Volume expansion by IL growth with fuel and Al matrix consumption

1. Introduction

3.1. Fuel meat swelling

3.1.3. Modelling of fuel meat swelling

3.1.3.2. Volume expansion by IL growth with fuel and Al matrix consumption

IL formation and its effect on fuel meat swelling during the irradiation period are distinguishable to the fuel swelling of the foil in the monolithic type. Thus, IL growth requires more complicated modeling in the fuel meat swelling. IL growth induces the volume consumption of U-Mo fuel and Al matrix. Thus, in the fuel meat swelling model, U-Mo fuel volume decreasing by IL growth and increasing by the fuel swelling should be dealt with simultaneously.

IL growth correlations in terms of IL thickness formed during irradiation as well as the corresponding IL volume are available in Eqs from (4) - (16). When IL volume is calculated, the consumed volumes of U-Mo fuel and Al matrix can be obtained by the mass balance; the consumed U-Mo fuel mass is equal to be the U-Mo fuel mass in the newly formed IL. For Al matrix, the same principle is employed.

The extent of volume expansion due to IL formation as well as volume consumption of fuel and Al matrix are depending on the compound stoichiometry of IL. Based on the available literature, IL has a composition of U(Mo)Alx where x has a range from 2 to 4 [40]. IL composition during irradiation is comparable to the composition of U-Al compounds. Only three compounds in the U-Al system were reported during in-pile irradiation; UAl2 formed from U-Al alloy melting, UAl3 and UAl4 formed by peritectoid reactions. Although they co-exist in UAlx fuel, the dominant phases were UAl3 andUAl4

[81][82]. During irradiation, all of these U-Al intermetallic fuels are amorphous. It was also reported that IL had a two- or three- layered structure depending on annealing time [83] from the out-of-pile annealing test for U-10Mo/Al fuel samples as shown in Fig. 13. The ratio of Al to (U+Mo) (x_{IL) for the} phase for the thickest layer (L1) was 3.

However, it is known that the interdiffusion between U-Mo and Al was enhanced by the fission [40]. In addition, for the irradiation period that is much longer than the out-of-pile test time, the amount of Al atoms diffused into the U-Mo fuel is much larger with fission-enhanced interdiffusion. Thus, x_IL can be increased in the overall IL. Indeed, the ultimate interaction product in UAlx/Al dispersion fuels was UAl4 [84]. Thus, for the modeling of fuel meat swelling in the irradiation condition, x_IL was assumed to be 4.

45

Pore

U rich phase

Fig. 13 Interaction layers in U-10Mo/Al fuel in the original contact plane of the diffusion couple for 5 hr and 550^oC.

The consumed fuel and Al matrix volume can be calculated for the given IL volume formed during irradiation by using densities for U-Mo fuel, Al matrix, and IL as follows:

c IL f

f IL

f IL

V M V

M

 



⁽³⁰⁾

c IL Al

Al IL IL

Al IL

V x M V

M

 



⁽³¹⁾

where  is the density, V is the volume in cm³, M is the molecular mass. Superscript c stands for the consumption, and subscripts represent the material; f for fuel and others for IL and Al.

Interaction layer volume (V_IL) that is given by using Eq. (11) includes a portion of volume expansion by swelling by U fission in the IL, since it is converted from the IL thickness which is predicted by the in-pile correlation. Thus, the contribution of volume expansion by U-fission in the IL should be excluded in the calculation of U-Mo fuel and Al matrix consumption, because these consumption reactions are not affected by fission-induced swelling in the IL. To get rid of this portion, it was assumed that fission induced swelling in the IL was approximately the same as that of UAlx. This assumption could be justifiable because Mo content in the fuel does not significantly alter the fission induced swelling of UAlx. The swelling rate for UAlx is available in the literature [85] as follows:

IL d 0 IL

V 6.4 F

V

 

  

  (32)

46

where F_d^IL is the fission density in 10²¹ fission/cm³-IL. It is to be noted that F_d^IL is the fission density of IL, which is the amount of fissioned ²³⁵U atom in a unit volume of IL. F_d^IL is calculated from the fission density of the U-Mo at the same time, which is shown as:

IL

d d

F   F

(33)

where



is the atomic ratio of fissile uranium atom number density in the IL to that in the fuel, calculated from:

235

IL IL

U U

f

U f IL Mo

N M 1

N M (1 wt )

  

  ⁽³⁴⁾

where M235U is the atomic weight of U²³⁵, and wt_Mo is the Mo weight fraction in the fuel. The ratio



is defined by the number density of U in the IL and fuel. For typical IL of U(Mo)Al4, with 10 wt%

Mo,



is 0.27.

Then, for the consumed volume of U-Mo fuel, the correction for Eqs. (30) and (31) by subtracting the portion of the volume expansion by fission induced swelling in the IL should be done as follows:

c IL f

f IL

f IL 0 IL

M 1 V

V V [1 ]

M 100 V

 

 

     (35)

and for Al matrix,

c IL Al

Al IL IL

Al IL 0 IL

M 1 V

V x V [1 ]

M 100 V

 

 

     (36)

It is to be noted that 1/100 is multiplied to

0 IL

V V

 

 

  which is calculated from Eq. (32) since the swelling quantity is given as a percentage, not a fraction.

After consumption of U-Mo fuel by IL growth, the fission-induced fuel particle swelling is considered by using the correlations described in Eqs. (1) and (29). Then, the time-dependent U-Mo fuel particle volume with considering the consumption as well as the fission-induced swelling is calculated as follows:

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0 c

f f f

0 f

1 ΔV

V (V -V )[1+ ]

100 V

 

  

  (37)

where V^c_f is given by Eq. (35), and V⁰_f is the as-fabricated U-Mo fuel particle volume, which is calculated by:

0 03

f

V 4 r 3

 

(38)

where r0 is the as-fabricated U-Mo fuel particle radius. Similarly, the time-dependent Al volume per fuel particle is given as:

0 c

Al Al Al

V  V  V

(39)

where V^c_f is given by Eq. (36), and V⁰_Al is the as-fabricated Al matrix volume per U-Mo particle, which can be calculated by using the as-fabricated fuel volume fraction as follows:

0 0

Al 0 f

f

V 1 1 V

v

 

  

  ⁽⁴⁰⁾

where V_f⁰ is given in Eq. (38) and v⁰_f is the as-fabricated fuel volume fraction in the meat. v⁰_f can be calculated if the U loading (or the uranium density in the meat) and the fuel alloy density are known.

A simple formulation to obtain as-fabricated fuel volume fraction is expressed by the definition as follows:

0 U

f

f Mo

L 1

v = ρ (1-wt )

⁽⁴¹⁾

where L_U is the U-loading in the unit fuel meat volume in gU/cm³.