19.

20. 21. 22.

23.

24. 25. 26.

W. Poenita, "Fast neutron capture cross section measurements, evaluations, and model calculations of fission product nuclei," in: Proceedings of Special Meeting on Neutron Cross Sections of Fission Product Nuclei, Bologna, Italy (1979), p. 85. S. Joly et al., Nucl. Sci. Eng., 70, 53 (1979). G. Reffo et al., Nucl. Sci. Eng., 80, 630 (1982). E. Cornelis, "Average capture cross section of fission product nuclei i~176176176 in: Proceedings of an International Conference on Nuclear Data for Science and Technology, Antwerp, Holland (1982), p. 222. M. Mizumoto et al., "Neutron radiative capture and transmission measurement of 1~ and l~ in: Proceedings of an International Conference on Nuclear Data for Science and Technology, Antwerp (1982), p. 226. W. Poenitz, "Fast neutron capture cross section measurement with ANL LIST," in: Proceedings of Special Meeting on Fast Neutron Capture, Argonne, US (1982), p. iii. Nucl. Sci. Eng., 82, 400 (1982). M. Mizumoto et al., J. Nucl. Sci. Tech., 16, 711 (1979).

EFFECT OF FUEL AND CLADDING ECCENTRICITY, CLADDING OVALITY, AND FUEL-PELLET CHIPPING ON THE TEMPERATURE FIELD IN A FUEL ELEMENT A. S. Shcheglov

UDC 621.039.54

In most computer codes the cross section of the fuel element is treated by using the method of solid coaxial cylinders. In calculations of the thermal conductivity of the fuelcladding system, expressions are used which are valid for flat, rough surfaces which are in contact or which have a gap between them. However, axial symmetry is not characteristic of an actual fuel element because of the effect of the waviness and ovality of the surfaces, the eccentricity of the fuel and cladding, cracking, chipping of the fuel pellet, entry of a minute fuel particle into the gap, axial asymmetry of the volume heat removal from the outer surface of the cladding. The temperature distribution in a fuel element can be found with consideration for these factors only if two-dimensional (in cross section) methods of fuel-element design are used [i]. In this paper we present the results of calculations of the temperature field in a fuel element with eccentricity of the fuel and cladding, chipping of the fuel pellet, ovality of the cladding, and axial symmetry of volume heat release in the fuel with consideration for thermomechanical deformations. For these calculations we used a code developed by the finite-element method [2]. Thermal expansion was calculated for the case of plane deformation of a cylinder with free end faces. We did not take into account axial heat leakage in the temperature calculations, but we did allow for radiant heat transfer within the central hole (if the emissivity of the fuel is 0.7, allowing for radiant heat transfer leads to a decrease of 20 K in the peak temperatures inside the central hole in the relevant case of fuel and cladding eccentricity). We used the following subintervals: ~/28 for the angle, and (Rf. o - Rf.i)/14 for the radius of the fuel pellet; and the corresponding number of nodes in the mesh of triangular finite elements was 522 (by symmetry, we examined half the cross section of the fuel ele ment). Here and subsequently, R is the radius, subscripts "f" and "c" indicate association with the fuel and cladding, and subscripts "i" and "o" indicate association with the inner and outer surfaces. Testing was carried out by comparing the reuslts of the numerical cal -~ culations with known analytical results. On a ES 1055 computer, the computing time for a cycle of solutions of the thermal problem (with iterations over the temperature dependence of the thermal conductivity If of t h e Translated from Atomnaya Energiya, Vol. 67, No. 3, pp. 204-207, September, 1989. Original article submitted March 24, 1988. 682

0038-531X/89/6703-0682512.50

9

1990 Plenum Publishing Corporation

Y Fig. I. Eccentricity of fuel and cladding; ~ and r are the current angle and current radius.

Fig. 2. Chipping of fuel pellet: AR is the depth of the chip, a n d ~ j is the chipping angle. fuel) and the problem of the corresponding thermal expansions (in each problem the initial temperature level was set at 300 K) was 6-20 min. In the calculation for the cases of cladding ovality or fuel and cladding eccentricity, during a cycle we adjusted the cladding geometry or the eccentricity in order to fulfill the condition of contact between fuel and cladding. Let us consider the conditions for anchoring the fuel and cladding which are necessary to avoid movement and rotation of them as a whole, and hence a singularity in the matrix of coefficients. By symmetry it follows that the mesh nodes lying on the axis of symmetry are anchored in the direction perpendicular to this axis. Let us assume that movement of a point on the inside surface of the cladding with ~ = ~/2 in the direction of the axis of symmetry is forbidden. Henceofrth, ~ is an angle in the plane of the cross section (Fig. i). The choice of the corresponding anchoring of the fuel is not physically obvious and is not unique. We will assume that movement of a point on the inside surface of the fuel with ~ = ~/2 in the direction of the axis of symmetry is forbidden. Let us consider the fuel element of a water-moderated, water-cooled reactor: Rc. o = 4.6 mm, Rc. i = 3.9 mm, Rf. o = 3.8 mm, and Rf. i = 1.15 ~n at an initial temperature of 300 K. Let us assume that if (W/m.K) depends on the temperature T (K): If = 4.82.10s/(351 + T) + 2.434.10 -14 T4; the thermal expansion coefficient of the cladding is ~c = 5-6"10-6 (l/K); the thermal expansion coefficient of the fuel is ~f = 5.544.10 -6 to 3.5.10 -s T; the thermal conductivity of the cladding is 20.5 W/m.K; the coolant temperature is 600 K; the coefficient of heat transfer from cladding to coolant is 3.5.104 W/m2-K; the gaseous medium under the cladding has a thermal conductivity of 0.28 W/m.K; the pressure is greater than 1 MPa: B + g = 8 ~m [3], where B is the effective height of the rough layer of the fuel and cladding, and g is the sum of the lengths of the temperature jumps at the surfaces of the fuel and cladding. We will assume that the specific heat release corresponding to the initial fuel density is qvl = l'10s W/m3; in this case the linear thermal load is qs = 412 W/cm. We con-

683

TABLE 1. Calculated Values of Temperature and Motions of Points of the Fuel Element for Zero and a Specified Eccentricity of the Fuel and Cladding

Parameter

0for all t961 1073 692,4 640,6 53 8,2 9,5

Tf i,K Tf'o,K Tc~i,K T.c.o,K ~f.o, ~m ~.i,~ m c.o, ~m

e=69

8=

(p=o

q~= g/2

t 729 768 708 647,6 39 8,4 9,8

i 923 1065 690 639,7 5i 8,0 9,4

2099 1322 68i 635,6 63 7,9 9,3

TABLE 2. Calculated Values of the Temperature of Points of the Fuel Element in Case of Chipping of a Fuel Pellet for ~i

=

~/14

I ~= 0 Tem~.,K n

i962 i072

Tf.i Tf.o

692,4 640,6

Tc.i Tc.0

~=5u17 ~=25~/28 ~=27~/28 i976 t079

t971 1073 693,1 640,9

692,2 640,3

t977 t042

m=

u

1978

684,3 637,0

(Js) 676,1 634,8

TABLE 3. Calculated Values of Temperature and Motions of Points of the Fuel Element for a Cladding Ovality W = 61.5 im Parameter

Tf i,K .... Tf[o,K Tci,K T.'o,K

~}o, ~m &c[i, pm

sider the initial period of operation, not take creep strain into account.

~= 0

cp= g/2

1952 1261 674 633,1 54,1 8,75

i853 781 72t 652 46,5 7,4

assume that there is no cracking of the fuel, and do

FUEL AND CLADDING ECCENTRICITY e = 69 pm This value corresponds to the case in which after thermal expansion the fuel touches the cladding at the point ~ = 0 (we may assume that contact between the fuel and cladding at the point ~ = 0 is a condition of fuel anchoring). Table 1 gives the calculated values of the temperature at points of the fuel element and the corresponding values of the temperature in a calculation using the model of solid coaxial cylinders (i.e., when e = 0). The average fuel temperature is equal to 1514 K for e = 69 Bm and to 1545 K at e = 0. Table 1 gives the calculated motions A of points of the fuel element. CHIPPING OF THE FUEL PELLET Let us specify chipping as illustrated in Fig. 2. Let us consider the case ~l = ~/14 for AR = 0.19 mm; then qs = 410.5 W/cm. Table 2 gives the results of our calculation of the

684

TABLE 4. Calculated Values of the Temperature of Points of the Fuel Element for e = 0, qv = qv2 f(r)f(9) Temp,, K

(p=O

1980 t093 698 643,2

Tf. i Tf.o Tc.i Tc. o

=

~/2

i9J9

|076 692 640,6

t856 1059 686 638

TABLE 5. Calculated Values of the Temperature of Points of the Fuel Element for e = 68 ~m, qv = qv= f(r)f(~) Temp., K Tf. i Tf.o Tc. i Tc. 0

(P=O

1738 775 715 65t

1873 t069 690 639

1985 t29t 675 633

temperatures of points of the fuel element. The calculated average fuel temperature is equal to 1554 K, i.e., it increases by i0 K in comparison with the case in which chipping is absent. For ~ i = ~/7 and the same AR, we obtain an average fuel temperature at 1568 K. CLADDING OVALITY Let us specify the ovality of the inner and outer surfaces of the cladding as Rc. o = R c + W c o s (2~), where W = 0.5 (Rc.ma x - Rc.min). Let us adopt the value W = 61.5 ~m corresponding case in which, after thermal expansion, the fuel touches the cladding at the point = ~/2. Table 3 gives the results of the calculation of temperature and motions of points of the fuel element. The average fuel temperature is 1497 K; when W = 50 ~m, it is equal to 1515 K. Thus, we may conclude that if the average gap between the fuel and the cladding is preserved, the axial asymmetry of the fuel-cladding system (eccentricity, ovality, and waviness of the cladding leads to a slight decrease in average fuel temperature. This decrease is due to the fact that because of the inconstant heat flux across the gap, more than half the heat is transferred through the portion of the gap with higher thermal conductivity than in the axisymmetric case. In fuel pellets without a central hole,* this leads to a decrease in maximum temperature. In pellets with a central hole, the maximum fuel temperatures increases; this increase may lead, for example, to an increase in the yield of gaseous fission products. If the average gap between fuel and cladding increases (through chipping of the pellet), the average and maximum fuel temperatures rise. In calculating the temperature field in the fuel element, one must take into account the change in the volume heat release in the fuel over an angle ~ and over the current radius r (see Fig. i) [4]. CHANGE IN qv OVER r WHEN e = 0 Let us assume that qv = qv2 "f(r), where f(r) = 0.8 + 0.2 Rf.i)/(Rf. o - Rf.i)] when qv= = 1-057"109 W/m3 (corresponding of the internal layers in the fuel leads to a decrease in the The calculated average fuel temperature is 1535 K (i.e., i0 K qvl = const); Tf. i = 1921 K, Tf. o = 1077 K, and Af. o = 52 ~m.

cos 8; 8(r) = (~/2).[i - (r to qs = 412 W/cm). Blocking temperature field in the fuel: lower than in the case qv =

*In a pellet without a central hole, if qs = 412 W/cm and e = 0 the calculated values of the maximum and average fuel temperatures are 2244 and 1609 K, respectively; if E = 64 ~m these values are 2225 and 1575 K.

685

TABLE 6. Calculated Values of the Temperature of Points of the Fuel Element for g = 71 ~m, qv = qv2 f(r) (i - 0.i cos~ Temp., K Tf.i Tf.o Tc.i Tc.o

r

i630 758 702 644,9

(p = • / 2

~886 1068 691 639,9

CHANGE IN qv OVER ~ W H E N

2i3i t370 686 637,8

e = 0

Let us assume qv = qv1(~), where F(@) = 1 + 0.i cos ~ . Here the temperature of the inside surface of the fuel is 2023 K if ~ = 0, and 1895 K of ~ = ~. The temperature of the outside surface of the fuel is 1088 K of ~ = 0 and 1056 K if ~ = ~. The average fuel temperature is 1544 K. CHANGE IN qv OVER r AND @ WHEN e = 0 Let us assume qv = qv2 f(r)f(@)" Table 4 gives the results of the calculation of the temperatures of points of the fuel element. The average fuel temperature is equal to 1533 K, and Af. o = 55 ~m if ~ = 0 and 50 pm if @ = ~. CASE qv = qv2f(r) f(@) FOR ~ = 68 pm Let us consider this case, in which we have the maximum possible eccentricity of the fuel and cladding, corresponding to the case of contact between the fuel and cladding with = 0. In this case the average fuel temperature is 1493 K; Table 5 gives the results of the calculation of temperatures of points of the fuel element. Here Af. o = 41 pm when @ = 0 and 59 ~m for ~ = ~. CASE qv = qv2f(r)(l - 0 . 1 c o s ~ )

FOR ~ = 71 Bm

Let us consider this case, in which we have the maximum possible eccentricity. The average fuel temperature is 1511 K; Af. o = 37 pm if ~ = 0 and 66 pm if ~ = ~. Table 6 gives the results of a calculation of the temperatures at point of the fuel element. We can see that this case is most dangerous for the high level of its maximum temperature (210 K higher than in the case e = 0, qv = qv2 f(r), i.e., in a calculation using solid coaxial cylinders) and apparently for its elevated yield of gaseous fission products. Based on the results presented here, we may conclude that allowance must be made for the axial asymmetry of the fuel-cladding system within the framework of two-dimensional calculations (in the cross-sectional plane of the fuel element). In particular, allowance for nonaxisymmetric factors is necessary in the processing of experimental results from fuel elements (temperature distribution in the fuel element, yield of gaseous fission products, redistribution of the heat flux through the cladding) and in the analysis of uncertainty factors in fuel-element calculations. Allowance for the effect of fuel and cladding eccentricity on the temperature distribution in the cladding is especially important in calculations of the behavior of fuel elements under accident conditions involving overheating and swelling of the cladding. Taking fuel cracking into consideration within the framework of the two-dimensional model thus is all the more important [i]. LITERATURE CITED io 2.

3. 4.

686

R. Williford, D. Lanning, and C. Mohr, "A model for the effective thermal and mechanical properties of cracked UO 2 pellets," Nucl. Tech., 56, 340-350 (1982). O. K. Zenkevich, The Finite-Element Method in Engineering [in Russian], Mir, Moscow (1975). A. S. Shcheglov, "A method of calculating the thermal conductivity of a contact between the fuel and cladding of a fuel element," At. Energ. Rubezhom, No. 8, 9-15 (1988). N. N. Ponomarev-Stepnoi, Heat Release in a Nuclear Reactor [in Russian], Energoatomizdat (1985), p. 160.