Excerpt from the Proceedings of the COMSOL Users Conference 2007 Grenoble

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1 Excerpt from the Proceedings of the COSOL Users Conference 007 Grenoble Evaluation of the moderator temperature coefficient of reactivity in a PWR V. emoli *,, A. Cammi Politecnico di ilano, Department of Nuclear Engineering *Corresponding author: via Ponzio 34/3 ilano ilan (Italy), vito.memoli@mail.polimi.it Abstract: The moderator temperature coefficient (briefly TC) plays an important role within the power thermal reactor dynamics. In order to run a reactor safely, a negative moderator coefficient temperature is necessary to reach stability during changes in temperature that can be caused by a step insertion of reactivity. Thus, TC calculation is a key point in the reactor design process. The aim of the present work is to implement a simple model for the TC estimation in a PWR. The model is based on the well-known twogroup diffusion theory, which allows to perform criticality calculations with good approximation. The two-group diffusion PDE system has been defined and solved in COSOL ulti-physics which provides a finite-elements solution for any kind of geometry. The reactor has been modeled in -D as a cylindrical homogeneous core along with a radial reflector. The calculations of TC have been carried out performing several static criticality calculations for different temperature variations and different reflector thicknesses. The change temperature effect on group constants in the core zone has been evaluated by means of simple relations obtained starting from the definitions of resonance escape probability and absorption macroscopic cross section. Two core sizes have been analyzed: a large core, H=370cm and W=340cm, and a small core with H=W=00cm. Keywords: moderator temperature coefficient, PWR, two-group diffusion theory. Introduction A nuclear reactor should be designed with a negative TC. This requirement makes sure that, for example, positive excursion of power are quenched by the negative feedback. In particular in a PWR a negative moderator temperature coefficient has an important effect on the control of the reactor following changes in demand for power from the turbine. In a thermal reactor, an increase in the moderator temperature affects the multiplication factor in two ways: (a) the temperature at which thermal cross-sections are computed is changed, and (b) the physical density of the moderator changes because of thermal expansion. The main purpose of this work consists in implementing a simple model for the evaluation of the moderator temperature coefficient in a PWR, which accounts for the effect (b), ordinarily the more significant one. The model [,], which is discussed in the next section, is based on the two-group diffusion theory and the TC can be obtained by running several static criticality calculations. From the calculated multiplication factors the TC can be calculated as: δρ δt δt k k () where k and k are the multiplication factors for two different temperature (T, and T, ) and δt = T, -T,. In this paper we have focused our attention on the influence of the reflector thickness and the magnitude of the moderator temperature change on the moderator temperature coefficient estimation.. odeling The reactor model which has been developed consists of a homogeneous cylindrical multiplying core surrounded by a reflector. The effective multiplication factor is determined by solving the eigenvalue equations derived from the two-group diffusion theory. In order to reduce the computing time we have replaced the original 3-D geometry with a -D one. The loss of neutrons in the axial direction has been taken into account by means of the axial buckling B z which can be calculated as B z π = H 0 ()

2 Excerpt from the Proceedings of the COSOL Users Conference 007 Grenoble where H 0 is the height of the cylinder plus the reflector saving which has been taken as one diffusion length in the reflector medium [3,4,5]. That means: H 0 =H + δ (3) with δ L (4) b where L b is the neutron diffusion length in the blanket (reflector). The diffusion length has been calculated as the square root of ratio between the diffusion coefficient and the absorption cross section in the reflector. According to the previous hypothesis the twogroup diffusion equations, in the core region, become: D B z D x, y + a, + = ( ν f, + ν f, ) k eff D + = B z D x, y a, where and are respectively the fast and the thermal neutron fluxes. D i is the diffusion coefficient and a,i is the absorption cross section (where index i= stands for fast group and i= for thermal groupp), is the removal cross section (i.e. the probability per centimeter that a fast neutron is slowed down to the thermal group). The fudge factor /k eff, the inverse of the multiplication factor, makes the equations time independent. The two group diffusion equations have been processed in the PDE module of COSOL ulti-physics where the direct method UFPACK has been set as eigenvalue solver. In order to validate the numerical code the numerical solution has been compared with the well-known analytical one in the case of D reflected core[5]. In Fig. the calculated thermal and fast fluxes are shown. Figure. thermal and fast normalized neutron fluxes evaluated with Comsol for a -D reflected core (core diameter 40 cm, reflector thickness 0cm) As can be seen the agreement with the analytical solution is good. The fast flux shows a peak in the core region because of the presence of fuel which can be regarded as a source of fast neutrons due to fission of U-35. The thermal flux increases in nearby reflector region where the moderation is the main effect. Within the two-group diffusion theory, the effects of temperature change on nuclear quantities can be simply calculated from the definition of resonance escape probability, removal cross section from the fast group to the thermal one and the absorption cross section. oreover, the assumption of homogeneous core requires the estimation of the effect on the fission cross section. In fact for a homogeneous core, the density variation of the moderator due to a temperature change should have an impact on the fuel which is dispersed in the liquid moderator. In general the resonance escape probability, p, can be expressed by the following relation: p ρ I F ξρσ exp. (5) = s where ρ F and ρ are respectively the fuel and moderator densities, I is the resonance integral[3,4,5], ξ is the lethargy (average log energy loss) and σ s is the microscopic scattering cross section.

3 Excerpt from the Proceedings of the COSOL Users Conference 007 Grenoble The latter relation, which refers to the one energy group theory, reduces to the two-group resonance escape probability definition given by p= + a, (6) in the hypothesis of low neutron absorption. By calculating the variation of the resonance escape probability with respect to the temperature variation we obtain the following relation: δp δρ = log = log βδt (7) p p ρ p where β is the thermal expansion coefficient of the moderator. In order to evaluate the removal cross section variation with respect to temperature change, δt Μ, the two-group resonance escape probability definition can be rewritten as: p = a, (8) p Thus, making the derivative of the previous one and using (7) the removal cross section variation is given by: log( / p) δ β T = δ (9) p The temperature variation effect on the thermal absorption cross section is simply given by : δ a, = β a, δt. (0) Parameter Group Group a(cm - ) D(cm) (cm - ) νf(cm - ) Table. Two-Group diffusion constants in the core region Parameter Group Group a(cm - ) D(cm) (cm - ) Table. Two-Group diffusion constants in the reflector region. The above data have been used to evaluate the multiplication factor k. The new multiplication factor k is obtained by updating the cross sections, a,, f,i to the new ones i.e. + δ, a, +δ a,, f,i +δ f,i. Two core sizes have been taken into account. A large core with H=370cm and W=340cm and small core with H=00cm and W=00cm. For the reactor geometry an unstructured fine mesh has been used as it is shown in Fig. which corresponds to the large core with a reflector thickness of 0cm. In this case the number of the cells is equal to 95. In the same way we can evaluate the variation of the fission cross section in the thermal region: δ f, = β f, δt. () 3. Results In Table and are shown the typical nuclear constants for a PWR[3]. The thermal expansion coefficient [6] of the moderator, β = / C, corresponds to the usual PWR operating conditions i.e. pressure of 5Pa and water temperature of 30 C which corresponds to the water mean temperature between core inlet and core exit. Figure. esh of the core geometry. This choice can be justified by Fig.3 which shows the k eff estimation as a function of the element number. The k eff remains practically

4 Excerpt from the Proceedings of the COSOL Users Conference 007 Grenoble constant for meshes with an number of elelemnts greater or equal to oderator Temperature Coefficient as a function of Reflector Thickness ultiplication factor versus number of mesh elements TC(pcm/ C) Keff Number of elements Figure 3. k eff estimation as a function of the mesh size. In Fig. 4 the fast and thermal neutron fluxes as a function of radial distance is shown for the large core with a reflector thickness of 0cm Reflector Thickness (cm) Figure 5. oderator temperature coefficient versus reflector thickness for PWR big core for δt =4 C. In the case of small core the effect of the reflector thickness on the TC is much more significant. The TC %-variation between and reflected (0cm) reactor is about 7.7%. oderator Temperature Coefficient vs Reflector Thickness TC (pcm/ C) Reflector Thickness(cm) Figure 4. Fast and thermal neutron flux as a function of the radial distance in large core. The effect of the reflector thickness is shown in Figs. 5 and 6 for large and small core respectively. A moderator temperature change of 4 C has been considered as reference value. In the case of large core, the reflector thickness has a small influence on the moderator temperature coefficient. The variation of the TC between and reflected (0cm) reactor is about 0.4 %. Figure 6. oderator temperature coefficient versus reflector thickness for PWR small core for δt =4 C. In both cases the curve reaches a plateau due to the fact that after a certain thickness value (about 0cm) the reflector behaves practically as an infinite reflector [3,4,5]. The TC as function of temperature change is shown in Figs. 7 and 8 for and reflected reactor in the two core size cases.

5 Excerpt from the Proceedings of the COSOL Users Conference 007 Grenoble oderator Temperature Coefficient as a function of temperature chenge 4. Conclusions -55 TC(pcm/ C) δt ( C) reflected 0cm Figure 7. oderator temperature coefficient versus temperature change for large core. TC(pcm/ C) oderator Temperature Coefficient versus Temperature Change Reflected 0cm The influence of both the reflector thickness and temperature change magnitude on the moderator temperature coefficient in a PWR was investigated for two different size. To carry out the criticality calculations a -D homogeneous reactor has been modeled on the basis of the twogroup diffusion theory. Simple relations have been used to quantify the temperature change effect on the nuclear constants of absorption cross section, removal cross section and fission cross section. On the basis of the results it can be asserted that the reflector thickness has a negligible effect on the moderator temperature coefficient estimation for large core. Instead the small core introduces an uncertainty on the determination of the TC which can reach a value of about 8%. Within the range of 3 C to 5 C the TC estimation does not change significantly δt ( C) Figure 8. oderator temperature coefficient versus temperature change for small core. The percentage variation of the TC as a function of the temperature change is shown in Fig. 9. The results refer to the large core only, The percentage variation is practically negligible in a range 3 C δt 5 C which is the range of investigation recommended by the nuclear standard (ANS, 997). %-variation %-variation of the TC versus temperature change δt ( C) reflected 0cm 5. References. ANS,997. Calculation and measurement of the moderator temperature coefficient of reactivity for water moderated reactor power reactors. American Nuclear Society, American National ANSI/ANS C. Housidias et al., Calculation of the moderator temperature coefficient of reactivity for water moderated reactors, Annals of Nuclear Energy, 8, (00) 3. J.J. Duderstadt, L.J. Hamilton, 976. Nuclear Reactor Analysis. John Wiley & Sons, New York. 4. J. L. EE, 964. Two Group Reactor Theory, Gordon and Breach Science Publishers, Inc. 5. J.R. Lamarsh, 966. Introduction to nuclear Reactor Theory. Addison-Wesley Publishing Company Inc, Reading, assachusetts. 6. NIST, Standard Reference Database 3. Physical and Chemical Properties Division. Figure 9. Percentage Variation of TC as a function of moderator temperature change for and reflected reactor.