FIRST RESULTS ON THE EFFECT OF FUEL-CLADDING ECCENTRICITY I. Panka, A. Keresztúri Hungarian Academy of Sciences KFKI Atomic Energy Research Institute, Reactor Analysis Department H-155 Budapest, P.O.BOX 49, Hungary panka@sunserv.kfki.hu ABSTRACT In the traditional fuel-behaviour or hot channel calculations it is assumed that the fuel pellet is centered within the clad. However, in the real life the pellet could be positioned asymmetrically within the clad, which leads to asymmetric gap conductance and therefore it is worthwhile to investigate the magnitude of the effect on maximal fuel temperature and surface heat flux. In this paper our first experiences are presented on this topic. 1. INTRODUCTION Traditionally, the effect of fuel-cladding eccentricity is neglected or it is not taken into account properly in the fuel-behaviour or hot channel calculation. Therefore we have decided to deal with this problem. It must be mentioned, that only the radial effects are considered, the axial heat conduction is neglected in this investigations. First, we were intended to solve the problem with available traditional codes, like FRAPT-6, FRAPTRAN, but these codes were not able to calculate the problem. Another attempt was to use an older code TOODEE, but there were some problems in adopting this code. Finally we have decided to elaborate a new code. In the first part of the papers we show briefly this code. It follows its verification and some results will be presented.. THE CODE First of all we define the problem to be solved. In Fig. 1, the eccentric geometry can be seen (everything in cm), where R 1 is the clad inner radius, R is the radius of the UO pellet, d is the fuel pellet displacement (the measure of eccentricity, d=0.007 cm, which is considered as a maximal conservative value), R UO is the distance of UO surface from the center of new coordinate system, dgap is the fuel-clad radial gap.
R =0.3785 dgap alfa R UO R 1 = 0.3865 d = 0.007 dclad = 0.0685 Figure 1: The eccentric fuel-cladding geometry In this case, the azimuthally depending gap size can be calculated as dgap = R - R = R - dcos( α) R d sin ( ). (1) 1 UO 1 θ If the nominal gap size is dgap0 = (R1 - R ) = 0.008 << R UO, an it is true in our case, then the size of the gap can be approximated by dgap = dgap0 - d * cos( θ ) () A hypothetical (d=0.007 cm) angular depended gap size can be seen in Fig..
0.016 0.014 dgap 0.01 0.01 dgap (cm) 0.008 0.006 0.004 0.00 0 0 5 10 15 0 5 30 35 40 45 Theta (grid points) Figure : The angular dependence of the gap size in case of d=0.007 cm, azimuthally 40 points are considered and here the gap is given in mm. Our task is to solve the time independent heat conduction equation for the entire pin: T r 1 T + r r 1 + r T θ,,, q& + = 0, (3) λ,,, where q& : q.,,, is not zero only in the region of pellet and λ is different for the pellet, for the gap and for the clad and λ is depending on the temperature in general case. It must be mentioned that in the gap region only the heat conduction is considered and we are not dealing with the thermal contact resistance and with the radiation, these effects are firstly neglected. The code solves equation (3) with the following assumptions: We use standard finite difference scheme for the discretization of eq. 3. In any material, many grid points can be given in both direction Generally, the λ coefficients are temperature dependent, and we used VVER related date, but in case of gap we considered only He gas and we used MATPRO data. It is possible to give constant λ values, it is important for verification The heat generation can be given in radial and azimuthal directions The clad outer surface temperature is fixed, in case of verification we used very bigλ for clad (see Chapter 3) The sparse linear system are solved by Gauss-Seidel iteration algorithm There is an other iteration cycle because of the temperature dependence of the λ coefficients. Of course, if we have the temperature field we can calculate other quantities, like heat flux, etc.
3. VERIFICATION OF In the literature, we found a theoretical investigation [1] for solving the following problem using some simplifications. These simplifications were: Neglecting the heat conduction in the clad, the temperature at inner surface of the clad is given: T 0 In the pellet region constant λ and heat generation is considered In case of gap: gap conductance is used, but calculating the gap size approximated theta dependence is considered: = h n d 1 d h( d, θ ) 1 + cos( θ ) + cos(θ 1 ( / 0) 0 0 ), (4) d dgap dgap dgap where h n is the nominal gap conductance. Our investigation pointed out that this approximation can lead different results as we used the correct gap size (see later and Fig. 3). 1000 1/dgap (1/cm) 900 1/dgap (correct) 1/dgap_approximated average 1/dgap nominal (1 Dimensional): 1/dgap0 00 100 0 0 5 10 15 0 5 30 35 40 45 Theta (grid points) Figure 3: The angular dependence of the 1/(gap size) in case of d=0.007 cm, azimuthally 40 points are considered, and these values are proportional to the gap conductance values. The theta dependence is taken into account by the following function:
,,, q& F( r, θ ) = T ( r, θ ) T0 1 4λ and r uo R uo n, (5),,, q r F( r, θ ) = & an cos( nθ ) 4λ uo n= 0 R. (6) uo Note, that here R UO is the radius of UO, earlier it was signed as R. For the determination of the a n coefficients, an infinite set of linear equations can be written, and the solution is based on stopping at a finite n. This means, that this solution is quasi theoretical. In the following some results will be given. First, for the one dimensional case, and it follows the two dimensional case. In all cases, two sub-cases are considered: pellet with central hole and pellet without central hole. In case of eccentricity, the results will be given for the cases: using a gap size coming from an approximated gap size (see equation (4)) or using the correct gap size, the results will be different. Unfortunately, for the two dimensional case it is not given formula for the case pellet with central hole. In this case, the results can be compared to the results of the case: using pellet without central hole, but the central hole is taken into account by such a material which have very small lambda and in this region there is no heat generation. In all cases, we used constant parameters:,,, W q & =.13 (relating for UO 3 (R uo ) only), cm W λ gap = 0. 0066, cmk λ = 0. W uo 0369 cmk o T = 317 C, clad inner surface temperature. 0
Case 1: One dimension. Pellet without central hole: Theoretical 0 0.05 0.1 0.15 0. 0.5 0.3 0.35 0.4 0.45 Figure 4: Fuel temperature in Case 1 Case : One dimension. Pellet with central hole: Theoretical 0 0.05 0.1 0.15 0. 0.5 0.3 0.35 0.4 0.45 Figure 5: Fuel temperature in Case As it can be seen, in 1-dimensional case the agreement is perfect.
Case 3: Two-dimensional problem. Pellet without central hole: 0 0.05 0.1 0.15 0. 0.5 0.3 0.35 0.4 0.45 Figure 6: Angular depended fuel temperature in Case 3, theoretical one 0 0.05 0.1 0.15 0. 0.5 0.3 0.35 0.4 0.45 Figure 7: Angular depended fuel temperature in Case 3, using approx. gap size
0 0.05 0.1 0.15 0. 0.5 0.3 0.35 0.4 0.45 Figure 8: Angular depended fuel temperature in Case 3, using correct gap size Figs. 6-8 shows qualitatively good agreements between the calculations, but if we use the correct gap size the maximal and minimal values are different as if we used the approximated gap size (see Eq.( 4) and Fig. 3) Table 1: Temperatures of the fuel pellet, without central hole T, center T, outer surface of UO -dim, approx. gap size, theoretical -dim, approx. gap size, -dim, correct gap size, Max. Min. Max. Min. 73.6 73.6 366. 506.6 75.1 (+0. %) 763.1 (+5.5 %) 1-dim 777.4 (+7.4 %) 75.1 (+0. %) 763.1 (+5.5 %) 777.4 (+7.4 %) 357.4 (-.4 %) 343.6 (-6. %) 486.1 (+3.7 %) 498.6 (-1.6 %) 580.8 (+14.6 %) 486.1 (-4.1 %) This table shows the maximal and the minimal temperature values at the fuel center and at the outer surface of the UO pellet. From the point of view of verification the bold one is important: at the center there is good agreement, but at the outer surface the difference is approx. %. This value is not large, and this difference can be explained by a various reason (e.g. numerical approximations, etc). However, it can be a possible reason that the theta dependence is not taken into account properly by Eq. (5). The problem is that the theta dependence of the gap size is very strange if we use approximated gap size given in Eq. (4), see Fig 3. Because in Eq (5) only cosine functions are used it is not enough to describe the theta dependence, consequently we are even needed to use sinus function.
Note, that we performed calculations for the case if cos(θ) is not considered in Eq. (5), and in this case the differences were below 0.3 %. The other results (given in Table 1) show how different values can be get using different approximations. E.g. in case of using the correct gap size the maximal and minimal temperate values are different as if we used approximated gap size. The maximal temperature of fuel is smaller in all cases than the values coming from the one-dimensional case. Another observation can be that the angular dependence of the temperature at UO outer surface is very big, which means more than 60 % difference to the average in heat flux. Case 4: Two-dimensional problem. Pellet with central hole: 0 0.05 0.1 0.15 0. 0.5 0.3 0.35 0.4 0.45 Figure 9: Angular depended fuel temperature in Case 4, using correct gap size 0 0.05 0.1 0.15 0. 0.5 0.3 0.35 0.4 0.45 Figure 10: Angular depended fuel temperature in Case 4, using approx. gap size
0 0.05 0.1 0.15 0. 0.5 0.3 0.35 0.4 0.45 Figure 11: Angular depended fuel temperature in Case 4, using approx. gap size, and a material with very small lambda and there is no heat generation in the central hole Table : Temperatures of the fuel pellet, with central hole T, center T, outer surface of UO -dim, approx. gap size, -dim, approx. gap size, hole filled with material having small lambda -dim, correct gap size, Max. Min. Max. Min. 709.7 665.3 494.7 356. 709.8 (+0.01 %) 760.5 (+7. %) 1-dim 738.8 (+4.1 %) 665.1 (-0.03 %) 690.0 (+3.7 %) 738.8 (+11.1 %) 494.7 (+0.0 %) 576.3 (+16.5 %) 481.8 (-.6 %) 356. (+0.0 %) 34.7 (-3.8 %) 481.8 (+35.3 %) In this case, similar observations can be get as in case 3, but now we were not able to used theoretical considerations. The verification is based on the results of the previous case. 3. RESULTS FOR THE ENTIRE PIN In this section the entire fuel pin will be calculated. First we use constant values and than we show the temperature dependent caser. Firstly, λ clad =0.199 W/cmK T0= o C (at outer surface of clad)
Case 5: Two-dimensional problem. Pellet with central hole, constant lambda values, calculation of the full fuel pin. 50 00 0 0.05 0.1 0.15 0. 0.5 0.3 0.35 0.4 0.45 0.5 Figure 1: Angular depended and 1-dimensional fuel temperatures in Case 5, using correct gap size. 1.30 1.5 1.0 1.15 Relative heat flux (-) 1.10 1.05 1.00 0.95 0.90 0.85 0.80 0.75 0.70 0 5 10 15 0 5 30 35 40 45 Theta (grid points) Figure 13: Angular depended relative heat flux in Case 5, using correct gap size.
Case 6: Two-dimensional problem. Pellet with central hole, temperature dependent lambda values, calculation of the full fuel pin. 50 00 0 0.05 0.1 0.15 0. 0.5 0.3 0.35 0.4 0.45 0.5 Figure 14: Angular depended fuel temperature in Case 6, using correct gap size. 1.30 Relative heat flux (-) 1.5 1.0 1.15 1.10 1.05 1.00 0.95 Heat flux, iter No. 1 Heat flux, iter No. 9 Heat flux, iter No. 10 Constant lambda values 0.90 0.85 0.80 0 5 10 15 0 5 30 35 40 45 Theta (grid points) Figure 15: Angular depended relative heat flux in Case 6, using correct gap size. From the point of view of DNBR calculations, the most important results can be seen in Fig. 15. This figure shows the angular dependent relative heat flux at the clad outer surface can be more than 1.5 (approx. 1.7 in this case) corresponding to the average relative heat flux. Additionally, the maximal fuel temperature is grater than in 1-dimensional case, but tis difference is not so big: approx. %, see Fig 14.
4. SUMMARY A new code, was presented with its verification and some results were showed. The results pointed out that the angular dependence of the gap size can be important, especially in case of the heat flux calculation at the outer surface of the clad: the angular dependent relative heat flux at the clad outer surface can be more than 1.5 (approx. 1.7 in our case) corresponding to the average relative heat flux. In the future, we are going to continue this work. We plan to take into account the uncertainties of the angular dependence of the gap sizes, considering a hydraulic module, etc. Another task can be the extension of this work for the time dependent case, which can be important in case of Reactivity Initiated Transients. REFERENCES [1] O. McNARY and T.H. BAUER, The effect of asymmetric fuel-clad gap conductance on fuel pin thermal performance, NED, 63 (1981) 39-46