The influence of direct energy deposition on BWR stability

Annals of Nuclear Energy 28 (2001) 1447±1456 www.elsevier.com/locate/anucene The in¯uence of direct energy deposition on BWR stability T.H.J.J. van der Hagen * Interfaculty Reactor Institute, Delft University of Technology, Mekelweg 15, 2629 JB Delft, The Netherlands Received 03 October 2000; accepted 24 November 2000 Abstract Direct energy deposition into the coolant of a BWR via high-energy s and neutrons is always neglected in reduced-order BWR dynamics models. The correctness of such a simple approach is investigated by comparing reactor stability predictions of the March±Leuba± Cacuci±PeÂrez reduced-order model, with a model where direct heating has been included. It is shown that neglecting direct energy deposition leads to a strong underestimation of the stability of the reactor. # 2001 Elsevier Science Ltd. All rights reserved. 1. Introduction Instabilities in boiling water reactors (BWRs) are brought about by an interplay between thermal hydraulics and neutronics. These two processes are physically coupled via heat transfer from the position of ®ssion to the coolant and via reactivity e€ects caused by temperature and density changes in the coolant. The majority of the ®ssion energy appears as kinetic energy of ®ssion fragments and is deposited in the fuel at the position of ®ssion. Consequently, the temperature of the fuel rises and the heat is transported to the coolant via conduction and radiation processes. Fission neutrons, both prompt and delayed, however, transfer energy directly to the moderator, i.e. the coolant, as they gradually slow down. In this manner, the fuel heat transfer process is in fact bypassed. Also gammas have a long range; some of them escape completely from the reactor, others deposit their energy in fuel and construction materials. All in all, in general some 90% of the total energy per ®ssion is deposited in the fuel itself, about 4% in the coolant, about 5% is * Corresponding author Tel.: +31-15-278-2105; fax: +31-15-278-6422. E-mail address: [email protected] (T.H.J.J. van der Hagen). 0306-4549/01/$ - see front matter # 2001 Elsevier Science Ltd. All rights reserved. PII: S0306-4549(00)00130-4 1448 T.H.J.J. van der Hagen / Annals of Nuclear Energy 28 (2001) 1447±1456 carried away by neutrinos, and the remainder, about 1%, is deposited in various other reactor materials (El±Wakil, 1978). It is clear that the dynamics of the energy-deposition processes di€ers: fuel heat transfer tends to damp power ¯uctuations and leads to a phase change between power ¯uctuations and coolant enthalpy ¯uctuations, whereas ¯uctuations in direct energy deposition are undamped and are in-phase with power ¯uctuations. Many researchers are using so-called reduced-order models to obtain a better understanding of the underlying physics of BWR dynamics. In doing so, they elaborate on the original work by DeShong and Lipinski (1958), Kramer (1958), Thie (1959) and, later on, by March-Leuba, et al. (1986a,b). Direct energy deposition is tacitly neglected in all these original and subsequent models (Prasad et al., 1995; Rao et al., 1995; MunÄoz-Cobo et al., 1996; Park et al., 1986; ; Uehiro et al., 1996; Karve et al., 1997; Lin et al., 1998; Van Bragt and Van der Hagen, 1998; Nayak et al., 2000). This paper focuses on the validity of neglecting direct energy deposition. Studies were performed by incorporating the direct-energy-deposition e€ect into the March± Leuba±Cacuci±PeÂrez reduced-order model. In our approach, the transfer of ®ssion energy to the coolant is split into two terms: a term that takes into account direct coolant heating (mainly brought about by neutron moderation), and a term that takes into account coolant heating via heating of fuel, cladding and internal structures. This last term does not only consist of heating by kinetic energy of ®ssion fragments, but it also incorporates heating by gammas that are released at the position of ®ssion but are absorbed in the fuel, cladding or structures at another position. These heat transfer e€ects are lumped in the ``fuel-to-coolant'' heat transfer process and are described by an e€ective heat transfer time constant. The paper serves two purposes: the in¯uence of direct energy deposition on reactor stability is explained and quanti®ed, plus, a better understanding of BWR physics and dynamics in particular is o€ered. 2. The extended March±Leuba±Cacuci±PeÂrez reduced-order model This section discusses the basis of the study, i.e. the March±Leuba±Cacuci±PeÂrez reduced-order model (March±Leuba, 1986; March±Leuba et al., 1986a,b). To suit our purposes, the model is extended by incorporating direct energy deposition. We will use the frequency-domain version of the model. Fig. 1 displays the interaction between the several processes in graphical form. Doppler-feedback is neglected here, since it has no in¯uence on the dynamics at frequencies relevant for BWR stability (March±Leuba et al., 1986a). The newly added direct-energy branch is indicated by a thick line. The zero-power transfer function from reactivity ¯uctuations to (normalised) power ¯uctuations is given by H1 (using the one precursor-group approximation): H1 s†  P s† l‡s ˆ  s† s s ‡ l ‡ † 1† T.H.J.J. van der Hagen / Annals of Nuclear Energy 28 (2001) 1447±1456 1449 Fig. 1. Schematic diagram of the March±Leuba±Cacuci±PeÂrez reduced-order model extended by a directenergy-deposition branch (Doppler feedback neglected). where the symbols have their conventional meaning: Laplace variable s, reactivity  normalised power P, neutron generation time  s†, precursor decay constant l (s 1) and delayed neutron fraction . Since the resonance frequencies of modern BWRs are typically around 0.5 Hz (i.e. in the so-called plateau region of the reactivity-topower transfer function) H1 can be approximated with a high degree of accuracy by 1= . We will use this simpli®cation. A fraction of the power ¯uctuations is directly coupled to variations of the heat current into the coolant channel; a fraction 1- is coupled via ¯uctuations of the fuel temperature via H2: H2 s†  qfuel ! coolant s† 1 ˆ P s† 1 ‡ sf 2† where q refers to normalised variations of the heat current from fuel to coolant and where f is the e€ective fuel time constant (s). In this model, the heat transfer mechanism is described by an e€ective fuel temperature, which gradually changes with time from the initial to the asymptotic value, after a power change. Axial e€ects and the temperature dependence of fuel heat-capacity and fuel heat-conductivity are neglected. A previous paper of the present author discusses the applicability of using a ®rst-order approximation of fuel heat transfer by comparing the stability features with those resulting from a model using a 2nd-order heat-transfer model (Van der Hagen, 1998). March-Leuba et al. used a static gain from power to fuel temperature, equal to a1 divided by a2 (in their terminology). We incorporate this term in the static gain of transfer function H3 in order to have the static gain of the combined (direct and indirect) heat transfer identical to theirs. 1450 T.H.J.J. van der Hagen / Annals of Nuclear Energy 28 (2001) 1447±1456 Channel thermal hydraulics±i.e. the e€ect of ¯uctuations of both direct energy deposition and fuel heat-current on void reactivity  ±are described by an oscillator with coecient K (s 2), and dynamic characteristics  and !0 : H3 s†   s† K ˆ 2 P s† ‡ qfuel ! coolant s† s ‡ 2!0 s ‡ !20 3† where  is the damping constant and !0 the characteristic frequency (rad/s) (March± Leuba et al. express these parameters in the coecients a3 and a4). The system shows a damped oscillatory impulse response when 0 <  < 1 and !0 > 0. The decay ratio of the channel ¯ow (DRC, de®ned as the ratio of two consecutive maxima of the impulse response of the system) and the channel resonance frequency !C † are: ! 2 DRC ˆ exp p 4† 1 2 p 5† !C ˆ !0 1 2 Finally, the feedback loop is closed via the relation: 6† ˆ The closed-loop transfer function of this system is equal to G s†  ˆ P ˆ  1 H1 H1 H3 ‡ H2 † 1 ‡ sf † s2 ‡ 2!0 s ‡ !20 † 1 ‡ sf † s2 ‡ 2!0 s ‡ !20 † K 1 ‡ s f † 7† For < 1 this transfer function has three zeroes and three poles. The system reduces to a second-order system for the hypothetical case of being equal to 1, where all energy is directly deposited in the coolant [which would be the case for reactors with in®nitely thin fuel rods and for homogeneous suspension reactors, like the KEMA Suspension Test Reactor (Kersten, 1985)]. For equal to 0, the system returns to the conventional BWR model. 3. Reactor stability A convenient method for calculating the linear stability threshold as a function of a parameter is the Routh±Hurwitz criterion (see text books on linear system dynamics, or Hetrick, 1993). For stability of a 3rd-order system, it is sucient and necessary that the following inequalities are obeyed: T.H.J.J. van der Hagen / Annals of Nuclear Energy 28 (2001) 1447±1456 1451 p3 > 0 8† p2 > 0 9† p0 p3 >0 p2 p1 p0 > 0 10† 11† where pi are the coecients of the polynomial form of the denominator of G(s), the so-called characteristic polynomial: p3 s3 ‡ p2 s2 ‡ p1 s ‡ p0 : Eqs. (8)±(11) lead to the following set of inequalities: f > 0 12† 1 ‡ 2!0 f † > 0 Kf 1 8 > > > > > <K> > > > > > :K <  
2!0 f > 2!0 1 ‡ 2!0 f ‡ !20 f2 )   2!0 1 ‡ 2!0 f ‡ !20 f2 1
for 04 < 1 ‡ 2!0 f f 1 2!0 f   2!0 1 ‡ 2!0 f ‡ !20 f2 1
< 41 for 1 ‡ 2!0 f f 1 2!0 f 13† 14† condition for dynamic stability† K <1 2!0 condition for static stability† 15† The conditions expressed by Eqs. (12) and (13) are ful®lled for all physical systems. Eqs. (14) and (15), however, show that the reactor is conditionally stable. As expected, the reactor becomes statically unstable when the static gain of the loop (being K=2!0 ) becomes larger than unity. It should be noted that the plateauregion approximation of the reactivity-to-power transfer function is no longer valid for these (in®nitely) small frequencies, unless one neglects the dynamics of the delayed-neutron precursors l ˆ 0†. Eq. (15) shows that a reactor with negative void-reactivity feedback (K<0) cannot become statically unstable. The boundary of dynamic stability is more interesting, since it depends on K and on [see Eq. (14)]. Fig. 2 displays the stability boundaries of the reactor in the parameter space K. The example of the ®gure was constructed for f ˆ 5 s, ˆ 0:0056, DRc=0.4 and !0 ˆ  rad/s. In this plane, the dynamic stability boundary consists of a hyperbola with a ver
tical asymptote at ˆ c  1= 1 ‡ 2!0 f . For g-values below this critical value, 1452 T.H.J.J. van der Hagen / Annals of Nuclear Energy 28 (2001) 1447±1456 Fig. 2. Parameter space for a reactor with direct-energy-deposition fraction . The dynamic stability boundary (hyperbola) and the static stability boundary are shown. Direct energy deposition has a stabilising e€ect on a reactor with negative void-reactivity feedback (K<0). the reactor becomes dynamically unstable for too small K-values. Direct energy deposition has a stabilising e€ect on the reactor since the stability boundary moves to stronger negative K-values (i.e. stronger negative void-reactivity feedback) for increasing . For larger than c , the stabilising e€ect of the direct energy deposition becomes that strong that the reactor is stable for all values of K, provided that K<0. In our example, c =0.21, which is much larger than the direct-energy-deposition fraction in a reactor. In fact, is smaller than c for all practical cases. The in¯uence of on reactor stability becomes stronger for smaller channel decayratios:  increases with decreasing DRC which leads to a smaller value of c ; the hyperbola becomes steeper with decreasing c . Fig. 2 shows that neglecting direct energy deposition leads to a conservative estimate of reactor stability: the reactor is in fact more stable than predicted. The next section gives quantitative values of the di€erence between this conservative estimate of the reactor decay-ratio and the actual decay ratio. The roots of the characteristic equation (obtained by equating the characteristic polynomial to zero) can be investigated by the root-locus method (Hetrick, 1993). In this method, the position in the s-plane of the poles and zeroes of the open-loop transfer function is studied as a function of a system parameter (usually the feedback gain, K). The system becomes unstable when a pole (or a pair of poles) crosses the imaginary axis and enters the right half-plane. In our case, the open-loop transfer function, F, is equal to: K F ˆ H1 H3 ‡ H2 † ˆ 1 ‡ s f † s2 ‡ 2!0 s ‡ !20 † 1 ‡ sf † 16† T.H.J.J. van der Hagen / Annals of Nuclear Energy 28 (2001) 1447±1456 1453 Fig. 3. Sketch of root loci of the open-loop transfer function as a function of K (for K<0) ( denotes a pole; * a zero) Increasing the negative feedback strength leads to an unstable system for cases where < c . Clearly, in general F has three poles (a pair of complex conjugate poles plus a real, negative pole) and a real, negative zero. Fig. 3 illustrates the behaviour of the poles as a function of feedback strength (for K<0). The two complex conjugate poles move to the right half-plane with increasing feedback strength, leading to an unstable system, for cases where the asymptote lies at a positive value, which is the case for < c . The vertical asymptote moves to the left half-plane for > c indicating that the reactor is stable for all feedback strengths. For ! 0 the zero moves to 1; the vertical asymptote moves to 1 and ®nally Ð when equals 0 Ð becomes tilted, crossing the real axis in !0 ; 0† at an angle of 60 . A pole-zero cancellation occurs for equal to unity; the vertical asymptote now coincides with the position of the two conjugate poles (at !0 ). Increasing the feedback strength now leads to a higher oscillation frequency, but does not a€ect the damping of the system. 4. Reactor decay-ratio In this section, a quantitative comparison is made of the stability of a BWR with and without direct energy deposition. The stability is expressed in the reactor decayratio, DRR. The decay ratio of the reactor system can be determined by calculating the position of the two conjugate poles of the closed-loop transfer function, as given by Eq. (7). DRR is then equal to DRR ˆ exp 2Refpg=Imfpg† 17† 1454 T.H.J.J. van der Hagen / Annals of Nuclear Energy 28 (2001) 1447±1456 Fig. 4. The reactor decay-ratio as a function of the direct-energy-deposition fraction , for several values of the channel decay-ratio, DRC. The feedback gain coecient K was set to 0.5 s 2. The value of c is indicated by a vertical dashed line segment for every value of DRC. where Re{p} denotes the real part of pole p, and Im{p} the absolute value of the imaginary part. Fig. 4 gives the decay ratio of the reactor as a function of the direct-energydeposition fraction, for several values of the decay ratio of the thermal-hydraulic Fig. 5. Enlargement of part of Fig. 4: the reactor decay-ratio as a function of the direct-energy-deposition fraction , for several values of the channel decay-ratio, DRC. The feedback gain coecient K was set to 0.5 s 2. 1455 T.H.J.J. van der Hagen / Annals of Nuclear Energy 28 (2001) 1447±1456 Table 1 Reactor decay-ratios obtained by neglecting or including direct energy deposition for three thermalhydraulic channel decay-ratios, DRC =0% =4% DRC=0.05 DRC=0.1 DRC=0.2 0.49 0.32 0.73 0.49 1.15 0.79 channel. The ®gure was constructed for the following typical BWR-values: f ˆ 5 s; =0.0056; K= 0.5 s 2 and !0 ˆ  rad/s. It can be noticed that the reactor has an oscillatory impulse response, even for the case where the hydraulic channel is critically damped (DRC=0). As discussed in the previous section, increasing stabilises the reactor as indicated by a lower decay ratio. Increasing beyond its critical value c has a destabilising e€ect on systems with a low DRC, ultimately possibly leading to a higher decay ratio than was the case without direct energy deposition. The practical importance of the dependence of reactor stability on can best be seen from Fig. 5, which gives an enlargement of the part of Fig. 4 that is relevant for practical BWRs. Table 1 gives a quantitative comparison of reactor decay-ratios for systems with and without direct energy deposition. It can be seen that the decay ratio of a conventional BWR (typically with DRC=0.05 and =4%) is overestimated by some 50%: the actual reactor decay ratio is 0.32, whereas an estimation based on neglecting direct energy deposition would render a value of 0.49. 5. Conclusions Direct energy deposition appears to have a strong stabilising e€ect on BWR systems. For practical cases, with 4% direct coolant heating, this e€ect is estimated to be as strong as 50% on the reactor decay-ratio. A higher fraction of direct heating would stabilise the reactor even more. This means that conventional reduced-order models, that without exception neglect direct energy deposition, yield a strongly conservative estimate of BWR stability. Regarding the simplicity of the model used, quantitative results should be handled with care. Nevertheless, the magnitude of the e€ect justi®es a further study with advanced BWR models. References DeShong Jr., J.A., Lipinski, W.C., 1958. Analyses of experimental power-reactivity feedback transfer functions for a natural circulation boiling water reactor. ANL-5850, 52 p. El-Wakil, M.M., 1978. Nuclear Heat Transport. American Nuclear Society, Inc, La Grange Park, Illinois, USA. 1456 T.H.J.J. van der Hagen / Annals of Nuclear Energy 28 (2001) 1447±1456 Hetrick, D.L., 1993. Dynamics of Nuclear Reactors. American Nuclear Society, Inc, La Grange Park, Illinois, USA. Karve, A.A., Rizwan-Uddin, Dorning, J.J., 1997. Stability analysis of BWR nuclear-coupled thermalhydraulics using a simple model. Nuclear Engineering and Design 177, 155±177. Kersten, J.A.H., 1985. The aqueous homogeneous suspension reactor project. N.V. KEMA, Arnhem, The Netherlands, 42 p. Kramer, A.W., 1958. Boiling Water Reactors. Addison-Wesley Publishing Company, Inc, Massachusetts. Lin, Y.N., Lee, J.D., Pan, Chin, 1998. Nonlinear dynamics of a nuclear-coupled boiling channel with forced ¯ows. Nuclear Engineering and Design 179, 31±49. March-Leuba, J., 1986. A reduced-order model of boiling water reactor linear dynamics. Nuclear Technology 75, 15±22. March±Leuba, J., Cacuci, D.G., PeÂrez, R.B., 1986. Nonlinear dynamics and stability of boiling water reactors: Part 1 Ð qualitative analysis. Nuclear Science and Engineering 93, 111±123. March-Leuba, J., Cacuci, D.G., PeÂrez, R.B., 1986. Nonlinear dynamics and stability of boiling water reactors: Part 2 Ð quantitative analysis. Nuclear Science and Engineering 93, 124±136. MunÄoz-Cobo, J.L., PeÂrez, R.B., Ginestar, D., EscrivaÂ, A., VerduÂ, G., 1996. Non linear analysis of out of phase oscillations in boiling water reactors. Annals of Nuclear Energy 23 (16), 1301±1335. Nayak, A.K., Vijayan, P.K., Saha, D., Venkat Raj, V., Aritomi, Masanori, 2000. Analytical study of nuclear-coupled density-wave instability in a natural circulation pressure tube type boiling water reactor. Nuclear Engineering and Design 195, 27±44. Park, G.-C., Podowski, M.Z., Becker, M., Lahey Jr., R.T., Peng, S.J., 1986. The development of a closedform analytical model for the stability analysis of nuclear-coupled density-wave oscillations in boiling water nuclear reactors. Nuclear Engineering and Design 92, 253±281. Prasad, R.O.S., Doshi, J.B., Kannan, Iyer, 1995. A numerical investigation of nuclear coupled density wave oscillations. Nuclear Engineering and Design 154, 381±396. Rao, Y.F., Fukida, K., Kaneshima, R., 1995. Analytical study of coupled neutronic and thermodynamic instabilities in a boiling channel. Nuclear Engineering and Design 154, 133±144. Thie, J.A., 1959. Dynamic behavior of boiling reactors. ANL-5849, 51 p. Uehiro, M., Rao, Y.F., Fukuda, K., 1996. Linear stability analysis on instabilities of in-phase and out-ofphase modes in boiling water reactors. Journal of Nuclear Science and Technology 33 (8), 628±635. Van Bragt, D.D.B., Van der Hagen, T.H.J.J., 1998. Stability of natural circulation boiling water reactors: Part I Ð description stability model and theoretical analysis in terms of dimensionless groups. Nuclear Technology 121, 40±51. Van der Hagen, T.H.J.J., 1998. Fuel heat transfer modelling in reduced-order boiling water reactor dynamics models. Annals of Nuclear Energy 25 (16), 1287±1300.