Basis for the modelling

Following this interpretation, the model proposed is based on the local description of the fuel in two zones, the original matrix, where the Xenon diffusion coefficient is negligible (classic diffusion), the restructured zone characterized by a higher Xenon diffusion coefficient (preferential diffusion).

Before the grain subdivision, Xenon is located in interstitial position or tri-vacancies sites.

It can agglomerate to form nanometric bubbles (5-10 nm). The lattice energy increases with defects accumulation. An energy threshold is defined above which restructuring can operate in order to relax the internal stresses. Two bubble populations are to be considered: the nano-bubbles in the original matrix, able to be completely destroyed by fission spikes, and the nano-bubbles on dislocation surfaces able to grow, thanks to the preferential diffusion in this zones.

The model accounts only for the second population. The first population is integrated in the gases spread in the original matrix. The average bubble radius is then growing with the expansion of the dislocation surfaces. An apparent diffusion coefficient is defined as :

where D^vo, is the Xenon volumic diffusion coefficient in the original crystal, D^ is the Xenon diffusion coefficient in diffusion "short-cuts", f^s is the fraction of Xenon present in the restructured surfaces.

3 - DESCRIPTION OF THE MODEL

3.1 - Diffusion coefficient used in the mddel

3.1.1 - Vacancy diffusion coefficient in the U02 matrix

The expression used [18] is very close to the formulation developped for the Uranium atom autodiffusion coefficient. The displacement of an atom A in the sub-lattice A create a vacancy site. Therefore, the migration of the defect is indeed equivalent to the autodiffusion of the atomic specie. The expression retained for the vacancy diffusion coefficient is then:

D^v = 1.09.1<r*.exp'' 52900]

~ ~ ~ ~ (1)

with D^v in m²/s, and the temperature T in Kelvin. Nevertheless, if the thermal activation is preponderant in the hottest part of the fuel pellet, it must be different in the rim.

Unfortunately, no athermal term or mixed term are clearly available in the open literature.

3.1.2 - Xenon volumic diffusion coefficient

The Xenon diffusion coefficient in the fuel matrix under irradiadon can be described with three terms representing the temperature contribution, the activation by fission fragments collidings (athermal term) and the influence of the vacancies creation under irradiation (mixed term) [19],[20] and [21] :

with the temperature contribution :

D,, = 7.6.10-'°.exP[-^p] (2)

the athermal contribution :

ath

and finally the mixed term for vacancies :

D^ath=6.10-⁴⁰.F (3)

_»!ZOj

^C4)

D^, D^a[h and D,^ are in m²/s, F is the local instantaneous number of fissions in the fuel per volume unit

3.1.3 - Xenon diffusion coefficient in the perturbed fuel zone

No works are presently available to evaluate the diffusion coefficient in the restructured zone. Therefore the assumption made is that this coeffient is the previous coefficient D^xc multiplied by a fitting factor A,:

Where D^Xes Is the diffusion coefficient in the perturbed zone (m /s) and \ is fitted to 100.

3.2 - Detailed steps in the resolution 3.2.7 - Evaluation of the lattice energy

The average lattice energy is assumed to evolve linearly with the local burnup. It is obviously strongly related to the local temperature, depending on the capacity of the defects to be recovered. The lattice energy is formulated as follow :

E ' = C t e * B U * f ( T ) (6) Where Cte is a constant, presently fitted at 10⁴ and f(T) represents the ratio between the steady state bubble volumic concentration at a temperature T and the same at 573.15 K.

This reference temperature is representative of the average cladding temperature at which the defects can be considered quasi motionless. So, f(T) can be formulated :

573.15

5~ ^Xe

The different terms in this expression are detailed further in the presentation when the evaluation of the bubble volumic concentration versus time is needed.

3.2.2 - Evaluation of the subdivided area

Assuming the fuel partly restructured with an intact zone and a zone where grains are subdivided, the global energy per volume unit can be written as follow :

E = E *^L + E * ( 1 - ) ( 8 )

f

v v

_{vo vo}

where V⁰ and V^r are respectively the total volume (stressed zone with an energy E* -t-restructured zone with an energy E^f) and the volume restructured where stresses are relaxed. As far as the lattice energy E is below the threshold energy Es (for an initiation of the restructuration), then E = E*, the lattice accomodate the distortions generated by the fission spikes. E is a fonction of the temperature and cumulated burnup. When E is higher than E^s, the fuel fraction restructured is evaluated as follow :

Vv0 (9)

where a^s is the average subgrain size after relaxation and S the surface by volume unit concerned by the relaxation (subdivided area). The energy threshold, E^s is experimentally fitted (a value of 6.0 10⁸ J.m³ have been fixed). Using the standard expression of the energy formation of a spherical cavity :

E^{c f}= 2 * S * y (10)

with Y as the surface tension parameter taken equal to 0,626 J.rn². One can then estimate the energy variation related to the formation of the S surface, writting the difference between the zone energy before and after the subdivision:

then

_P * n —E-r 11 — ~—i / — z. y.cio ^r > ( — ? -v H?

VVA I

(11)

after integration it comes :

S = —, 1 — Cte a^s T

(12)

(13)

if S=0, E*=E^S and Cte = 2y + — ( E^s- E^f) ; one can then establish the following expression for E > E^s:

O — 1 —

J-(E^s-E^f)

(14)

The lattice energy after relaxation must be lower than the threshold energy E^s. Nevertheless there is presently a lack in the data provided, for an accurate evaluation of this energy level. Then, in first approximation, the problem can be simplified assuming E^f = E^s The expression of S becomes (in m²/ m³):

(15)

3

*>

2T

9v4-— ^- fF* — F ^I *\"^ S/

The surface fraction subdivided f. is deducted as follow

(16) as the average subgrain size is assumed about 7.5 10"8 m

3.2.3 - Determination of the bubble volumic concentration

The evolution of the volumetric concentration g of the gas bubble, hi the rim zone, can be written with three components : the bubble creation term T,, the bubble thermal annealing term T² and the resolution term due to fission spikes T³:

dg

dt = T,-(T²+T³) ⁽¹⁷⁾

Presence of dislocations in the matrix favours Xenon precipitation in trapping sites. Then Tj can be formulated as follow :

T^z^IVC^ (18) where

zd number of free sites next to the dislocation (3 to 6), p^d dislocation density in fresh fuel before irradiation,

D^xe Xenon volumic diffusion coefficient in the matrix (m²/s),

Cx^b volumic concentration of intragranular gas.atoms in the lattice (atom/m³)

The second term limits the number of germs because of the partial temperature annealing of the defects on which atoms can be trapped. This term becomes very significative at high temperatures but is negligible in the rim where temperatures are lower.

T² = k^e.z^nv.D^v.g (19)

k^e constant for defects thermal elimination rate (fitting factor)

z^nv volumic concentration for vacancy clusters in the matrix (m~³) (10²⁵ m"³) D^v vacancies average diffusion coefficient (mVs)

g volumic concentration of gas bubbles (bulles/m³)

The last term is related to the bubble average radius. Its expression is based on the TURNBULL theory which consider a complete resolution of the bubbles with a radius below the minimum value required to be stable. This resolution is activated by the fission spikes when the fission fragment interacts with a resident bubble. If b is the probability for a bubble to be destroyed, the third term is related to the fission density and to the number of bubbles :

T³ = b.F.g (20)

With this three terms, the detailed expression to calculate the time evolution of the bubble concentration within the fuel is:

^ = z^d.p^d.D^x,C^hxbe -(k^e.z^nv.D^v +b.F).g (21)

dt

'—s——' '——s;——

If At is the time step, the g concentration at instant t is then : g =

g-Bt

(22)

3.2.4 - Determination of the Xenon volume fraction in the restructured volume

The following expression contains a generation term, characteristic of the gas flow drained toward this fuel zone, reduced of a resolution term corresponding to the macroscopic destruction of the smallest gas bubbles.

j/-i

- ⁽²³⁾

where

G^ Xenon volumic concentration in the restructured zones (moles/m³)

Cxbc volumic concentration of intragranular gas atoms in the lattice (moles/m3) D^xe Xenon volumic diffusion coefficient in the matrix (m²/s),

d average distance of resolution for gas atoms in the lattice (m) b probability of resolution for the bubbles

F fission rate (fiss/cm3/s)

then, the Xenon concentration at instant t is obtained by :

.{l - exp(-B².At)} (24)

A <; "n p

hb

remark : if t -> ~, G ->

B² d.b.F

The mole fraction f present in the short cuts is then :

(25) 3.2.5 - Determination of the Xenon mole concentration in bubbles

As for the bubble concentration, the time evolution of the Xenon concentration in the bubbles can be written as the combination of three components : the bubble alimentation by the two diffusion processes described above (bulk diffusion and short cutsXT,), the gas contained in the bubbles created during the time step At (T²) and a resolution term (T³), macroscopic (TURNBULL) and microscopic (NELSON).

b T " (26)

dt

where C\e the Xenon atom concentration within the bubbles (moles/m3)

For the formulation of the. first term one can refer to OLANDER [23]. The retained expression is :

T¹ = 4icJLD^iB.(CS-C^bXe).g (27)

where

R average bubble radius (m)

D^app apparent Xenon diffusion coefficient (m²/s)

C¹^ total intragranular gas concentration (lattice+bubbles) (moles/ m³)

Cx^e gas concentration in the bubbles (moles/ m³) g concentration of bubbles (bub/ m3)

If N⁰ is the minimum number of atoms for a bubble to be stable (a value of 2 is retained), and (Ag.At) the number of bubbles created during the time step At, the population of Xenon is increased of the following term :

(28) Avogadro number = 6,022045 10²³

The last term T³ is the resolution term. Depending upon its size, a bubble can be totally or partly destroyed. In the first case, the fission fragment collides with a bubble and the gas atoms are spread in the surrounding matrix. As given above this term is :

r = b.F.g (29) This formulation is applied if R < R^ where R^ is the maximum critical size of the bubble

(R^bc fitted to 1.5 10 m). Above this size, the microscopic NELSON approach is more representative with the resolution of gas atoms in the bubble when a fission fragment collides with it. The bubble content can then decrease as collidings proceed, depending upon the alimentation term. The expression of the NELSON term is [23] :

T³=b^re.Ti^re.m (30)

where b^re is the probability per time unit for a gas atom in a bubble to get enough impact energy to reenter the matrix :

n"^{x 4 4}

(31) E^³* creation energy of the fission fragments (J)

E¹™" minimum energy for a Xenon atom for an effective resolution (J) Z atomic number for Xenon = 54

e elementary electronic charge = 1.60219 10-19 Coulomb H^ff. fission fragments free path (m)

and T|^re , the resolution yield:

k Boltzman constant = 1.3806624 10'²³ T gas temperature (K)

7 surface tension for the bubble (J/m²) R average bubble radius (m)

Finally, the evolution of the gas concentration in the bubble is ruled by the following equation :

—— = 4TC.R.D .(C^Xe — C^Xe).g + N^Q ——— — T³ (33)

dt X i f R < Rb c T3=T3'

i f R > R^{b c} T³=T³"

The gas concentration in bubbles at instant t is (in moles/m³):

A L ^ , .> (34)

A3

with:

A³=47i.R.D^app.^g(_^At+T³ Ag.At C^{lOl . VT}

-„, Xe + N0 ————

3.2.6 - Average bubble radius

Solving the reduced Van Der Waals equation describing the thermodynamic state of the gases in the bubbles, one can evaluate the average bubble radius :

p.(V^eff-mB) = m.k.T (35)

or

y I i ^..N-..^/. -T<»..•.>. ^ _ _ _ (36)

R where

P^ex[ is the hydrostatic pressure (Pa) y surface tension for the bubble (J/m²) a ratio coefficient

G shearing coefficient (Pa)

b lattice parameter (for UO², b = 5.475 10'¹⁰ m) B the atomic volume of the Xenon atom = 85.10"³⁰ m³ m number of Xenon atoms per bubble

2v G.b The tension — at the bubble surface contributes to its stability. The second term a.——

R R represents the distortion generated in the cristal lattice by the bubbles formation. The I/a is proportionnal to the number of vacancies flowing from the lattice towards the bubble.

These vacancies participate to the bubble growing and a diminution of the internal

pressure, a must be proportionnal to ————. Nevertheless, because we have no D^Xe + D^v

realistic data on the vacancies participation to the bubble stabilisation, a is taken presently equal to zero.

Dans le document Advances in Fuel Pellet Technology for Improved Performance at High Burnup | IAEA (Page 186-194)