MSR Reactor Physics Dynamic Behaviour. Nuclear Reactor Physics Group Politecnico di Torino. S. Dulla P. Ravetto

Transcription

1 MSR Reactor Physics Dynamic Behaviour Nuclear Reactor Physics Group Politecnico di Torino S. Dulla P. Ravetto Politecnico di Torino Dipartimento di Energetica Corso Duca degli Abruzzi, Torino Tel Fax Atelier GEDEON Cadarache, June 2002

2 Summary of presentation L Review of literature L Slug Àow model L Consistent point kinetics L Discussion of dynamical effects of motion L Open questions

3 Spatial kinetic model for circulating-fuel reactors [Meghreblian, R., Holmes, D., Reactor analysis Fleck J., Kinetics of circulating reactors at low power, Nucleonics, 12 (10), (1954)], standard diffusion equation with delayed neutrons, convection terms added to the precursors equations, slug Àow along the 5 axis, multidimensional problem reduced to one dimension making the assumption that the Àux and the concentration of precursor have the same spatial transverse shape.

4 In cilindrical geometry it is assumed that: o Ehc ' )E5c a f 4 f - o Ehc ' 7 E5c a f 4 f - and the diffusion equations take the form., Y Y )E5c ' R u 2 k l Y 2 )E5c n > 2 )E5c Y5 2 n de qb& qo )E5c n & DP s S b 7 E5c Y Y 7 E5c n T Y Y5 7 E5c ' q DP s )E5c b 7 E5c f 5 M Y Y 7 E5c n T Y Y5 7 E5c ' b 7 E5c M 5 M M is the core height EM M is the external loop lenght

5 Point-kinetic model for circulating-fuel reactors: [Ash, M., Nuclear reactor kinetics Schultz, M. A., Control of nuclear reactors and power plants MacPhee, J., The kinetics of circulating fuel reactors, Nucl. Sc. Eng., 4, , (1958)], standard expression of the equation for the amplitude function for neutrons?e, convection and delay terms added to the precursors equations, the way to calculate the coef cients for the equations is not explained

6 .?E ' B&E q,?e n S b S E S E ' b S E n q,?e S E * S n S E u e ub * S S E * S is the Àow rate out of the core at time ( S is the residence time of the fuel in the core) S E u e ub * S is the Àow rate into the core at time ( u is the residence time in the external loop)

7 The kinetics of circulating-fuel reactors Balance equations in multigroup diffusion for neutrons: [ } Yx } Y ' 8 ( } 8 x } P } x } n Rc} DP s} 3 E q n P } 3 $} x} 3n } 3 7 } n [ b c} + initial and boundary conditions same structure as for the solid-fuel case

8 Geometrical structure of a circulating-fuel reactor S ' M SJoe 7 SJoe u ' u,jjr 7,JJR 7 SJoe 7,JJR ' D,JJR D SJoe D SJoe ' Z- 2 SJoe Fission products are moved through and outside the core by the motion of the ssile material.

9 , Balance equations for precursors: Y Y ' b n q [ } DP s} x } 8 E streaming term describes motion of delayed neutron precursor through the core boundary conditions are needed they must account for decay outside the core in cylindrical geometry: E5 ' fc oc ' eb u D SJoe ] D SJoe E5 ' Mc oc u _D + initial conditions

10 In slug-àow conditions the velocity eld is maintained by externally-driven devices and is one-dimensional (axial): 8 E ' Y Y5 the delayed-precursor streaming has some important physical consequences:

11 i) the delayed-precursor equations cannot be eliminated in steady-state con guration. solid fuel: the concentration of the precursor can be expressed as a function of the ssion term and substituted into the diffusion equation. [ b ' q DP s} x } } circulating fuel: the equation for the precursor is still differential for the space variable and the concentration can not be made explicit and substituted. [ 8 E ' b n q DP s} x } }

12 ii) the multiplication eigenvalue depends on delayed neutron characteristics k eff λ=0.01 1/s β=0.007 L=0 cm L=100 cm mean velocity [cm/s] k eff λ=0.01 1/s β=0.007 L=0 cm β= mean velocity [cm/s] k eff λ=0.01 1/s L=100 cm β=0.007 β= mean velocity [cm/s]

13 C 4 C 5 C 6 C 1 C 2 C 3 PHI 1 PHI 2 PHI 3 iii) the space distribution of delayedprecursor is not following the neutron distribution and is completely different from standard solid-fuel systems x 10 8 z [cm] x z [cm] x 10 8 z [cm] x 10 z [cm] x 10 z [cm] x 10 z [cm] z [cm] z [cm] z [cm]

14 iv) the role of delayed emissions is reduced by space-redistribution and external recirculation (reduction of effective q) q cess ' b kx n m l [ b kx n m l n E q kx n m DP s xl Ratio q/q as a function of and & when ' Sf and ff cm/s. Results for a critical system are also shown. [s] $ [cm/s] & Sf & ' fbdffb ff Sf & ' fb.feh ff Sf & ' fbbf2h ff Sf & ' ffff ff

15 v) factorization schemes should be applied to both neutrons and precursors vi) point kinetics model needs a speci c formulation vii) the numerical solution can take advantage of the slower time-scale of the motion of the delayed precursors In an implicit scheme two time intervals are introduced for the time discretization: { for the neutron equation integration { 2 to spatially move delayed precursors For the whole interval { 2 delayed neutrons are kept frozen

16 Discrete equations for the Àux (balance phase) x?n } ' x } EE? n { x?n } x? } ' 8 ( } 8 x?n } P } x?n } { [ } 3 } n Rc} DP s} 3 E q n P } 3 $} x?n } 3 n 7?n } n [ b c}?n q [?n DP s} 3 2 } 3 '? eb { n x?n } 3 n x? } 3 e b { { } ' c 2c c C( ' c 2c c S

17 Update the position of the precursors through time interval { 2 (convection phase):?e E E{ 2 ' J,_ E interpolation may be needed E ' [?e E 3E{ 2 7E 3E{ 2 3?e

18 Point kinetic model for circulating fuel systems, consistent with Henry factorization procedure., the time-dependent diffusion equations for neutrons and delayed precursors are considered: ; A? Yx } Y } Y ' Y5 ( Y } Y5 P -c} x } n [ C [} n E q } EDP s } 3 x } 3 n P } 3 $}x } 3n n C[ } 3 '}n } 3 ' P } 3 $}x } 3 n } 3 ' -[ c} b n 7 } ' } ' c 2c c C A= Y Y ' Y Y5 b n q ' c 2c c - C[ EDP s } 3 x } 3 } 3 ' +initial and boundary condition

19 , a reference con guration is introduced _ f ' _5 ( _ }cf _5 P -c}cf x }cf n [ C n E q } EDP s } 3 cf x [} } 3 cf n n C[ } 3 '}n } 3 ' P } 3 $}cfx } 3 cf n } 3 ' P } 3 $}cfx } 3 cfn -[ c} b cf n 7 }cf ' } ' c 2c c C f ' f _ cf _5 b cf n q +boundary conditions ' c 2c c - C[ EDP s } 3 cf x } 3 cf } 3 '

20 , a physically consistent de nition of the neutron and delayed precursor importance x and is de ned, as the total number of ssion neutrons produced within the system, the balance equations for the importance functions are shown to be the adjoint to the balance equations for neutrons and precursors, having de ned the inner product as: E c ' Cn- [ k? m? l ' Cn- [ ]?E%? E%_% where Eh ' Ex c c x C c c c - Eh ' Ex c c x C c c c -

21 , The system of equations for the importance takes the form: 5 ( _ }cf _5 P -c}cf x }n [ C C[ n E q EDP s }cf? x? n P }$} 3 cfx } 3 n n [} } 3 ' } 3 '}n [ - P }$} 3 cfx } 3 n EDP s }cf q n 7 } ' f ' } ' c 2c c C f _ _5 n b ' b ' c 2c c - C[ c? x?, A time separation is introduced, to derive the point kinetic model x } E5c ' x }cf E5 E ( E5c ' cf E5C E

22 The new point kinetic system of equations is: _ \ _ ' 4 7 q n -[ b K n 7 ' \ _K _ ' q n 4 Eb n > c n > 1c K n j ' c c - $ equivalent structure to the point kinetic model for solid-fuel systems $ kinetic parameters have different de nitions $ unconventional terms appear in the delayed neutron equations: 4 : perturbation of production term > c : perturbation of Àuid velocity > 1c : perturbation of recirculation time

23 normalization factor I ' n E q -[ ' C[ C[ x? c? b cf n }' C[ x?? EDP s }cf x }cf l effective delayed neutron precursors K ' I C[ x? c? c f C E effective neutron source 7 ' I C[ x? 7? l effective delayed neutron fractions q ' I C[ [ - x? c? b cf c q ' q '

24 reactivity 4 7 ' 4 f n 4 Reo, where 4 f ' I C[ 7? x?cf l + 4 Reo ' C[ _x? I _5 B( _x?cf? n _5 C[ C. [ n x? E q? B EDP s } x }cf n }' C[ x? BP -c? x?cf l n < C[ n x? BP }$? x }cf l > }' prompt neutron lifetime \ ' I C[ x?? x?cf

25 generalized precursor lifetime \ ' C[ G G x? cf l unconventional terms 4 ' I ( ' c 2c c - c? cf G q C [ }' BEDP s } x }cf. > c ' G C[ G B _ cf _5 x? c? cf > 1c ' f EM c fem C[ G c? c f x?

26 apparent precursor source ; A? j ' A= f Ef c fem 1 cf n B1 Kcf c C[ G c? cf x? if A - c f Ef cfem 1 cf n B1 K E A - C[ G c? cf x? if : A - [Lapenta, G., Mattioda, F., Ravetto, P., Point Kinetic model for Àuid fuel systems, Annals of Nuclear Energy, 28, , (2001)]

27 Open questions $ Dependence of the multiplication eigenvalue on thermal-hydraulics parameters $ Detail of thermal-hydraulic models for steady-state and transient calculation $ Non-linear effects, feedbacks and possible instabilities for different material and geometrical con gurations $ Adaptation of existing models and methods for transient calculation (point, quasi-static, ) $ Effect of the uncertainty of the parameters (fractions and decay constants) of delayed neutron precursors for minor actinides perturbative study

28 $ Instabilities due to concentration and precipitation of some material within the Àuid $ Characteristics of typical safety-related transients (neutronic, thermal-hydraulic) $ Role of the external source in subcritical systems (space-energy distribution) $ Role of delayed emissions: comparisons critical-subcritical systems $ Transport vs. diffusion $ Study of projection techniques to reproduce integral parameters position of problem of adjoints and subcritical systems $ Generalization of perturbation theory boundary perturbation effects $...