An improved model for fission product behavior in nuclear fuel under normal and accident conditions

Journal of Nuclear Materials North-Holland, Amsterdam

195

120 (1984) 195-212

AN IMPROVED MODEL FOR FISSION PRODUCT NORMAL AND ACCIDENT CONDITIONS *

BEHAVIOR IN NUCLEAR

FUEL UNDER

J. REST Materials

Science and Technology Divrsion, Argonne National Laboratory, Argonne, Illinois 60439, USA

Received

1 August

1983; accepted

23 November

1983

A theoretical model for the prediction of the behavior of the gaseous and volatile fission products in nuclear fuels under normal and transient conditions has undergone substantial improvements. The major improvements have been in the atomistic and bubble diffusive flow models, in the models for the behavior of gas bubbles on grain surfaces, and in the mode1 for iodine solubility. The theory has received extensive verification over a wide range of fuel operating conditions, and can be regarded as a state-of-the-art model based on our current level of understanding of fission product behavior. Because of existing uncertainties in both materials properties and mechanisms of fission product response, any verified mechanistic description of fission product release entails assumptions in these areas. The diffusivity of fission gas bubbles during transient conditions, the effect of irradiation induced grain boundary gas-atom re-solution on intragranular diffusive flow rates, and the effect of iodine solubility on iodine release rates are aspects of fission product behavior that are currently clouded with uncertainty. The assumptions that have been made in the theory have been evaluated and are discussed in relation to the mode1 verification, uncertainties in the existing data base, and other theoretical descriptions of fission product behavior.

1. Introduction

and fission-gas bubble/gas-atom interactions. In addition, the evolution in time of the average bubble sizes is calculated. The intragranular single gas atoms are characterized by number density; the intragranular, grainface, and grain-edge bubbles are characterized by number density and the average number of atoms per bubble, L&(t). (The index “i” indicates an intragranular, grain face, or grain-edge bubble type.) Models [l-3] are included for the effects of the following key variables: production of gas from fissioning nuclei, bubble nucleation and coalescence, bubble irradiation induced re-solution, gasmigration, bubble/channel formation on grain faces, temperature and temperature gradients, interlinked porosity on grain edges, microcracking, experimentally derived steadystate bubble mobilities, and phenomenological modeling of bubble mobilities during transient nonequilibrium conditions. These models are used to calculate fission gas release and the swelling due to retained fission gas bubbles in the lattice, on grain faces, and along the grain edges for steady-state and transient thermal conditions. As the noble gases have been shown to play a major role in establishing the interconnection of escape routes from the interior to the exterior of the fuel [4,5], a

A theoretical model (FASTGRASS) has been used for predicting the behavior of fission gas and volatile fission products (VFPs) in UO,-base fuels during steady-state and transient conditions [1,2]. This model represents an attempt to develop an efficient predictive capability for the full range of possible reactor operating conditions. Fission products released from the fuel are assumed to reach the fuel surface by successively diffusing (via atomic and gas-bubble mobility) from the grains to grain faces and then to the grain edges, where the fission products are released through a network of interconnected tunnels of fission-gas induced and fabricated porosity. The model provides for a multi-region calculation and uses only one size class to characterize a distribution of fission gas bubbles. Whereas the behavior of fission gas at the grain face and grain edge is based entirely on this single-size-class description, the description of intragranular fission gas behavior includes the kinetics of fission gas atom generation and migration * Work supported

by the U.S. Nuclear

Regulatory

Commis-

sion.

0022-3115/84/$03.00 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)

B.V.

196

J. Resr / An rmproved model for fission product behauror

realistic description of VFP release must a priori include a realistic description of fission gas release and swelling. In addition, as the VFPs are known to react with other elements to form compounds, a realistic description of VFP release must include the effects of VFP chemistry on VFP behavior. A mechanistic description of VFP behavior was developed by modifying the FASTGRASS fission gas analysis to include theoretical models for the effective production rates of the relevant VFPs, the chemical interactions between the various VFPs, the interaction of the VFPs with the fission gas bubbles, and the migration of the VFPs through the solid UO, fuel. (See ref. [2] for a detailed discussion of VFP modeling.) In the present treatment, the VFPs I, Cs. and their major reaction products (CsI, Cs,MoO,, and Cs,UO,) have been included. The formation of Cs,MoO, and Cs,UO, can have a crucial effect on the reactions involving CsI, which are of major concern for deducing the form of the iodine released in LWR power plant accident scenarios. As the available data are not sufficient to describe the kinetics of VFP chemical reactions, the approach to modeling VFP chemistry used here is to assume that the kinetics of the relevant VFP chemical reactions occur fast enough that a quasi-chemical equilibrium is maintained. The model calculates the density and chemical form of retained fission products as a function of fuel morphology as well as the amount and chemical form of the released fission products. At present, no attempt is made to describe the behavior of fission products after release from the fuel (e.g., the behavior of the fission products in the fuel-cladding gap). The effects of a steam environment, fuel liquefaction and gross fuel melting on fission gas and VFP behavior are currently not considered. An important feature of all physical models of fission product release and swelling is the calculation of the effective fission product deposition rate at the boundaries of the fuel grains. A previous treatment [3] of diffusive flow to a spherical boundary for conditions of changing fission rates and fuel temperatures has been examined and found to be deficient in that relatively small time steps are required in order to obtain consistent results. The use of very small time steps degrades the calculational execution efficiency and is thus undesirable. The inaccuracies in this approach have been identified with the assumption that the fission products are uniformly distributed throughout the grain at the beginning of each time step. This assumption results in an artificial surge of fission products to the grain boundaries at that time. A more realistic model for describing the diffusive

flow of fission products based on the work of Matthews and Wood for the diffusive flow of gas atoms [6,7] has been incorporated into the theory. In this model, the problem of diffusive flow to a spherical boundary is approximated by describing the concentrations of the diffusing species in terms of quadratic functions in two concentl;ic regions. A variational principle is used to calculate the radial distribution of the concentration. This approach provides an improved degree of accuracy as well as shorter running times. The diffusive flow of fission gas bubbles to the grain boundaries is also considered. The calculation of the diffusive flow of gas bubbles is treated in an analogous fashion to that for gas and VFP atoms by assuming that the gas bubbles are not being nucleated and are essentially noninteracting during the calculational time increment. The effects due to bubble nucleation and interaction are included in the analysis subsequent to the diffusive flow calculation. The coupling of the diffusive flow problem to other processes affecting fission product behavior (e.g., gas atom re-solution, gas atom and VFP trapping by gas bubbles, gas bubble nucleation and coalescence, and chemical reactions between VFPs) is accomplished by solving for the intragranular densities of VFPs, gas atoms, and average-sized gas bubbles using a rate theory approach; the various fluxes due to diffusive flow, gas atom re-solution, and atom-bubble, VFP-bubble, and bubble-bubble interactions are treated as time-dependent coefficients in the rate equations. This method provides a realistic and fast calculation of intragranular fission product behavior. Similar equations describing intergranular (grain face and grain edge) fission product behavior are coupled to the equations for intragranular fission product behavior and the complex of nonlinear differential equations is solved by a finite-difference technique. Additional improvements in modeling fission product behavior include a more precise model for irradiation induced re-solution, the use of a lenticular (instead of spherical) bubble geometry on the grain boundaries in the calculation of intergranular gas bubble swelling and saturation, and an improved model for iodine solubility. Section 2 of this paper contains a detailed description of the recent major improvements to the theoretical FASTGRASS model. In sections 3 and 4, results of a relatively extensive verification effort are shown and discussed for steady-state and transient conditions, respectively. Because of existing uncertainties in both materials properties and mechanisms of fission product response, any verified mechanistic description of fission

191

J. Rest / An improved model for fission product behavior

product release entails assumptions in these areas. The diffusivity of fission gas bubbles during transient conditions, the effect of irradiation induced grain boundary gas-atom re-solution on intragranular diffusive flow rates, and the effect of iodine solubility on iodine release rates are aspects of fission product behavior that are currently clouded with uncertainty. The assumptions that have been made in the theory have been evaluated and are discussed in section 5 in relation to the mode1 verification, uncertainties in the existing data base, and other theoretical descriptions of fission product behavior.

generation rates and temperatures has previously been handled by obtaining the general solution of eq. (la) as the sum of the solutions to two separate problems: ProbIem

I

Eq. (la) is solved

Ca = 0

at t = t,

Cg = 0

at r = d,/2

acs

%=O

Problem

atr=O

with the boundary

conditions

for 0 < r G d,/2,

(le)

for t < t < t, + h,

(19

fort,
2

2. Improved models 2. I. Diffusive

flow

with boundary

In the modeling of fission gas behavior in nuclear fuel, it is necessary to calculate the loss of gas to the grain boundaries by diffusion of single gas atoms and gas bubbles. To simplify the problem, grains within the fuel are assumed to be spherical. If the only sink for gas atoms is the boundary itself, the concentration of gas atoms, C’s, within the spherical grain satisfies the equation

(Ia) where Dg is the gas atom diffusion coefficient the rate of generation of gas atoms. In general, is solved with the boundary conditions Cs = 0

at t = 0

Cg=O

atr=d,/2

ac, %=O

atr=O

for 0 Q r < d,/2, fort,gt
and K, eq. (la)

(lb)

(Ic) (ld)

where d, is the grain diameter, and h is the time interval between times t, and t, -t h. An analogous equation describing the intragranular diffusion of gas bubbles to the grain boundaries can also be defined. However, the problem of calculating the diffusive flow of gas atoms and bubbles to a spherical boundary is not so straightforward if real irradiation histories are to be followed, with changing gas generation rates and temperatures: In this case, no general analytical solution is possible. Additional complications are provided by the processes of gas atom re-solution, gas atom trapping by gas bubbles, gas bubble nucleation, and bubble coalescence. The problem of calculating diffusive flow of fission gas to the grain boundaries for the case of changing gas

att=t,

C,(r,t,)=C,(t,) Cg(r,

to)=0

ac,( r, t)

conditions forO
atr=d,/2

=0

(2b)

fort,,
atr=O

(2c)

fort,
(24

ar

where cs( to) is the average grain at t = to.

concentration

within

the

Physically, Problem 1 describes the behavior of the gas generated subsequent to the change of irradiation conditions, at to, and Problem 2 deals with the behavior, after t,, of the gas generated prior to the change of conditions. The genera1 solution to eq. (la) for changing gas generation rates and temperatures is the sum of the solutions of Problems 1 and 2. The diffusive flow of fission gas bubbles to the grain boundaries is also considered. For this case, it is assumed that gas bubbles are not being nucleated and are essentially non-interacting from t = t, to t = t, + h. The solution for the flux of bubbles to the grain boundary is obtained in an analogous fashion to that for the gas atoms. The coupling of the diffusive flow problem to other processes affecting fission gas behavior (e.g., gas atom re-solution, gas atom trapping by gas bubbles, and gas bubble nucleation and coalescence) is accomplished by solving for the intragranular densities of gas atoms and average-sized gas bubbles, given equations of the following form: dY -_L= dt

-aiY,2 - biYi +

ci.

The meanings of the variables scribed in table 1.

used in eq. (3) are de-

198

J.

Table 1 Definition

of variables

in eq. (3):

Rest

dY,/dr

/

An

improved

= - o,Y; -

biY,

model

for

fissron

product

hehovlor

+ c,

Y,

a,Y2

b,Y,

c

Density of intragranular gas atoms

Rate at which gas atoms are lost owing to gas bubble nucleation

Rate at which gas atoms are lost owing to diffusive flow to the grain boundaries and diffusion into gas bubbles

Rate at which gas atoms are gained owing to gas atom resolution and fission of uranium nuclei

Density of intragranular gas bubbles

Rate at which gas bubbles are lost owing lo bubble coalescence

Rate at are lost flow to and gas

Rate at which gas bubbles are gained owing to bubble nucleation and diffusion of gas atoms into bubbles

An analogous set of equations is used to characterize the average-sized bubbles on the grain faces and along the grain edges. The full set of coupled equations is solved incrementally as a function of time. Wood and Matthews [6] have identified inaccuracies in the above approach for calculating the diffusive flow of gas atoms to a spherical boundary, i.e., eqs. (la, e, f, g) and (2a-d). Furthermore, the calculated release is heavily dependent on the number of time steps taken during a given irradiation period and is always above the true release. The inaccuracies found were identified with the assumption Cs(r, to) = cs( to), (i.e., eq. (2b)), in Problem 2, above. The assumption that the gas is uniformly distributed throughout the grain at the beginning of each time step results in an artificial surge of gas to the grain boundaries at that time. Matthews and Wood [7] have suggested an approach to solving the problem of diffusive flow to a spherical boundary which uses an approximation describing the intragranular concentrations of the diffusing species in terms of quadratic functions in two concentric regions. A variational principle is used to calculate the radial distribution of the concentration. Note that this approach is in sharp contrast to the approach described above, which assumes a uniform concentration of the diffusing species within the grain. The concentration of gas atoms in a spherical grain described by eq. (la) is written as C:(r) at a time t. After a small time interval 81 the concentration becomes C,(r). Using the backward Euler approximation, for small St, eq. (la) may be replaced by

which gas bubbles owing to diffusive the grain boundaries atom re-solution

tional principle

equivalent

to eq. (4):

which assumes that Dirichlet boundary conditions are to be applied. Matthews and Wood showed that an approximate solution to the problem may now be obtained by choosing a trial function that satisfies the boundary conditions and minimizes the integral in eq. (5) in terms of free parameters in the function. Many types of trial function could be chosen, but Matthews and Wood claim that piece-wise functions are easier to handle than global functions. Quadratic functions are attractive as they allow an exact representation of eq. (la) for long times. To meet their objectives of a realistic level of accuracy with a minimum of computer storage and running time, Matthews and Wood split the spherical grain into two concentric regions of approximately equal volume. In each region, the gas concentration is represented by a quadratic function. In the central region I (see fig. l), the concentration function is

(4) Euler’s

theorem

may now be used to obtain

a varia-

Fig. 1. Configuration

of the two-zone

model used in ref. [7].

199

J. Rest / An improved model for fission product behavior

constrained to have dCs/dr = 0 at r = 0. In the outer region II, the concentration function is constrained to a value of Cs = 0 at r = dJ2. The two functions are also constrained to be continuous at the common boundary of the two regions. This leaves three free parameters. Matthews and Wood chose these to be the concentrations C,, C,, and C, given respectively by the radius ratios p, = 0.4, p2 = 0.8 and p3 = 0.9 where p = 2r/d,. These positions are the midpoint radius of region I, the boundary between the regions, and the midpoint radius of region II, respectively. Thus, the trial function are as follows: For region I: Cs = Cr(O.64 - p2)/0.48

+ C,( p2 - 0.16)/0.48,

(6)

for region II: c, = 5C,(10p2

=

(4

- lop2 - 8).

+ Gr)

C, + (@s/d,2

-

+ q&t

3

4 = q8D,/d,2 + q,P

3

F2 = @s/d,2

+ q,P,

4 = K,q, + (Ci’e + C,Oq,)/k F’ = q&/d,2 X2

=

K,q,,

F7

=

q,,D,/d~

X,

=

J&q,,

+ e/St, +

(Cd

+

+

q12/6t,

+ (C&g

+

C&7

+

C_%,)/k

and

C&2

)/St.

The flux of gas atoms to the boundary atoms/cm3 . s) is given by

F $dJ2.

(7)

+ q&Q)G

(9c)

F,C,)/F,,

F, = @s/d,2

J = - 19~ + 9) + lOC,(lSp

Eqs. (6) and (7) are substituted for Cs in eq. (5) and an extremum is found by differentiating with respect to C,, C, and C, in turn. A set of three linear equations is thus obtained: (G’s/d,2

G

where

(in units

of

(10)

g

From eqs. (10) and (7), the gas atom flux to the grain boundaries is (11) d,2

q, = 4.552,

q2 =0.06935,

qs=

q4 = 0.02167,

q5 = 0.09102,

q6 = 37.78,

Eqs. (6)-(11) are coupled to eq. (3) for the assessment of fission gas behavior within the grains and for the determination of the fission gas atom and gas bubble flux to the grain boundaries. In particular, the gas atom flux to the grain boundaries, eq. (11) is one term in the expression for the rate at which gas atoms are lost, given by the term biq in eq. (3) and defined in table 1. The treatment for the diffusive flow of gas bubbles is handled in an analogous fashion. For proper coupling of the diffusive flow process to other processes affecting fission gas behavior, (e.g., gas atom re-solution, gas atom trapping by bubbles, and gas bubble nucleation and coalescence) information is required on the average concentration of fission gas within the grains. Matthews and Woods [7] determined that the best expression for the average concentration within the grains, cg, is given by

q7 = 0.07614,

qs = -38.72,

q9 = 0.008456,

c, = 0.2876Cr + 0.2176C,

q10 =0.01008,

qll = 87.04,

q12 = 0.08656,

= J&q, + (($7, (@s/d;

+ C%,)/k

+ q&Q)

+ ( @s/d,2 = K,q,, (q&/d;

C, + (@s/d,2

+ q,/‘ar)

+ (C:,, + q,/St)

(8a) + q+)G

C,

+ C%, + &7,)/k C, + (Q$d,Z

(8b) + q,z/‘ar)

= J&q,, + (C,oq, + C,oq,,)/k

G (8~)

where Cp, Ci, Ci are values of the concentrations at the evaluation points at the start of the time increment. The various q coefficients are integrals which when directly evaluated are, to 4 figures, -4.552,

q13 =0.1083.

The set of eqs. (8) can be directly the concentrations C,, C,, and C,: X, c,=

solved

to obtain

+ 0.4216C,.

(12)

At the end of an iteration, the concentrations C,, C,, C, in eq. (12) are scaled by imposing the condition that the average concentration calculated by use of eq. (12) is equal to the average concentration calculated by use of eq. (3) i.e., that

F2C2

F

,

(94

1

C’b)

c, = Yi .

(13)

The modified C,, C,, C, then become the initial values of these concentrations (i.e., Ct, Ct, C,“) to be used for the next iteration. The diffusive flow of fission gas

200

J. Rest / An rmprouedmodel for fission product behavior

bubbles is treated analogously to that for fission gas atoms, but with K, = 0 in eq. (la). This method of coupling diffusive flow to other processes affecting fission gas behavior (e.g., gas atom re-solution, gas atom trapping by gas bubbles, bubble nucleation and coalescence) is computationally efficient and has been benchmarked against various analytical solutions. The results of analyses performed with the theoretical FASTGRASS model [S] demonstrate that the calculations are remarkably stable with respect to time step changes when the Matthews-Wood diffusive flow model is used. A maximum deviation of - 2.5% for time steps that differ by a factor of 500 obtained with the new diffusive flow model is in sharp contrast to a - 80% maximum deviation obtained by use of eqs. (la, e, f, g) and (2a-d). In addition, as the time step is decreased, the new theoretical results approach a constant limiting value much faster than the results obtained using the previous theory. As discussed above, the use of eqs. (la, e, f, g) and (2a-d) leads to calculated intragranular releases always above the true release. In order to maintain reasonable agreement between theory and experiment, the gas-atom re-solution rate used in the theory was increased from 2 X 1O-5 s-’ (at a fission rate]= 4 X lOI cme3 SK’) to 8 x lop5 s-‘. This value of gas atom resolution rate is more consistent with what appears to be the most reliable experimental estimate [9] of - 2.7 X 10e4 s-’ for i= 4 X lOI cmp3 s- ‘. The model for gas-atom re-solution used in the theory is discussed in the following section. 2.2. Irradiation induced re-solution The model for gas atom re-solution from gas bubbles used previously [2,3] is a simple approximation to single gas atom knockout from a bubble of radius R, where the average distance a “knocked” atom travels (re-solution distance) is X. The rate b (in s-‘) at which gas atoms are ejected from the bubble is given by b=b,Rj/(R+h),

(14)

where j is the fission rate (cmb3 s-l), and b, is a measurable property of the material (b, = 2 X lo-” cmp3). Although eq. (14) is a reasonable approximation for R z+ h, eq. (14) is in error by about a factor of 2 when R - h. For values of R < X,- the error introduced in using eq. (14) is of the order h/R. In order to remedy the deficiencies inherent in eq. (14), a more precise model was developed for the determination of the gas-atom re-solution rate from bub-

bles the cal out

of radius R. In this approach, b is calculated with assumption that gas-atom re-solution from a spheribubble is isotropic and proceeds by the knocking of single atoms [lo]. Thus,

(154 where cos 13= ( R2 - X2 - r2)/2rh. integration of eq. (Isa) results in

A straightforward

b=F(F2-F,),

(I5b)

where

, and 2 6R-A+(R-A)‘+1 16X i

F,=(R-X)

. 8

Because of the variation of X with R [lo], it is necessary to ascribe to this quantity some constant value averaged over the irradiation time of interest in order to facilitate the solution for b. The value of X used in eqs. (15a) and (15b) is discussed in section 5.1. 2.3. Intergranular bubble swelling Fission gas can migrate from the grain faces to the grain edges by diffusion (random and biased) and via short-circuit paths created by grain-face channel formation (i.e., interlinkage of grain-face bubbles). Channel formation on the faces is a function of the area1 coverage of the face by the fission gas. A realistic approach for calculating grain-face saturation by fission gas is to deal directly with the calculated fission-gas bubble distributions. Previous analysis [l-3,6] have been based on the swelling of spherical bubbles. However, because cos B = y&2 y # 0 ( y = UO, surface energy, ysb = grain boundary energy), it is more reasonable to assume that the bubbles are lenticular-shaped pores, containing m gas atoms and having a radius of curvatur p, which are joined in the plane of the boundary with dihedral angles 28 = 100” (see ref. [ll]). The fractional swelling due to these bubbles is given by the expression AV I/ = 4?rp3f (8)/3Y, wheref(B)

= 1- 5

(16) cos

e +

f

cos38,

P = ( 4;;;y)“2.

M is the number

(17) of gas atoms in the bubble,

and Y is

J. Rest / An improved model for fission product behavior

the number of bubbles on the grain face per unit volume. Eq. (17) was derived by assuming equilibrium and using the idea1 gas law. The projected area1 coverage of the grain face by these bubbles per unit volume is given

201

24

,1773K

20-

-

by A = w( p sin B)‘Y.

(18)

For fixed values of m and Y, eqs. (16)-(18) result in values of AV/V and A which are - 0.86 smaller and - 1.74 larger, respectively, than those calculated assuming spherical bubbles. 0

2.4. Iodine solubility

4

0

12

16

20

24

28

TIME (WEEKS)

In a previous paper [2], iodine release predictions were examined for two different assumptions about the diffusion of atomic iodine: (a) it diffuses intragranularly through the solid UO, (in solid solution), and (b) it diffuses with CsI in fission gas bubbles (in the vapor phase). Based on a comparison of the calculations with the data of Tumbull and Friskney [12] it was concluded that assumption (b) was more reasonable as well as more consistent with the assumption of quasi-equilibrium. However, subsequent to the incorporation of the improved diffusive flow model, it became apparent that assumption (b) was too restrictive (i.e., that the iodine solubility was zero). As a reliable calculation (and measurement) of iodine solubility is not currently available, the mode1 for iodine solubility is characterized by the equation

PITOT=a[l],+(l--a)[lhv

(19)

and [I] refer to total iodine and atomic iodine concentrations, respectively, and the subscripts a and b refer to iodine residing in solid solution and in fission gas bubbles, respectively. Eq. (19) is more general that the approach used in ref. [2] in that a can have a value between 0 and 1. As in ref. [2], CsI can migrate only within fission-gas bubbles. The solubility coefficient a in eq. (19) can be determined by comparison with experiment and is discussed in section 3. where

[UTOT

3. Comparison conditions

of theory and experiment:

steady-state

3.1. Fission product release Fig. 2 shows predicted fractional release of the stable fission gases as a function of time, and compares it with the isothermal release data for ‘33Xe from ref. [12]. The

Fig. 2. Predicted fractional release of the total stable noble gases at 1733 k40 K (solid curves), compared with ‘33Xe data of Turnbull and Friskney 1121 (symbols).

three predicted curves reflect the f40 K uncertainty in irradiation temperature. The agreement between prediction and data is good. It should be noted that FASTGRASS currently calculates only the behavior of the stable fission gases, and a comparison between the predicted fractional releases of stable gases and data for ‘33Xe (with a 5.25-day half-life) may be affected by qualitative and quantitative differences in behavior between the stable gases and ‘33Xe. The fact that the data for 133Xe release follow the predicted release of the stable gases supports the proposition delineated in this paper that the stable fission gases play a major role in establishing the interconnection of escape routes from the interior to the exterior of nuclear fuel. After steadystate concentrations of ‘33Xe have been achieved (presumably, within the first lo-12 weeks of irradiation and prior to any substantial bubble interlinkage), the 133Xe follows any pathways to the fuel exterior that have been created by the stable gases, and is released. Fig. 2 should be compared with predictions reported earlier (fig. 3 of ref. [2]), where the calculated release values were underpredicted for the last 7 weeks of the irradiation. The improved predictive capability shown in fig. 6 is primarily due to the model improvements described in section 2. The theory accurately describes fission gas saturation effects in that very little gas release is predicted (and observed) to occur prior to - 12 weeks of irradiation. At - 12 weeks of irradiation, the grain faces are saturated with fission gas, leading to interconnection and release. Subsequent to gas release the face channels “heal” and the accumulation of gas on the boundaries

J. Rest / An improoed model for fissron product behavior

202

begins anew. This physical picture of gas behavior on the grain faces is consistent with recent experimental observations [13]. Fig. 3 shows predicted fractional release of iodine as a function or irradiation time, and compares these results with the data of Turnbull and Friskney [12]. Again, to reflect the experimental uncertainty in temperature reported in ref. [12], three predicted curves are given, corresponding to irradiation temperatures of 1733 & 40 K. The circles in fig. 3 represent the fractional release of iodine (i3’I + 1331) calculated from the data by taking into account the respective fission yields of 13’1 and ‘331. The prediction of the theory is that the majority of the iodine released during this experiment occurred as CsI. Unfortunately, no experimental information on the chemical form of the released iodine is available at this time. Fig. 3 was generated using a value of 0.6 for the solubility coefficient, a, defined in eq. (19). Thus, approximately 40% of the atomic iodine is assumed to be within the fission gas bubbles, with the other 60% existing in solid solution within the UO, lattice. This is in contrast to the results reported previously [2], where the assumption that both atomic iodine and CsI diffuse predominantly in ,gas bubbles gave the best overall agreement with the data. The primary reason for this result was that the xenon (and krypton) gas bubbles had a predicted average bubble diameter of - 25 A and diffused to the grain boundaries at a faster mass transfer rate than did the diffusing iodine atoms. However, the effect of the improved diffusive flow model (section

2) is a predicted average intragranular bubble diameter of - 40 A. Bubbles of this size are predicted to diffuse at a slower mass transfer rate than the diffusing iodine atoms. From the above discussion, it can be seen that the rate of VFP migration to the grain boundaries is predicted to be very sensitive to the kinetics of fission gas bubble behavior. FASTGRASS utilizes an effective generation rate for the VFPs which takes into account precursor effects as well as radioactive decay. The use of an effective generation rate is reasonable after the attainment of quasi steady-state conditions in the fuel such that the VFP concentrations are relatively stable. It is estimated that for (13’1 + ‘331), steady state would occur after several months of irradiation. The fact that the FASTGRASSpredicted iodine release agrees with the data (as shown in fig. 3) supports this contention. In addition, the experimental observation that iodine release is qualitatively similar to noble gas release (e.g., compare the total iodine release data in fig. 3 with the ‘33Xe data in fig. 2. Also, see ref. [14]) provides additional support for the hypothesis that the noble gases create the majority of the escape paths for VFPs such as I and Cs as well as for certain short-lived noble gas isotopes, such as ‘33Xe. Fig. 4 shows the predictions of the theory for total iodine release at 1733 K as a function of irradiation time for various values of the iodine solubility coefficient, a. As in fig. 3, the circles in fig. 4 represent the fractional release of iodine (i3rI + 1331) calculated from the data. The results shown in fig. 4 demonstrate that

1 0 DATA

o.31 *

* I773K

*

&

/ 0

1733K

*

TIME (WEEKS)

1693K

J 33

Fig. 3. Predicted fractional release of “‘I+ Is31 at 1733 +40 K (solid curves), compared with data of Turnbull and Friskney (symbols).

,a:0

TIME(WEEKS)

Fig. 4. Predicted fractional release of iodine at 1733 K for various values of the iodine solubility coefficient, a (solid curves), compared with the data of Tumbull and Friskney [12] (symbols).

203

J. Rest / An improved model for fission product behavior

the predicted iodine release is a strong function of the iodine solubility. Because of the experimental temperature uncertainties for the data shown in figs. 3 and 4, it is difficult to suggest a value for a based on agreement between theory and experiment. Clearly, additional experimental and theoretical work is needed in order to further resolve the question of iodine solubility in UO, fuels. Fig. 5 shows predicted fission gas release as a function of fuel burnup, and compares these results with the data of Zimmermann [15]. Uranium dioxide fuel with a fission rate of 1Or4 f cmv3 s-r was used in these experiments. A temperature gradient of 1OOO’C s-’ and grain diameters between 1 and 10 pm were used for the calculation. Four different sets of calculated curves were generated using average fuel temperatures of 1250,1500, 1750 and 2000 K. The use of relatively small grain diameters for the calculation of the low-temperature Zimmermann data agrees with the results obtained by other authors [16]. Presumably, the use of relatively small “effective” grain diameters is required in order to simulate, to some degree, subgrain-boundary formation which may have occurred in this fuel [17]. The 1250 and 1500 K data are bracketed by predictions based on 1 and 2.5~pm grain sizes, respectively. The 1500 and 2000 K data are bracketed by predictions based on 2.5 and 5+m, and 5 and lo-pm grain sizes, respectively. Again, agreement between theory and data is reasonable. Fig. 6 shows calculated end-of-life gas release for fuel irradiated in the Carolinas-Virginia Tube Reactor (CVTR), the H.B. Robinson (HBR) No. 2 Reactor, and the Saxton Reactor, compared with the measured values. Also shown in fig. 6 are the predicted and measured

MEASURED

Fig. 6. Comparison

FISSION GAS RELEASEIW

of theoretical

predictions

with end-of-life

gas release.

end-of-life releases for the Tumbull and Zirmnermann experiments corresponding to figs. 3 and 5, respectively. The diagonal line indicates perfect agreement between theory and experiment. To supply FASTGRASS with the proper operating conditions for the CVTR, HBR, and Saxton irradiations, FASTGRASS was coupled to an experimental LWR fuel behavior code generated by making suitable modifications [18] to the LIFE fuel performance code. As is evident from fig. 6, the theory predicts the data reasonably well for fission gas release between 0.2 and - 100% and for bumups between 0.7 and 10 at!% (- 7000-100 tXKl MWd/MT). The largest differences between predictions and measurement occur for the CVTR irradiations. These differences are attributed to uncertainties in power history and to uncertainties in the LIFE calculation of fuel temperatures. 3.2. Average size of fission gas bubbles

ZIMMEAMANN DATA ---CALCULATIONS

0

2

4

6

8

IO

BURNUP (%I

Fig. 5. Predicted fractional fission gas release at 1250, 1500, 1750 and 2000 K (dashed curves), compared with the data of Zimmermann [15] (solid curves).

Fig. 7 shows predicted average bubble size compared with the data of Come11 et al. [19]. The data of ref. [19] were obtained from transmission electron micrographs of thin foils which were prepared from an irradiated UO, pellet (which had acquired a bumup of 3.2 x 102’ f me3 in 40 days) so that the temperature dependence of the intragranular bubble size could be determined. Intragranular bubbles were observed in material where the estimated irradiation temperature lay in the 860-158O’C range. Fig. 7 shows that the theory under-predicts the bubble size measured by Cornell for fuel temperatures

J. Rest / An improved model for fission product behavior

204

:

16 0 CORNELLDATA A PREDICTED

.

0

a

.A

ii00

_ 1200

1600

2000

TEMPERATURE(KI

Fig. 8. Predicted retained fission gas at 3 and 12 at% burnup (solid circles), compared with the data of Zimmermann (151 at burnups above 3% (open circles).

states that this gas originates from intragranular pores and bubbles with diameters down to 10 nm and from gas which was retained on the grain boundaries. After grinding, the powdered fuel was dissolved in nitric acid. The fission gas released during the dissolution process was in solution within the fuel matrix or in very small intragranular bubbles, and is thus called “gas in the matrix”. Fig. 9 shows predicted retained fission gas in the matrix at 3 and 12 at% bumup (solid circles) as a function of irradiation temperature compared with the data of Zimmermann [15] for burnups greater than 3% (open circles). Again, the agreement between theory and experiment is quite good. Fig. 10 shows predicted retained gas in pores at 3 and 12 at% burnup (solid circles) as a function of irradiation temperature compared with the data of Zimmermann for burnups greater than 3% (open circles). In contrast to the reasonable agreement between theory and experiment for the total retained gas and the gas retained in the matrix (figs. 8 and 9, respectively), the results for the fission gas retained in pores are consistently below the average of the measured values. The reason for this discrepancy is not clear in that the retained gas in pores plus the retained gas in the matrix should equal the total retained gas (as do the predicted Zimmermann

01

900

1200

1400

1600

TEMPERATUREPC)

Fig. 7. Predicted average bubble size (triangles), the data of Cornell [19] (circles).

compared

with

greater than below - 1400°C. For temperatures 1400°C, the predicted average bubble sizes are in reasonable agreement with the experimental observations. The discrepancy between the predicted (average) and measured bubble size for fuel temperatures below 14OO’C could result from a discrepancy between the measured bubble diameters and the actual average size of the distribution, owing to the presence of small bubbles below the limit of experimental resolution. 3.3. Retained fission gas Fig. 8 shows predicted total retained fission gas at 3 and 12 at% bumup (solid circles) as a function of UO, irradiation temperature, compared with the unrestrained data of Zimmermann [15] (open circles). Zimmerman’s data in fig. 8 are for burnups greater than 3 at% and presumably include the entire burnup range covered by the experiments. The predicted values of total retained fission gas are in reasonable agreement with the data. As shown in fig. 8, above 3% burnup and at temperatures of about 1600 K and higher, there is no influence of bumup on the retained gas concentration. In addition, the amount of retained gas decreases as the irradiation temperature increases; at low temperatures there is a relatively high fission gas retention. In order to evaluate the retained fission gas, Zimmermann ground the irradiated fuel in a ball mill to particle sizes noticeably smaller than 1 pm. The fission gas released during the grinding is called “gas in pores”.

Fig. 9. Predicted retained fission gas in matrix at 3 and 12 at% burnup (solid circles), compared with the data of Zimmermann [15] at burnups above 3% (open circles).

J. Rest / An improved model for fission product behavior 12

T----I-

I

205

I

I

I

C.BURNUP RANGE 2-3% D: " " 3-4% E: I0 4-5% q

. -DATA ---CALCULATIONS

TEMPERATURE

IK)

IOC

Fig. 10. Predicted retained fission gas in pores at 3 and 12 at% burnup (solid circles), compared with the data of Zimmermann [15] at burnups above 3% (open circles).

results). (Note that Zimmermann’s data do not, in general, obey this sum rule.) Given that the predicted results for the total retained gas and the gas retained in the matrix are in reasonable agreement with the data (figs. 8 and 9), the predicted results for fission gas retained in pores should agree with a consistent set of data obtained for these conditions.

) TEMPERATURE iK1

Fig. 12. Predicted swelling rates (dashed curves) as a function of fuel temperature compared with the data of Zimmermann [20] (solid curves) for burnups of 2-391, 3-44;. and 4-5%.

3.4. Fission gas swelling Figs. 11 and 12 show predicted rates of swelling due to retained fission gas as a function of irradiation temperature (dashed lines) compared with the results

z 1

IS-

-DATA ---CALCULATIONS

2 ap 5 2

IO-

: (3 s I: 2

obtained by Zimmermann [20] (solid lines), for UO, fuel irradiated over the burnup ranges of O-l, 1-2, 2-3, 3-4, and 4-5 at%. Zimmermann obtained the swelling results shown in figs. 11 and 12 by comparing the external volume changes of the UO, with calculated values for UO, densification (i.e., irradiation-enhanced sintering of oxide fuel). In general, the predicted values of swelling rate obtained using FASTGRASS theory agree reasonably well with the results obtained by Zimmermann. Fig. 11 shows a very strong temperature dependence of the swelling rate at low bumups. However, with increasing burnup (fig. 12), the swelling rate and the temperature dependence diminish, owing to the saturation of the fission gas swelling rate caused by the enhanced release of fission gas at increased values of fuel bumup (see fig. 5).

5-

4. Comparison of theory and experiment: transient conditions 4.1. Microcracking TEMPERATURE iK1

Fig. 11. Predicted swelling rates (dashed curves) as a function of fuel temperature compared with the data of Zimmermann [20] (solid curves), for bumups of O-l% and l-28.

The ability to determine whether microcracking will occur during a given thermal transient is an important element in the prediction of fuel temperatures and fis-

206

J. Rest / An improved model forfission product hehauror

sion product release [1,2]. Microcracking can reduce the thermal conductivity, F,, of UO, to - 50% of the F, value in dense fuel [13,21]. A change of this magnitude will have a strong effect on calculated temperature profiles. As an example, calculations of the centerline temperature of fuel that had undergone a thermal transient induced by a direct electrical heating (DEH) technique [13,21] vary by as much as 600 K, depending on whether or not microcracking is considered. As a frirst-cut approach to modeling ductile-brittle behavior of oxide fuels, a model based on the work of DiMelfi and Deitrich [22] has been incorporated into the FASTGRASS theory [1,2]. This model estimates the growth rate of a grain-boundary bubble under the driving force of internal pressuration. The volume growth rates due to crack propagation and to diffusional processes are compared to determine the dominant mode of volume swelling. Knowledge of the mechanical properties of UO, is not required. The FASTGRASS model was executed with a fuel behavior code [1,18] for the steady-state irradiation of a fuel rod in the HBR reactor in order to generate the required initial conditions for a transient analysis. The HBR fuel had average heat-generation rates of 22.4 and 17.7 kW/m in the first and second cycles, respectively, and reached a maximum burnup of 3.14 at%. Subsequently, FASTGRASS was executed with a transient temperature code [13,21] for a series of DEH tests. The calculational scenario is as follows (see fig. 13): Based on the DEH test operating conditions, the radial transient temperature profile is calculated and is subsequently used for the calculation of the fission gas response. In turn, the fission gas behavior results are used for the calculation of fuel microcracking. If micro-

*TEST 22 0 PREDICTIONS ATEST 32 A PREDICTIONS

600

01 0

0.2

0.4

0.6

08

IO

FRACTIONAL RADIUS

Fig. 14. Predictions of pore-solid surface area S, as a function of pellet radius for Tests 22 and 32, compared with the data of Gehl [13].

*TEST 24 0 PREDICTIONS ATEST 29 A PREDICTIONS

I

OO

I

0.2

0.4

0.6

0.8

1.0

FRACTIONAL RADIUS

Fig. 15. Predictions of pore-solid surface area .S, as a function of pellet radius for Tests 24 and 29, compared with the data of Gehl [13].

aTEST 33 0 PREDICTIONS ATEST 37 n PREDICTIONS

600

1

%% FRACTIONALRADIUS

Fig. 13. Interrelationship between fuel fracturing (microcracking), temperature scenario, and fission gas bubble response.

Fig. 16. Predictions of pore-solid surface area .S, as a function of pellet radius for Tests 33 and 37, compared with the data of Gehl [13].

J. Rest / An improved model for fission product behavior cracking occurs, the fission-gas release, retention, and swelling results are updated accordingly. Finally, the microcracking results are passed back to the transient temperature calculation where the thermal conductivity expression is modified, and the calculation proceeds to the next time step. Figs. 14-16 show the predictions of the theory for pore-solid surface area per unit volume, S,, as a function of pellet radius for DEH tests 22 and 32, 34 and 29, and 33 and 37, respectively, and measured values [13] of S, for the same tests (the measured pore-solid surface is assumed to be producted mainly by fuel microcracking). In general, considering the complexity (synergistic nature) of the phenomena and the relatively wide range of test conditions, the results of the theory are in remarkably good agreement with the data. For example, there is reasonably good agreement between the theory and data for both test 33 and test 37 which had heating rates of 22 and 234 k/s, respectively (fig. 16). The greatest discrepancy between theory and experiment occurs for test 22 (fig. 14) where the theory underprediets the data obtained near the center of the pellet by more than a factor of 2. The implication of this underprediction of fuel microcracking is that the calculated fuel temperatures will be low with a resultant underprediction of fission product release (see fig. 13). This scenario will be addressed further in the following section. 4.2. Transient fission gas release Fig. 17 shows the predictions of the theory for transient fission gas release for 10 transient DEH tests on irradiated UO, fuel. Nine tests were on fuel irradiated in the HBR reactor and one test was on fuel irradiated under relatively high-power, load-following conditions in the Saxton reactor [13]. The diagonal line in fig. 17 indicates perfect agreement between theory and observation. Except for test 22 (12% gas release measured), the predictions are in reasonable agreement with the measured values. There appears to be relatively uniform scatter of the predicted vs. the measured values on either side of the diagonal line, indicating random rather than biased uncertainties. Random uncertainties are most likely associated with the calculation of fuel temperatures. The complex synergism among radial heat flux, fuel microcracking, and fission gas response has already been discussed in section 4.1 above (see fig. 13). In addition, the actual transient temperature profiles for the DEH tests contain asymmetries due to nonuniform heating associated with the inhomogeneity of the DEH

207

4oL

: 309

L

IO

7.0

30

40

MEASURED FISSIONGAS RELEASE1%) Fig. 17. Predictions

experimentally

of the theory of transient gas release measured values from DEH tests.

vs.

test pellets. These asymmetries have not been quantified and were not included in the analysis of the DEH tests. The theory predicts that 2.3% gas release occurred during DEH Test 22, as compared to the measured value of 13. 1% (fig. 17). As discussed in section 4.1 above, and shown in fig. 14, the theory also underprediets (by more than a factor of two near the pellet center) the amount of pore-solid surface area generated during DEH test 22 by fuel microcracking. Based on the discussion of the synergisms involved in the determination of radial heat flux (represented pictorally in fig. 13), this under-prediction of fuel microcracking should lead to underprediction of fuel temperatures and, hence, to an underprediction of fission gas release. As relatively reasonable predictions for fuel microcracking were made for the other DEH tests (figs. 14-16), the predictions for fuel temperatures and fission gas release in those tests should also be reasonable (if the fission gas response theory is accurate); indeed, they are, as demonstrated in fig. 17. Fig. 18 shows the results of the theory for transient fission gas release from UO, fuel (solid line) as a function of time and temperature for the HI-1 hightemperature transient test (221 performed at Oak Ridge National Laboratory (ORNL), compared with the measured values ( + symbols) for 85Kr obtained from a downstream charcoal trap. Also shown in fig. 18 are the

J. Rest / An improved model for fission product behavior

208

flowing steam atmosphere reasonably well. Fig. 19 shows the results of the theory for 16 ORNL transient fission product release tests compared with the measured values [23-251. The temperatures were ramped to values of 500-1600°C and held for various lengths of time before test termination. The diagonal line in fig. 19 indicates perfect agreement between theory and experiment. In general, the agreement between theory and experiment is reasonable. A range of predicted values is shown for three tests in fig. 19 and correspond to reported uncertainties [26] in the fuel temperatures during the test. The temperature uncertainties in these tests are attributed to the combined heat from rapid cladding oxidation and higher levels of ohmic energy deposition. -5 0

IO

20

30

40 TiME (mid

Fig. 18. Predictions of the theory (solid curve) for noble gas

release as functions of time and temperature, compared with the s5Kr Data in Test HI-l. measured fuel temperatures obtained by thermocouple and optical pyrometer. The ORNL tests were performed with high-burnup LWR fuel (from the HBR reactor) to explore the characteristics of fission product release in a flowing steam atmosphere under a controlled loss-ofcoolant accident (LOCA) over the temperature range 1400 to - 2400°C. Earlier tests [24,25], conducted under similar conditions, were performed at temperatures of 500 to 1600°C. The results shown in fig. 18 indicate that the theory predicts the release of 85Kr during the HI-l test in a

2

ORNL TESTS : HBU A HT-I,HT-Z,HT-3,HT-4

5. Discussion Because of existing uncertainties in both material properties and mechanisms of fission product response, any verified theoretical description of fission product release entails assumptions in these areas. The effect of iodine solubility on fission product behavior was addressed in sections 2.4 and 3.1, above. The effects of irradiation induced gas-atom grain-boundary re-solution on fission gas diffusive-flow rates during steadystate conditions, and the diffusively of fission gas bubbles during transient conditions, are additional aspects of fission product behavior that are currently clouded with uncertainty. The assumptions that have been made in the current theory for these specific mechanisms of fission gas behavior are evaluated in this section and discussed in relation to model verification (sections 3 and 4), uncertainties in the existing data base, and other theoretical descriptions of fission product behavior.

l

5

5.1. Irradiation boundaries

IO

MEASURED FRACTIW

15

20

RELEASE i’%l

Fig. 19. Predictions of the theory for transient gas Release vs. experimentally measured values from the HBU, HT, and HI tests.

induced gas-atom

resolution

on grain

The importance of irradiation induced gas-atom resolution on grain boundaries was quantified first by Speight [27] and more recently by Rest [3] and Dowling et al. [28]. Speight proposed a modified diffusion model incorporating the influence both of a fine dispersion of intragranular gas bubbles and of a coarser distribution of lenticular intergranular bubbles decorating the grain faces. Assuming that re-solution of gas from these bubbles is irradiation-induced, Speight demonstrated that the simple gas release model of diffusion from a sphere proposed by Booth (291 was valid only if a lower effective diffusion coefficient and a non-zero gas atom concentration at the surface of the sphere were assumed to

209

J. Rest / An improved model for fission product behavior

account for the effects of re-solution from intragranular and grain-boundary gas bubbles, respectively. Rest showed that the dependence of fission gas release on the gas atom diffusion coefficient and on the gas atom re-solution rate was very strongly correlated with the strength of irradiation induced re-solution on grain boundaries. Dowling et al. used a finite-difference technique to compute exact solutions to the diffusion equation describing fission gas release from UO, nuclear fuel during steady-state reactor operation. The re-solution of grain boundary bubbles was considered in two alternative ways: One in which the affected atoms residing in grain boundary bubbles are deposited at a single resolution depth X, and the other in which these atoms are deposited uniformly over a depth 2X. The numerical calculations were insensitive to the re-solution model chosen except at low temperatures where, since gas atom diffusion is very slow, release is strongly controlled by re-solution. Under such low-temperature conditions, the grain boundary concentration is predicted to increase linearly with time, whereas at high temperatures, the t’/’ dependence characteristic of diffusional release is exhibited. The rate of accumulation of gas per unit area of boundary can be obtained using Speight’s [27] suggestion that the flux of atoms to the boundary from each grain is given by the flux in the absence of gas-atom re-solution from grain boundary bubbles modified by the fraction (1 - Cx/Co), where Cx and Co are the gas atom concentrations a distance X and a distance >> A from the boundary, respectively. Speight argues that an equilibrium situation is reached when the diffusion flux of gas atoms from the local concentration gradients is exactly balanced by the flux due to re-solution. If instantaneously there are N bubbles per unit area of face, the re-solution flux is Nb/2 into each grain, where b is the gas-atom re-solution rate. If the gas atom concentration is zero at the boundary and Cx a distance X from it, the diffusion flux is approximately D CA/X from Fick’s law. Thus, at equilibrium, when the two fluxes are assumed to be equal, (I-

CA/Co)

= (1 - NbX/ZDC’).

compared with the 1250 K data of Zimmermann (solid line; see fig. 5). A grain size of 2.5 p was used for the calculations shown in fig. 20. The value of b. used in eq. (15) for the gas-atom re-solution rate is 2 x 10-l’ cm3. This value of b, yields a gas atom re-solution rate probability of 8 X 10e5 s-’ for a fission rate B = 4 X 10t2f crnm3 s-‘. This value for the re-solution rate probability agrees reasonably well with the value of 2.7 X 10m4 s-’ measured by Marlowe and Kaznoff [8], and is in the center of the range reported by Turnbull and Cornell [30]. The value of X used in eq. (15) was taken to be h = 5 x 10m9 m. This value of X is within a factor of two of the value (lo-’ m) used by Dowling et al. [28], and is consistent with the value used by A.D. Brailsford (X - 10e9 m) in his study on the effect of gas re-solution on the growth and interlinkage of grain boundary bubbles [31]. The results shown in fig. 20 demonstrate that gasatom re-solution from grain boundary bubbles is a critical factor in determining low-temperature fission gas release; the predicted results are strongly dependent on the backward flux of gas atoms from grain boundary bubbles into the lattice. In addition, good agreement of the theory with the data is only obtained for small values of C,” (Ct g: 1). Physically, these results are interpreted as due to a small value of the penetration depth, X, (i.e., see eq. (20)) rather than to a fundamental difference in re-solution behavior between grainboundary and intragranular bubbles. The value of X used by this author (X = 5 x 10e9 m), and by other authors (28,311 is consistent with this conclusion. The variation in predicted results when different

(20)

Within the rate theory approach used in the present formulation [e.g., see eq. (3)], the rate of gas atom re-solution from grain boundary bubbles, bN/2, is multiplied by the factor Ct = CA/Co of eq. (20) in order to obtain the backward flux of gas atoms into the lattice. Fig. 20 shows predicted results (dashed lines) for fission gas release vs. bumup at various values of C,“,

BURNUP 1%)

Fig. 20. Predictions of the theory for fractional gas release at 1250 K with various values of the irradiation induced re-solution strength on grain boundaries, C,“.

210

J.

Rest

/

An

tmproved

model

for

fission

values of C$ are used for the higher temperature data of Zimmermann (e.g., the 1750 and 2000 K cases) is negligible. The reason for this, as pointed out earlier in the discussion of ref. [28], is that at higher temperatures, diffusional release processes dominate over irradiation induced re-solution effects. 5.2. Mobility of tions

fission

product

behavior

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gas bubbles under transient condi-

It has been shown that the experimentally observed small-bubble diffusivities are many orders of magnitude less than predicted by surface diffusion [l-4,32]. However, with the usual assumption of negligible fission gas solubility, the extent of gas release from transient heating experiments can successfully be modeled only if bubble diffusivities are estimated by assuming surface diffusion to be the rate-controlling process [l-4,32,33]. The model for bubble diffusion used in FASTGRASS theory [l-4] is unique in the sense that it relates the bubble diffusivities to the fuel yield stress, heating rate, and vacancy mobility, as well as to fuel temperatures and bubble radius. Recently, MacInnes and Brearley [34] have proposed a model for the release of fission gas from reactor fuel undergoing transient heating which utilizes an alternative release mechanism based on stationary bubbles and migration of gas by diffusion of single gas atoms. The crux of this model is that gas bubbles which have previously been regarded as infinite sinks for gas atoms can, in fact, accept only a few atoms before thermal emission of atoms dominates flow to the bubble. However, the successful application [34] of this thermal re-solution approach requires the assumption that the initial bubble radius is extremely small (< 0.5 nm), and values for the solution energy which are substantially smaller ‘than have been determined theoretically [35]. In this section, the effect of changing the bubble mobility upon gas release during a thermal transient will be examined. Fig. 21 shows the predictions of FASTGRASS theory for gas release from DEH test 33 as a function of fuel fractional radius, compared with the measured values (open circles). Also shown in fig. 21 are the predicted vs. measured values of total gas release during this test. The dotted curve and solid circle in fig. 21 show the predictions of the theory using gas atom diffusivities based on the observations of Cornell [36]. (The theory of bubble mobility [1,2] based on the assumption that surface diffusion is the rate-controlling process was used for the calculations shown in fig. 21.) The upper curve and open square in fig. 21 show predictions based on gas-atom diffusivities from Matzke

20-

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[37]. Considering that the Matzke gas atom diffusivities are - 100 times greater than those obtained by Cornell, the results in fig. 21 indicate that the results of the theory for DEH transient heating test 33 are relatively insensitive to intragranular single-gas-atom diffusion. The dashed curve and open triangle in fig. 21 show predictions based on the Cornell gas atom diffusivities and a constraint which precluded any biased bubble motion of the fission gas bubbles. The random motion of gas bubbles results in substantially lower predictions than those obtained without the constraint on biased bubble motion. Thus, the results of fig. 21 demonstrate that within the context of FASTGRASS theory, biased motion during transient heating conditions is a key mechanism of fission gas behavior. Fig. 22 shows the predictions of the theory for the same transient heating test conditions used in fig. 21, but with the assumption of 100% thermal re-solution of gas atoms from bubbles during the transient. Thus, gas atom diffusion is assumed to be the only mechanism whereby fission gas can migrate to the grain boundaries. The results shown in fig. 22 demonstrate that without any bubble motion, 100% thermal re-solution and relatively high gas atom mobilities are required in order to

211

J. Rest / An improved model for fission product behavior

1 0

LASER-SAMPLING DATA (KRYPTON) .....O PREDICTIONS USINGMATZKEDIFFUSIVIT#S _’ ,- &PREDICTIONSUSINGCORNELLDlFFUSlVlTlES

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Fig. 22. Transient gas release predictions of the theory, with the assumption of 100% thermal re-solution of intragranular gas and various values of gas atom mobility, compared with the measured values.

obtain agreement between the theory and the data. It should also be noted that the assumption of 100% thermal re-solution results in a prediction of zero microcracking for this test, in constrast to the substantial number of fractured boundaries observed (fuel microcracking was artificially simulated for these calculations in order to provide the correct temperature profiles; see section 4.2. above).

6. Conclusions Based on the relatively extensive comparison between theory and experiment described in this paper, the following conclusions can be made: (a) Fission gas behavior under steady-state conditions is relatively well understood. Good agreement between theory and experiment was obtained for fuel burnups between - 0.5 and 10 at%, and for gas release values between 0.20 and - 100%. (b) The noble gases play a major role in establishing the interconnection of escape routes from the interior to the exterior of nuclear fuel. The volatile fission products

generated within the fuel primarily follow these pathways to release. (c) As the VFPs are known to react with other elements to form compounds, a realistic theory of VFP behavior must include the effects of VFP chemistry. Iodine solubility is an important materials property in that its value can strongly influence the overall iodine release characteristics. induced re-solution on grain (d) Irradiation boundaries can have a substantial impact on fission gas release rates at lower temperatures, but is negligible at higher temperatures. Based on the comparison between theory and experiment, this effect appears to be relatively weak, owing to a small value of the gas atom penetration depth. (e) Microcracking (grain boundary separation) can occur extensively during transient heating conditions and can have a strong effect on fuel temperatures and fission product behavior. (f) The biased motion of relatively small bubbles appears to be a more likely mechanism of fission gas behavior during transient conditions than models based on stationary bubbles and thermal emission processes.

Acknowledgments The author wishes to thank Drs. E.E. Gruber, F.A. Nichols, and SW. Tam for many stimulating discussions. The author would also like to thank S.A. Zawadzki and M. Piasecka for their assistance in performing the calculations.

References [l] J. Rest, Nucl. Technol. 56 (1982) 553. [2] J. Rest, Nucl. Technol. 61 (1983) 33. [3] J. Rest, NUREG/CR-0202, ANL-78-53, tional Laboratory (June 1978).

Argonne

Na-

[4] J. Rest and S.M. Gehl, Nucl. Engrg. Des. 56 (1980) 233. [5] J.A. Turnbull and M.O. Tucker, Philos. Mag. 30 (1972) 47. [6] M.H. Wood and J.R. Matthews, J. Nucl. Mater. 89 (1980) 53-61. [7] J.R. Matthews and M.H. Wood, Nucl. Eng. Des. 56 (1980) 439-443. [8] J. Rest, NUREG/CR-2970, Vol. II. ANL-82-41, Vol. II Argonne National Laboratory (May 1983) p. 60. [9] M.O. Marlowe and A.I. Kaznoff, in: Proc. Internat. Conf. Nuclear Fuel Performance (British Nuclear Energy Society London) 1973. [lo] R.S. Nelson, J. Nucl Mater. 31 (1969) 153. [II] P. Nikolopoulos, S. Nazare and F. Thiimmler, J. Nucl. Mater. 71 (1977) 89-94.

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1211 PI

~231 v41 PI

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