT 1.1 with the new model are given in termsof the ratio between masses, the uncertainties donot affect the conclusions.

During the experiment, the low concentrationof cesium and tellurium as vapors, suggested theformation of compounds such as cesium carbon-ate, telluride and or manganate. The analysisresults for tellurium species in the aerosol particleindicate that tellurium concentrated at the surfaceof both, the manganese and silver based aerosols,but a higher correlation between the silver andtellurium deposition was observed to support theformation of Ag2Te (Allen et al., 1987). It wasalso observed that at temperatures around 1000K, the interaction of tellurium with SS coupons


Fig. 5. Nodalization used in the analysis of the Marviken 4 experiment.

was inhibited by either the metal oxide surface orpreferential interaction between tellurium vaporand silver aerosols.

The nodalization used for the calculations isrepresented in Fig. 5 along with the surface ofeach volume. It only contains the first elevenvolumes of the facility, because as it was observedin some preliminary calculations, from theeleventh volume on, the vapor pressure of tel-lurium was practically negligible. A detailednodalization of the pressure vessel and internalshas been made because it is in these parts of thesystem where the high temperatures take placeand where the chemical phenomena are of great-est importance.

For tellurium–SS interaction, the three avail-able options for calculating the parabolic ratehave been used: IFE=1 (self-diffusion coefficientindependent of product composition); IFE=2(self-diffusion coefficient dependent on productcomposition), and IFE=3 (presence of an oxidelayer). Fig. 6 represents the values for the rational

rate constant, K, by using the three model optionsfor the 10–15 m axial coordinate, which corre-sponds to a part of the pipe L01. It can bededuced that the values of K for IFE=1 andIFE=2 are practically the same, while the valueof K for IFE=3 is really much lower, whichindicates that when an oxide layer is present thereaction rate clearly decreases.

The comparison in terms of mass of telluriumdeposited has been made for the eleven volumesof the nodalization and for the total duration ofthe experiment. The results obtained for IFE=1,2 and 3 are represented in Fig. 7 along with thoseobtained when the model originally implementedinto the RAFT code (IFE=0) is activated. Theresults show that whatever the chosen option ofthe new model was, the mass of tellurium whichhas reacted with the SS is much lower than thatcalculated by the original model. The optionsIFE=1 and IFE=2 give similar results becausethe respective rational rate constants differ in afactor of two which is reduced to its square root


Fig. 6. Evolution of the rational rate constant for the interaction of tellurium with SS by using the three model options.

when the product layer thickness is calculated. Ifthickness is put in terms of the tellurium reactedwith the SS, the differences even decrease. WhenIFE=3, it could be logical to expect a loweramount of tellurium deposited due to the passiva-tion effect, which would be theoretically confi-rmed by the value of the rational rate constant.However, these predictions do not hold due to thefollowing reasons: the highest amount of tel-lurium deposited in the second volume whenIFE=3 is a consequence of the characteristics ofthe product formed FeTe1.451 with a melting pointlower than that of FeTe0.9, therefore, given thatthe surface temperature in this volume is initiallyabove the melting point of the FeTe1.451, the masstransfer in gaseous phase will be the rate deter-mining step. In volumes 3, 4 and 5, as the rationalrate constant is low when IFE=3, the thicknessof the product layer will be also small and in thenext time step its increase will be higher. Thedifference reached in volume 4 could seem to beexcesive when compared to that existing in othercontrol volumes, however it must be noticed that

this volume corresponds to the centreplate with0.3 cm length an a high flow velocity, so theresidence time in this volume is really small andthe passivation effects are not present. From vol-umes 6–11, the surface temperature makes theproduct formed to be FeTe2 with a higher tel-lurium content.

For tellurium–silver interaction, the differencesbetween both options of this module, i.e IPLA=1 (when the data from the self-diffusion coefficientare used), and IPLA=2 (when the data from theionic conductivity are used) have been also ana-lyzed. Fig. 8 represents the values of the rationalrate constant obtained by applying the boundaryconditions of the MARVIKEN 4 experiment. Asit was foreseen, the differences between both op-tions is small because the values of the parametersused are related through the Nerst–Einstein equa-tion and Haven’s factor. The difference is evensmaller when the mass of tellurium which hasreacted in each volume is analyzed. It is worthmentioning that it is in the vessel (volumes 1–5)where the highest interaction with silver aerosols


Fig. 7. Comparison of the mass reacted of tellurium with SS by using the three possible options of the new model and the originalmodel (no update) of RAFT 1.1.

occurs as can be seen in Fig. 9. The interactionreduces the vapor pressure of tellurium and there-fore its interaction with the SS.

The good agreement with the experimental re-sults after the inclusion of the new model in theRAFT 1.1 code can be observed in Fig. 10. Itshows the results obtained by applying the newversion of the code (MODELO in the figure),which includes the modules for the interaction oftellurium with SS and with silver aerosols; theresults obtained by applying the initial version ofthe code (RAFT 1.1 in the figure), and the resultsobtained by other users of RAFT (RAFT USA inthe figure) (MXIP, 1985). The represented vari-able is the ratio between the mass of telluriumdeposited as calculated by the code, and the ex-perimental value measured, so the best valuewould be 1. It is evident from the comparisonthat, in the reactor vessel (RV), the depositionpredicted by the new model (MODELO) is higherthan the experimentally measured, however theoverprediction does not exceed the factor 4, while

the underprediction of RAFT 1.1 and RAFTUSA is higher than 2. The improvement is clearlyappreciated in the pipes towards the pressurizer(PIPES.PRES) (volumes 6–11) where the newmodel fits the behavior of the data better; theoverprediction of the mass of tellurium depositedin this part of circuit decreases from a factor of8.37 (RAFT 1.1) to 1.32 obtained with the newmodel, so, it is worth mentioning that the betteragreement with the experimental data is conse-quence of the inclusion of a new model to addressthe reaction of tellurium with silver aerosols andthe inclusion of a new model to take into accountthe passivation effects in the interaction of tel-lurium with structural surfaces.

Perhaps the main improvement of the newmodel is that it provides a reliable way to estimatetellurium behavior and to distinguish and identifythe mechanisms which govern its transport in theRCS. Thus, tellurium interaction with silver aero-sols and its deposition is the main mechanism inthe reactor vessel while the interaction of tel-


Fig. 8. Evolution of the rational rate constant for the interaction of tellurium with silver aerosols by using the two model options.

lurium vapors with SS governs the transportalong the pipes to pressurizer.

7. Conclusions

The problem of major concern in environmen-tal impact analyses is the characterization of thesource term. In some applications, much of theuncertainty present in the calculations stems fromlack of knowlegde about the physical/chemicalprocesses taking place in the primary circuit andthe related parameters, such as the reaction ratesneeded in the transport calculations through theprimary system. In general, tellurium chemistryhas not been considered specifically and hence thepertinence of a detailed study of transport datafor it and its compounds.

A new model based on basic laws, experimentalevidences when available, and taking into accountthe limitations imposed by the accident condi-tions, has been developed for analyzing the inter-action of tellurium with silver aerosols and with

SS. The model considers the reaction rates to becontrolled by the mass transfer in gaseous phaseabove the melting point of the product formed,and by the diffusion through the product layer atlower temperatures. In this article only the calcu-lation of the reaction rates at temperatures lowerthan the melting point of the product is addressedgiven that it is the main contribution of the newmodel.

The reaction rates of tellurium with silver andSS have been calculated taking as a basis thetheoretical rate equations developed by Wagner.The self-diffusion coefficients of iron and silverions through the product formed as well as theionic conductivity (in some cases) have been cal-culated. These data along with the appropriatethermodynamic data obtained from the literatureare used to calculate the reaction rates.

The model has been programmed in FOR-TRAN with a modular structure. The input vari-ables needed are gas and surface temperatures,total pressure, partial pressure of tellurium, distri-bution of silver aerosols (mass and size) and the


Fig. 9. Mass of tellurium reacted with silver aerosols in the different volumes of facility.

thickness of the oxide layer when the option ofthe SS oxidized is chosen by the user. The modelhas been implemented into the RAFT 1.1 code inorder to assess its capacities and to perform theverification.

Although there are no data in the literaturewith which the reaction rates may be compared, ithas been decided to use the conditions of theMarviken ATT 4 experiment, which included sil-ver aerosols as well as internal structures of SS,for comparing the global results of the model withthe experimental data in terms of mass of tel-lurium deposited.

The module which controls the interaction withthe SS gives values for tellurium deposition verysimilar to the experimental ones in the pipes topressurizer while the inclusion of the module forthe interaction with silver aerosols improves theresults in the reactor vessel. This allows for theidentification of the mechanisms controlling thetransport: the interaction with aerosols in thereactor vessel and the reaction with the SS in theRCS.

Although the behavior of the new model is verysatisfactory, to obtain final conclusions, its valida-tion against specific separate effect experimentswithout any external influence should be desir-able.

The assumptions and results of the model seemto be valid, although improvements could bemade to account for the simultaneous presence oftwo product layers in the case of the tellurium-SSreaction if the coexisting atmospheres of telluriumin equilibrium with both phases were available inthe range of conditions of interest. Evenmore, thestudy of the kinetics of the formation of othertellurides (i.e. cadmium tellurides, tin tellurides)would be desirable.

8. Nomenclature

a1 thermodynamic activity of silver (iron)in the product formedthermodynamic activity of tellurium ina2

the product formed


Fig. 10. Comparison of the ratio calculated/experimental mass of tellurium deposited: Original code (RAFT 1.1), improved version(MODELO) and other results (RAFT USA).

C1 concentration of silver (iron) in theproduct formedconcentration of tellurium in theC2

product formedself-diffusion coefficient of cationsD1*self-diffusion coefficient of anionsD2*electronic chargee

f Haven’s factorF Faraday’s constantf2 weight fraction of tellurium in the

product formedBoltzman’s constantk

K rational rate constantparabolic rate constantKp

Tamman’s rate constantK %M2 atomic weight of telluriumn quantity of product formed

ionic densitynAg

defect fractionNd

PTe2 partial pressure of telluriumq cross sectionR gas constant

timett1 transference number of cationst2 transference number of anions

transference number of electronst3

T absolute temperatureV equivalent volume

jump frequencyWt

thickness of product layerxchemical valence of silver (iron)Z1

Z2 chemical valence of telluriumdensity of the product formedr

DHm activation enthalpy of diffusionDH0 molar activation enthalpy of formation

energy of a mol of point defectsconductivity of product layers

activation energy for an ionic dif-o

fusionactivation energy associated to the con-os

ductivitychemical potential of the anionsm2

ionic mobilitymAg

y frecuency


Acknowledgements

This work was financed by the Commission ofEuropean Communities (Contracts 3608-88-12 ELISP PC and 4175-90-12 EL ISP E) and by theOECD-LOFT-Espana Project.

Appendix A. Calculation of the Tamman andparabolic rate constants for the silver–telluriumsystem

The Tamman rate constant calculated from Eq.(2) (V=20.2 cm3/equivalent)

T51075 K:

K %�cm2

s�

=7.0442�10−5�exp� −0.14

8.616�10−5�T��ln

PTe2¦PTe2%

.

1075BTB1232 K:

K %�cm2

s�

=2.059�10−5�lnPTe2¦PTe2%

.

The parabolic rate constant Kp2 refers to the tel-

lurium which has been taken up by the specimenfrom the gas phase (Himmel et al., 1953),

K2p� g2

cm4s�

=2M2 f2rK

�Z2� , (A.1)

where M2 is the atomic weight of tellurium (127.6g mol−1); f2, the weight fraction of tellurium inthe growing telluride layer (0.3716); r, the densityof the growing telluride layer (8.5 g cm−3); K, therational rate constant, and Z2 the valence oftellurium. If the values are substituted the expres-sions will be

T51075 K:

K2p� g2

cm4s�

=1.4055�10−3�exp� −0.14

8.616�10−5�T��ln

PTe2¦PTe2%

.

1075BTB1232 K:

K2p� g2

cm4s�

=4.1082�10−4�lnPTe2¦PTe2%

.

Appendix B. Calculation of the Tamman andparabolic rate constants for the stainlesssteel–tellurium system

The Tamman rate constant calculated from Eq.(2) (V=12.6 cm3/equivalent)

K %�cm2

s�

=3.52�10−3�exp�−11 004.9

T��ln

PTe2¦PTe2%

.

The parabolic rate constant is calculated fromEq. (A.1). In this particular case M2 is the atomicweight of tellurium (127.6 g mol−1); f2, the weightfraction of tellurium in the growing telluride layer(0.6728); r, the density of the growing telluridelayer (6.77 g cm−3), and �Z2� the valence of tel-lurium. If previous values are substituted we ob-tain

K2p� g2

cm4s�

=16.22�10−2�exp�−11 004.9

T��ln

PTe2¦PTe2%

.

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