Time-domain linear and non-linear studies on density

Chemical Engineering Science 81 (2012) 118–139

Contents lists available at SciVerse ScienceDirect

Chemical Engineering Science journal homepage: www.elsevier.com/locate/ces

Time-domain linear and non-linear studies on density wave oscillations Davide Papini 1, Antonio Cammi n, Marco Colombo, Marco E. Ricotti Department of Energy, CeSNEF—Nuclear Engineering Division, Politecnico di Milano, Via La Masa 34, 20156 Milano, Italy

H I G H L I G H T S c c c c c

Fundamentals of density wave oscillations for boiling channel systems were studied. Theoretical lumped parameter model developed to predict the instability threshold. Theoretical results compared with numerical analysis by RELAP5 and COMSOL codes. A linear stability analysis led to the definition of the system eigenvalues. State-of-the-art advances on DWO theory development are presented.

a r t i c l e i n f o

a b s t r a c t

Article history: Received 9 February 2012 Received in revised form 23 May 2012 Accepted 5 June 2012 Available online 14 June 2012

Density wave oscillations (DWOs) are investigated in this paper as the most representative instabilities encountered in the boiling systems. This dynamic type instability mode constitutes an issue of paramount interest for the design of industrial systems and equipments, such as steam generators and boiling water nuclear reactor cores. Suited analytical and numerical modelling tools are useful for grasping the fundamental features of this oscillation mode and for predicting the instability threshold dependence on the main system parameters. A theoretical lumped parameter model – moving boundary type – was developed, based on the integration of mass, energy and momentum 1D equations. Homogeneous two-phase flow model has been assumed within the boiling region. Theoretical predictions on DWOs have permitted to investigate the delay propagation phenomena that, in respect of the constant-pressure-drop boundary condition across the channel, trigger the development of self-sustained flow rate oscillations. Instability threshold calculation has permitted to draw stability maps (Npch Nsub stability plane), representing e.g. the parametric effect of the inlet subcooling on the instability inception. Several sensitivity studies have permitted finally to identify in the proper simulation of two-phase frictional pressure drops the most critical issue for a correct prediction of the phenomenon. Theoretical calculations from analytical model were then successfully compared with numerical results obtained with the RELAP5 thermal-hydraulic code and the COMSOL multiphysics code. In addition to homogeneous flow model, a drift-flux model for the two-phase flow was also implemented with the latter approach. Topological characterization of DWO instability phenomena was completed by means of a linear stability analysis leading to the definition of the system eigenvalues, both dealing with the analytical model equations and with the thermal-hydraulic model developed in COMSOL. Linear analysis showed to be a quick and powerful tool generally for instability studies and in particular when addressing the influence of the two-phase friction models. & 2012 Elsevier Ltd. All rights reserved.

Keywords: Mathematical modeling Numerical analysis Non-linear dynamics Linear stability Multiphase flow Boiling channel instability

Abbreviations: ATHLET, analysis of thermal-hydraulics of leaks and transients; ATWS, anticipated transient without scram; BWR, boiling water reactor; DFM, drift-flux model; DWO, density wave oscillation; HEM, homogeneous equilibrium model; IRI, Interfaculty Reactor Institute (Delft University, The Netherlands); IRIS, international reactor innovative and secure; LOCA, loss of coolant accident; LWR, light water reactor; NASA, National Aeronautics and Space Administration; ODE, ordinary differential ¨ equation; PDE, partial differential equation; PDO, pressure drop oscillation; PANDA, PAssive Nachwarmeabfuhr und Druckabbau test Anlage (Passive Decay Heat Removal and Depressurization Test Facility); PSI, Paul Scherrer Institute (Switzerland); RAM, RAMONA (computer programme for BWR transient analysis in the time domain); RELAP, Reactor Excursion and Leak Analysis Programme; ROM, reduced order model; SIET, Societa Informazioni Esperienze Termoidrauliche (Italy) (Company Information and Experiences on Thermalhydraulics); SMR, small modular reactor; UVUT, unequal velocity unequal temperature n Corresponding author. Tel.: þ39 02 2399 6332; fax: þ 39 02 2399 8566. E-mail addresses: [email protected] (D. Papini), [email protected] (A. Cammi). 1 Present address: Nuclear Energy and Safety Research Department, Laboratory for Thermal-Hydraulics, Paul Scherrer Institut (PSI), 5232 Villigen PSI, Switzerland. 0009-2509/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ces.2012.06.005

D. Papini et al. / Chemical Engineering Science 81 (2012) 118–139

1. Introduction Two-phase flow instabilities are one of the most complex and debated topics within the thermal-hydraulic field. Amongst them, key role is played by DWO phenomena that can be established when vapour generation occurs in boiling systems with the boundary condition of pressure drop imposed across the channel (the so named constant-pressure-drop boundary condition) (Yadigaroglu, 1981). In the nuclear area, instability phenomena can be triggered both in boiling water reactor (BWR) fuel channels (where they are moreover coupled through neutronic feedbacks with the neutron field), and in once-through steam generators, which experience boiling phenomena inside parallel tubes (foreseen in all the Generation III þ and Generation IV new nuclear reactors (Cinotti et al., 2002)). When speaking about DWOs, the classical interpretation of the phenomenon ascribes the origin of the instability to waves of ‘‘heavier’’ and ‘‘lighter’’ fluids – leading to density perturbations throughout the channel – and respective delay propagations (Kakac- and Bon, 2008; Yadigaroglu, 1981). The difference in density between the fluid entering the heated channel (subcooled liquid) and the fluid exiting (low density two-phase mixture) triggers delays in the transient distribution of pressure drops along the tube, which may lead to self-sustained oscillations (with single-phase and two-phase pressure drop terms oscillating in counter-phase) (Yadigaroglu, 1981). Physical insight into the distinctive mechanism leading to DWO inception, under the mandatory constant-pressure-drop boundary condition, can be found e.g. in the basic work of Papini et al. (2011b), where parametric discussion concerning the instability threshold behaviour is didactically provided. Extensive attention is required in case of multi-channel systems (reactor cores, steam generators, etc.) because parallel channel instability is difficult to be detected. The total mass flow of the system remains in fact constant while the instability is locally triggered among some of the parallel channels. Thermally induced oscillations of the flow rate and system pressure are undesirable, as they can cause mechanical vibrations, problems of system control, and in extreme cases induce heat transfer surface burn-out. Large amplitude fluctuations in the heater wall temperature (so named thermal oscillations) usually occur under DWO conditions (Kakac- and Bon, 2008). Continual cycling of the wall temperature can lead to thermal fatigue problems which may cause tube failure. Two general approaches are possible to analyse the stability of a boiling channel: i. frequency domain, linearized models; ii. time domain, linear and non-linear models. In frequency domain (Lahey Jr. and Moody, 1977), the system stability is evaluated with classic control-theory techniques that are applied to the transfer functions obtained from linearization and Laplace-transformation of the governing equations. On the other hand, the models built in time domain permit either 0D ˜ oz-Cobo et al., 2002; Schlichting et al., 2010), based analyses (Mun on the analytical integration of conservation equations in the competing regions, or more complex but accurate 1D analyses (Ambrosini et al., 2000; Guo Yun et al., 2008; Zhang et al., 2009), by applying suited numerical solution techniques (finite differences, finite volumes or finite elements). In these models the steady-state is perturbed with small stepwise changes of some operating parameters simulating an actual transient, such as power increase in a real system. The stability threshold is reached when undamped or diverging oscillations are induced. Non-linear features of the governing equations permit indeed to grasp the

119

feedbacks and the mutual interactions between variables triggering a self-sustained density wave oscillation. Notwithstanding lots of lumped-parameter and distributedparameter stability models, both linear and non-linear, have been published since the 1960–1970s (Boure´ et al., 1973; Kakac- and Bon, 2008; Yadigaroglu, 1981), the study of density wave instability in single, parallel twin or multi-channel systems represents ˜ oz-Cobo still nowadays a topical research area. For instance, Mun et al. (2002) applied a non-linear 0D model to the study of out-ofphase oscillations between parallel subchannels of BWR cores. Based on a similar equation solution manifold, Farawila and Pruitt (2006) developed an extended reduced order model (ROM) for use as an accurate quantitative tool for simulating actual reactor situations, both dealing with global oscillation mode and regional oscillation mode. In the framework of the future development of nuclear power plants in China, Guo Yun et al. (2008) and Zhang et al. (2009) investigated DWO instability in parallel multichannel systems by using control volume integrating technique. Schlichting et al. (2010) analysed the interaction of PDOs and DWOs for a typical NASA type phase change system for space exploration applications. Finally, modern methods for non-linear stability analysis of boiling water nuclear reactors were proposed by Dokhane et al. (2005, 2007) and Rizwan-Uddin (2006). Stateof-the art advancements on stability and semi-analytical bifurcation analyses of BWRs can be found in the work of Lange et al. (2011). On the other hand, qualified numerical simulation tools can be successfully applied to the study of boiling channel instabilities. Whether the adoption of simplified analytical models is well established to study basic thermal-hydraulic phenomena, the use of complex numerical system codes still represents a reliable option, since accurate quantitative predictions can be provided with simple and straightforward nodalizations. Within this respect, it is worth mentioning that a BWR stability analysis methodology was developed at Paul Scherrer Institut (PSI) based on the BWR system code RAMONA (Hennig, 1999). A novel ¨ approach has been then proposed at Technische Universitat Dresden by considering both integrated BWR system codes (e.g., RAMONA) and reduced order models (ROMs) – i.e., the so called RAM-ROM method – as complementary tools to examine the stability characteristics of fixed points and periodic solutions of the non-linear differential equations describing a BWR loop (Lange et al., 2011). However, several numerical studies published on DWOs featured the RELAP5 thermal-hydraulic code as the main analysis tool (Ambrosini et al., 2000; Ambrosini and Ferreri, 2006; Colombo et al., 2012). Numerical works available from literature address a single channel configuration by working with an imposed DP, kept constant between two headers throughout the simulation. Amongst them, Ambrosini and Ferreri (2006) performed a detailed analysis about thermal-hydraulic instabilities in a boiling channel using the RELAP5/MOD3.2 version. The authors demonstrated the capability of the RELAP5 system code to detect the onset of DWO instability. The multipurpose COMSOL Multiphysicss numerical code can be applied to study the characteristics of boiling systems too. In this respect, linear and non-linear stability analyses have been provided by Schlichting et al. (2007), who developed a 1D drift-flux model applied to instability studies on a boiling loop for space applications. At last, the capabilities of the thermal-hydraulic system code ATHLET in calculating BWR natural circulation unstable behaviour have been successfully tested by Paladino et al. (2008), on the basis of largescale experiments in the PANDA facility at PSI. Final objective of the modelling on density wave instabilities developed in the present work was to prepare (pre-test analyses) and interpret (post-test analyses) an experimental campaign carried out at SIET labs (Piacenza, Italy). Experiments were

120


focused on parallel channel instability phenomena, directly investigated with a test section reproducing in full scale two helical tubes of the IRIS steam generator (Papini et al., 2011a). Due to the complexity of the helical geometry, the basic experimental investigation provided is of relevance for the diffusion of such helically coiled steam generators. As suggested also in the review work of Goswami and Paruya (2011), theoretical study on density wave instabilities was dealt with by following the sound ‘‘moving boundary’’ modelling approach, widely applied to the dynamics of steam generators (Abdalla, 1994; Jensen and Tummescheit, 2002; Li et al., 2008; Papini, 2011) and required to reasonably replicate real-time situations involving a continuous movement of the boiling boundary (if compared to fixed boundary problems). Main idea is to dynamically track the length of the different regions of the heat exchanger (subcooled liquid zone, two-phase zone and superheated zone) by means of lumped parameter model, described by ordinary differential equations (ODEs) obtained after analytical integration of the partial differential equations (PDEs) governing the fluid flow. Moreover, this type of modelling consists of low-order numerical systems that are well suited for control design issues. Nevertheless, it is fundamental to notice that, being the final goal the study of DWO phenomena in complex geometry (helical coil tubes), a good practice requires to start from the simplest configuration case, hence the developed modelling efforts started unavoidably from the simplifying case of vertical tube geometry. This paper is focused on a topological characterization of density wave instabilities, achieving a satisfying validation of the modelling in case of vertical tube channels, prior to investigate the influence of the helical coil tube geometry on instability occurrence. At the end, a benchmark of different theoretical and numerical calculations of the instability threshold is proposed. Both the studies of a single boiling channel configuration (experimentally provided using a large bypass tube connected to the heated channel to maintain the constant-pressure-drop boundary condition (Saha et al., 1976)) and a parallel channel configuration (experimentally provided connecting two or more channels, e.g. steam generator tubes, via common headers (Masini et al., 1968)) were addressed. Major attention is given to the mathematical aspects of the developed models, providing in details the derivation of the non-linear and linear nodal models (Section 2). Oscillating transient response and stability maps are presented and discussed in Section 3. Section 4 briefly introduces the numerical studies on DWOs using the RELAP5 and COMSOL codes, compared finally with the theoretical results from analytical model and with classical experimental findings. Though the basic nature of the provided models, main achievements richer than what previously published (Goswami and Paruya, 2011; Kakac- and Bon, 2008; Yadigaroglu, 1981) are finally resumed in Section 5.

2. Analytical lumped parameter model: fundamentals and development The analytical model proposed to study DWO instabilities is ˜ oz-Cobo et al. (2002), close to the IRIbased on the work of Mun Delft ROM developed by Van Bragt and Van Der Hagen (1998). Proper modifications have been considered to fit the modelling approach with steam generator tubes with imposed thermal power (representative of typical experimental facility conditions). The developed model is based on a lumped parameter approach (0D) for the two zones characterizing a single boiling channel, which are single-phase region and two-phase region, divided by the boiling boundary. Modelling approach is schematically illustrated in Fig. 1.

Fig. 1. Schematic diagram of a heated channel with single-phase (0 o zo zBB) and two-phase (zBB o zo H) regions. Externally impressed pressure drop is DPtot.

Differential conservation equations of mass and energy are considered for each region, whereas momentum equation is integrated along the whole channel. Wall dynamics is accounted for in the two distinct regions, following lumped wall temperature dynamics by means of the respective heat transfer balances. The model can apply to single boiling channel and two parallel channel configurations, suited both for instability investigation according to the specification of the respective boundary conditions: i. constant DP across the tube for single channel; ii. same DP(t) across the two channels (with constant total mass ˜ oz-Cobo et al., 2002). flow) for parallel channels (Mun The main assumptions considered in the provided modelling are: (a) one-dimensional flow (straight tube geometry); (b) homogeneous two-phase flow model; (c) uniform heating along the channel (linear increase of quality with tube abscissa z); (d) system of constant pressure (pressure term is neglected within the energy equation); (e) constant fluid properties at given system inlet pressure; (f) subcooled boiling is neglected.

2.1. Mathematical modelling Modelling equations are derived by the continuity of mass and energy for a single-phase fluid and a two-phase fluid, respectively. Single-phase flow equations read: @r @G ¼0 þ @z @t

ð1Þ


@ðrhÞ @ðGhÞ þ ¼ Q 000 @t @z

ð2Þ

Two-phase mixture is dealt with according to homogeneous flow model. By defining the homogeneous density rH and the reaction frequency O (Lahey Jr. and Moody, 1977; Wallis, 1969) as follows:

rH ¼ rf ð1aÞ þ rg a ¼ OðtÞ ¼

1 vf þxvf g

ð3Þ

Q ðtÞvf g AHhf g

ð4Þ

one gets: ð5Þ

@j ¼ OðtÞ @z

ð6Þ

Momentum equation is accounted for by integrating the pressure balance along the channel: Z H @Gðz,tÞ dz ¼ DPðtÞDP acc DPgrav DP f rict ð7Þ @t 0 As concerns the wall dynamics modelling, a lumped tworegion approach is adopted. Heated wall dynamics is evaluated separately for single-phase and two-phase regions, following the dynamics of the respective wall temperatures according to a heat transfer balance: dT dQ 1f 1f 1f 1f ¼ M h ch h ¼ Q 1f ðhSÞ1f T h T f l dt dt

ð8Þ

dT dQ 2f 2f 2f 2f ¼ M h ch h ¼ Q 2f ðhSÞ2f T h T f l dt dt

ð9Þ

1f

2f

2.1.1. Mass-energy model in the two-phase region The continuity equation for a homogeneous two-phase mixture – Eq. (6) – is integrated to calculate the average mass flux G2j in the two-phase region (to be used for the calculation of two-phase frictional pressure drops). The well-known expression of the total volumetric flux is reminded: ! x 1x j¼ G2f ðz,tÞ þ ð10Þ

rg

DPðtÞ ¼ P in P ex

G2f ðz,tÞ ¼

OðtÞ½zzBB ðtÞ þvf Gin ðtÞ vf þ xðz,tÞvf g

ð11Þ

Eq. (11) can be then used to calculate the mass flux at channel exit Gex (with z¼H), and the average mass flux in the two-phase region G2f (by applying the theorem of integral average):

O½HzBB þ vf Gin

ð12Þ

vf þ xex vf g

G2 f ¼

1 HzBB

Z

vf þ xex vf g

H

G2f ðz,tÞdz ¼

The gravitational term of pressure drops DPgrav can be represented as ð16Þ

The frictional term of pressure drops DPfrict can be represented as (by adding single-phase and two-phase contributions, both concentrated and distributed ones): 2 z G2in HzBB D 2 E G2f G2 DPf rict ¼ kin þ f BB þf F þ kex F2ex ex ð17Þ D 2rf D 2rf 2rf Left-hand-side term of the pressure drop balance – Eq. (7) – is treated by splitting the integration into two intervals, from 0 to the boiling boundary zBB, and from zBB to the channel length H, and applying for the second interval the Leibniz rule: Z Z H @G dG d H dzBB dz ¼ zBB in þ ð18Þ G2f dzþ Gin @t dt dt dt 0 zBB Mass flux integral along two-phase region is computed according to Eq. (13), giving: Z d H dzBB dxex dG dO þb2 þ b3 in þ b4 ð19Þ G dz ¼ b1 dt zBB 2f dt dt dt dt where the following coefficients are defined: vf g vf 2O½HzBB 2O½HzBB þ ln 1þ xex Gin b1 ¼ xex vf g vf g xex vf vf g xex b2 ¼

ð20Þ

vf g O½HzBB 2 vf ½HzBB 2O½HzBB þ ln 1þ x G ex in vf g xex vf vf g x2 vf g x2ex ex vf ½HzBB O½HzBB Gin þ ð21Þ xex ðvf þ vf g xex Þ xex vf g

b3 ¼

vf ½HzBB vf g ln 1 þ xex vf g xex vf

ð22Þ

b4 ¼

vf vf g ½HzBB 2 1 ln 1 þ xex vf g xex vf g xex vf

ð23Þ

Finally, the equation governing the rate of change of mass flux at the inlet of the channel is obtained substituting Eq. (19) into Eq. (7): ðb3 þ zBB Þ

dGin dzBB ¼ DPðtÞDP acc DP grav DP f rict ðGin þ b1 Þ dt dt dxex dO b4 b2 dt dt

ð24Þ

This equation permits to know the dynamics of the flow rate entering the respective channel, once they are defined: i. the dynamics of the boiling boundary zBB; ii. the dynamics of the exit quality xex; iii. the dynamics of the reaction frequency O (from heated wall model).

O½HzBB

xex vf g vf g O½HzBB Gin xex ln 1 þ xex vf g vf zBB

ð14Þ

The accelerative term of pressure drops DPacc can be represented as (more generally than homogeneous flow model assumption): " # x2ex ð1xex Þ2 G2 DPacc ¼ G2ex þ ð15Þ in aex rg ð1aex Þrf rf

rf

Integrating Eq. (6) from the boiling boundary position zBB(t) to an arbitrary position z, and accounting for Eq. (10), one obtains after some algebra:

Gex ¼

2.1.2. Integration of momentum equation Within Eq. (7), DP(t) is the difference between the inlet and outlet channel pressure:

DPgrav ¼ g rf zBB þg ð1aÞrf ½HzBB þ g arg ½HzBB

@rH @G ¼0 þ @z @t

121

ð13Þ

2.1.3. Boiling boundary dynamics Boiling boundary dynamics can be easily obtained by integrating the energy equation – Eq. (2) – in the single-phase region

122


(Van Bragt and Van Der Hagen, 1998): Z zBB Z zBB Q @ðGhÞ @ðrhÞ dz ¼ dz V ch @z @t 0 0

ð25Þ

Leibniz rule is applied to the right-hand-side term of Eq. (25). The strong hypothesis that local enthalpy changes simultaneously at all the axial locations (with average value h ¼ ðhin þ hf Þ=2) is assumed. It is just mentioned that this approximation is reasonable at low frequency. Integrating Eq. (25) and after some algebra, one obtains: dzBB 2Gin 2Q zBB ¼

¼ b8 dt rf hf hin V ch rf

ð26Þ

2.1.4. Exit quality dynamics Exit quality dynamics is obtained from the formula that relates the void fraction a to the quality, with homogenous flow model:

a¼

gx

1 þ g1 x

ð27Þ

where

g¼

rf vg ¼ vf rg

ð28Þ

The average void fraction a is computed as " # Z xex ðtÞ

1 g 1 1

a¼ a dx ¼ ln 1 þ g1 xex xex ðtÞ 0 g1 g1 xex

ð29Þ

If Eq. (29) is derived with respect to time, and two-phase mass conservation equation – Eq. (5) – is integrated along the boiling region, one has: da dxex ¼ b5 dt dt

ð30Þ

da Gin Gex a dzBB þ ¼ dt Hz BB dt rg rf ½HzBB

ð31Þ

with b5 ¼

"

g

g1 2 xex

# g1 ln 1þ g1 xex

þ xex 1 þ g1 xex

ð32Þ

ð33Þ

where b9 ¼ b6 þ b7 b8

Gin vf g xex O½HzBB

½HzBB vf þ vf g xex rf g b5

b7 ¼

g

"

1 g1 ½HzBB b5

#

ln 1 þ g1 xex

g1 xex

h

i 2f 2f Q 2f ðhSÞ2f T h T f l

ð39Þ

Rearranging Eq. (37) on the account of Eqs. (38) and (39), one has: i vf g 1 h 1f 1f 2f 2f b10 ¼ M h ch b10 þ M h ch b10 ð40Þ hf g AH In the above equations, specific empirical correlations are considered to calculate single-phase and two-phase heat transfer coefficients, respectively the Dittus-Boelter and the Kandlikar equations (Incropera et al., 2007). 2.2. Model development Model construction is based on the ODEs drawn in Section 2.1, and descriptive of the hydraulic and thermal behaviour of a single boiling channel. In case of single boiling channel modelling, boundary condition of constant pressure drop DP(t) between channel inlet and outlet must be simply introduced by specifying the imposed DP of interest within the momentum balance equation – Eq. (24). In case of two parallel channel modelling, the boundary condition is dealt imposing: (i) the same pressure drop dependence with time across the two channels; (ii) a constant total flow ˜ oz-Cobo et al., 2002): rate (Mun

DP1 ðtÞ ¼ DP2 ðtÞ ¼ DPðtÞ

ð41Þ

Gin,1 ðtÞ þ Gin,2 ðtÞ ¼ const

ð42Þ

With respect to single boiling channel case, the comprehensive model includes a set of 5 non-linear ordinary differential equations, in the form of dZi ¼ f i ðZÞ, dt

i ¼ 1,2,. . .,5

Z1 ¼ zBB , Z2 ¼ xex , Z3 ¼ Gin , Z4 ¼ T 1hf , Z5 ¼ T 2hf

ð43Þ

ð44Þ

With respect to parallel channel case, the thermal-hydraulic ˜ ozbehaviour of two channels is combined as suggested by Mun Cobo et al. (2002), such that the comprehensive model includes a set of 9 non-linear ordinary differential equations, in the form of: dZi ¼ f i ðZÞ, dt

ð35Þ

Z1 ¼ zBB,1 , Z2 ¼ zBB,2 , Z3 ¼ xex,1 , Z4 ¼ xex,2 , Z5 ¼ Gin,1 f 2f 1f 2f Z6 ¼ T 1h,1 , Z7 ¼ T h,1 , Z8 ¼ T h,2 , Z9 ¼ T h,2

i ¼ 1,2,. . .,9

ð45Þ

where the state variables are:

ð36Þ

2.1.5. Reaction frequency dynamics The reaction frequency dynamics is given as vf g 1 dQ dO ¼ b10 ¼ dt hf g AH dt

1 M 2hf ch

ð34Þ

with b6 ¼

2f

b10 ¼

where the state variables are:

Rearranging Eqs. (29)–(31), the dynamics of the exit quality is finally obtained: dxex ¼ b9 dt

following two coefficients (single-phase and two-phase wall portion, respectively): i 1 h 1f 1f 1f 1f b10 ¼ 1f ð38Þ Q ðhSÞ1f T h T f l M h ch

ð37Þ

thus, it depends on the model adopted for the heated wall. According to the two-node lumped approach chosen in this work – Eqs. (8) and (9) –, heated wall dynamics is governed by the

ð46Þ

Thermal power Q, inlet loss coefficient kin, and exit loss coefficient kex are defined as system inputs in both cases. Besides, in case of single channel modelling, also the imposed DP is treated as a system input. For the sake of clarity, all the coefficients introduced within the modelling equations – with respective meanings – are reported in Table 1. Non-linear features of the provided model are apparent. First, steady-state conditions of the analysed system are calculated by solving the whole set of equations with time derivative terms set to zero. Steady-state solutions are then used as initial conditions for the integration of the equations, obtaining


123

Table 1 List of the coefficients defined throughout the development of the modelling equations. b1 ¼

vf g vf 2O½HzBB 2O½HzBB þ ln 1 þ xex Gin vf vf g xex xex vf g vf g xex

relates the dynamics of the boiling boundary to the dynamics of two-phase mass flux (integral) vf g vf ½HzBB O½HzBB 2 vf ½HzBB 2O½HzBB O½HzBB Gin þ ln 1 þ xex Gin þ vf g xex xex vf g vf xex ðvf þ vf g xex Þ vf g x2ex vf g x2ex

b2 ¼

relates the dynamics of the exit quality to the dynamics of two-phase mass flux (integral) vf ½HzBB vf g ln 1 þ xex vf vf g xex

b3 ¼

relates the dynamics of inlet mass flux to the dynamics of two-phase mass flux (integral) vf vf g ½HzBB 2 1 ln 1 þ xex vf g xex vf g xex vf

b4 ¼

relates the dynamics of the reaction frequency #to the dynamics of two-phase mass flux (integral) "

g1 ln 1 þ g1 xex g

þ 2 x g 1 x 1þ ex ex g1 xex

b5 ¼

relates the dynamics of the exit quality to the dynamics of the average void fraction Gin vf g xex O½HzBB

½HzBB vf þ vf g xex rf g b5

b6 ¼

coefficient within exit dynamics# " quality

ln 1 þ g1 xex g

1 b7 ¼

g1 ½HzBB b5 g1 xex coefficient within exit quality dynamics 2Gin 2Q zBB

rf hf hin V ch rf

b8 ¼

boiling boundary dynamics b9 ¼ b6 þ b7 b8 exit quality dynamics vf g 1 dQ hf g AH dt

b10 ¼

reaction frequency dynamics i 1 h 1f 1f 1f Q ðhSÞ1f T h T f l 1f Mh ch

1f

b10 ¼

dynamics of wall temperature in the single-phase region i 1 h 2f 2f 2f Q ðhSÞ2f T h T f l 2f Mh ch

2f

b10 ¼

dynamics of wall temperature in the two-phase region 1

DPDP acc DP grav DP f rict ðGin þ b1 Þb8 b2 b9 b4 b10 b11 ¼ b3 þ zBB inlet mass flux dynamics (under constant DP boundary conditions)

1

DP acc,2 DPacc,1 þ DP grav,2 DP grav,1 þ DP f rict,2 DP f rict,1 þ Gin,2 þ b1,2 b8,2 Gin,1 þ b1,1 b8,1 þ b2,2 b9,2 b2,1 b9,1 þ b4,2 b10,2 b4,1 b10,1 b11,1 ¼ b3,1 þ zBB,1 þ b3,2 þ zBB,2 mass flux dynamics at the inlet of channel 1 (under parallel channel boundary conditions)

the time evolution of each computed state variable. Input variable perturbations can be introduced both in terms of step variations and ramp variations. The described dynamic model was solved through the use of the MATLAB software SIMULINKs (The Math Works, Inc., 2005). A suitable set of Matlab Functions has been implemented to compute at each time step the dynamic coefficients listed in Table 1, based on the values of the state variables at the previous time step. Integrator function block has been finally used to integrate the whole set of non-linear ODEs—Eqs. (43), (45). Graphical interface of the developed model is depicted in Figs. 2 and 3, with reference to single channel and two parallel channel configurations respectively.

To simplify the calculations, modelling equations are linearized with respect to the three state variables representing the hydraulic behaviour of a single channel, i.e. the boiling boundary zBB(t), the exit quality xex(t), and the inlet mass flux Gin(t). That is, linear stability analysis is presented by neglecting the dynamics of the heated wall (Q(t)¼ const). The initial ODEs – obtained after integration of the original governing PDEs – are (Papini, 2011): Mass-energy conservation equation in the single-phase region: dzBB ¼ b1 dt Mass-energy conservation equation in the two-phase region: dxex dzBB ¼ b4 ¼ b2 þb3 dt dt

2.3. Linear stability analysis

ð48Þ

ð49Þ

Momentum conservation equation (along the whole channel): Modelling equations can be linearized to investigate the neutral stability boundary of the nodal model. The linearization about an unperturbed steady-state initial condition is carried out by assuming for each state variable: 0

lt

ZðtÞ ¼ Z þ dZ e

ð47Þ

dGin ¼ b5 dt

ð50Þ

By applying Eq. (47) to the selected three state variables, as: zBB ðtÞ ¼ z0BB þ dzBB elt

ð51Þ

124


Fig. 2. SIMULINKs model scheme (single channel configuration).

xex ðtÞ ¼ x0ex þ dxex elt

ð52Þ

Gin ðtÞ ¼ G0in þ dGin elt

ð53Þ

the resulting linear system can be written in the form of:

dzBB E11 þ dxex E12 þ dGin E13 ¼ 0

ð54Þ


ð55Þ


ð56Þ

The calculation of the system eigenvalues is based on solving: E11 E12 E13 E ð57Þ 21 E22 E23 ¼ 0 E31 E32 E33 which yields a cubic characteristic equation, where l are the eigenvalues of the system:

l3 þ al2 þbl þ c ¼ 0

ð58Þ

Computational details of the linearization process can be found in Appendix A.

3. Analytical lumped parameter model: results and discussion Single boiling channel configuration is considered for the discussion of the results obtained by the developed model on DWOs (Papini et al., 2011b), and this configuration is referenced throughout this section. For the sake of simplicity, and availability

of similar works in the open literature for validation purposes (Ambrosini et al., 2000; Ambrosini and Ferreri, 2006), typical dimensions and operating conditions of classical BWR core subchannels are considered. Table 2 lists the geometrical and operational values taken into account in the following analyses. 3.1. System transient response To excite the unstable modes of density wave oscillations, input thermal power was increased starting from stable stationary conditions, step-by-step, up to the instability occurrence. Instability threshold crossing is characterized by passing through damping out oscillations (Fig. 4(a)), limit cycle oscillations (Fig. 4(b)), and divergent oscillations (Fig. 4(c)). This process is rather universal across the boundary. From stable state to divergent oscillation state, a narrow transition zone of some kW was found in this study. The analysed system is non-linear and pretty complex. Trajectories on the phase space defined by boiling boundary zBB vs. inlet mass flux Gin are reported in Fig. 4 too. The operating point on the stability boundary (Fig. 4(b)) is the cut-off point between stable (Fig. 4(a)) and unstable (Fig. 4(c)) states. This point can be looked as a bifurcation point. The limit oscillation is a quasi-periodic motion; the period of the depicted oscillation is rather small (less than 1 s), due to the low subcooling conditions considered at inlet. With reference to the eigenvalue computation, by solving Eq. (58), at least one of the eigenvalues is real, and the other two can be either real or complex conjugate. The linear stability


125

Fig. 3. SIMULINKs model scheme (two parallel channels configuration).

Table 2 Dimensions and operating conditions selected for the analyses. Heated channel Diameter (m) Length (m)

0.0124 3.658

Operating parameters Pressure (bar) Inlet temperature (1C) kin kex

70 151.3–282.3 23 5

boundary is reached when the complex conjugate eigenvalues become purely imaginary (i.e., the real part is zero). Crossing the instability threshold is characterized by passing to positive real part of the complex conjugate eigenvalues, which is at the basis of the diverging response of the model under unstable conditions.

3.2. Description of a self-sustained DWO The simple two-node lumped parameter model developed in this work is capable to catch the basic phenomena of density wave oscillations. Numerical simulations have been used to gain insight into the physical mechanisms behind DWOs, as discussed in this section. The analysis has shown good agreement with some findings due to Rizwan-Uddin (1994). Fully developed DWO conditions in a single boiling channel are considered. By analysing an inlet velocity

variation and its propagation throughout the channel, particular features of the transient pressure drop distributions are depicted. The starting point is taken as a variation (increase) in the inlet velocity. The boiling boundary responds to this perturbation with a certain delay (Fig. 5), due to the propagation of an enthalpy wave in the single-phase region. The propagation of this perturbation in the two-phase zone (via quality and void fraction perturbations) results in further lags of two-phase average velocity and exit velocity (Fig. 6). All these delayed effects combine in single-phase pressure drop term and two-phase pressure drop term acquiring 1801 out-ofphase fluctuations (Fig. 7). What is interesting to notice, indeed, is that the 1801 phase shift between single-phase and two-phase pressure drops is not perfect (Rizwan-Uddin, 1994). Due to the delayed propagation of inlet velocity perturbation, single-phase term increase is faster than two-phase term rising. In some operating conditions, the superimposition of the two oscillations is such to create a total pressure drop along the channel oscillating as a nonsinusoidal wave. The peculiar trend obtained is shown in Fig. 8; respective oscillation shape has been named ‘‘shark-fin’’ shape. Such behaviour found indeed corroboration in the experimental evidence collected with the facility at SIET labs (Papini et al., 2011a). In Fig. 9 an experimental recording of channel total pressure drops is depicted. The experimental pressure drop oscillation shows a fair qualitative agreement with the phenomenon of ‘‘shark-fin’’ shape described theoretically. 3.3. Sensitivity analyses and stability maps In order to provide accurate quantitative predictions of the instability thresholds, and of their dependence with the inlet

126


HEM Model - Nsub = 3 - 136 kW Phase space trajectory 0.668

882

0.667

880

0.666 0.665

878

zBB [m]

Inlet mass flux Gin [kg/m2s]

HEM Model - Nsub = 3 - Ramp response to 136 kW 884

876

0.664 0.663

874

0.662

872

0.661

870 80

100

120

140

160

180

200

0.66 870

220

872

HEM Model - Nsub = 1.5 - Ramp response to 180 kW

710 708 706

876

878

880

HEM Model - Nsub = 1.5 - 180 kW Phase space trajectory 0.198

705 704

0.1978

703

0.1976

702

zBB [m]



712

874

Gin [kg/m2s]

Time [s]

701 130

132

134 136 Time [s]

138

140

704

0.1974 0.1972 0.197

702 0.1968 700 100

110

120 Time [s]

130

140

700

150

702

703

704

705

Gin [kg/m2s]

HEM Model - Nsub = 3 - Ramp response to 138 kW

HEM Model - Nsub= 3 - 138 kW Phase space trajectory

1100

0.9

1000

0.8

900

zBB [m]


701

800

0.7

0.6

700

0.5

600 50

100

150

200

Time [s]

0.4 400

600

800 Gin [kg/m2s]

1000

1200

Fig. 4. Inlet mass flux oscillation curves and corresponding trajectories in the phase space: (a) stable state; (b) neutral stability boundary; (c) unstable state.

subcooling to draw a stability map (as the one commonly drawn in the Npch Nsub stability plane (Ishii and Zuber, 1970)), it is first necessary to identify most critical modelling parameters that have deeper effects on the results.

Several sensitivity studies were carried out on the empirical coefficients used to model the two-phase flow structure. In particular, specific empirical correlations have been accounted for within the momentum balance equation to represent two-phase frictional


1.1

127

x 104 Gin

zBB

7 6

1

Pressure drops [Pa]

Non-dimensional value

1.05

0.95 0.9 0.85 0.8 225

5 Single-phase

4

Two-phase 3 2

230

235

1

240

Time [s]

247

Fig. 5. Non-dimensional inlet mass flux and boiling boundary. Nsub ¼8 and Q¼ 133 kW.

Gav-tp

249

250 251 Time [s]

252

Gex

8.1

254

x 104

1150

Total

1100 Pressure drops [Pa]

Mass flux [kg/m2s]

253

Fig. 7. Oscillating pressure drop distributions. Nsub ¼ 2 and Q¼ 103 kW.

1200 Gin

248

1050 1000 950 900 850 225

230

235

240

Time [s] Fig. 6. Mass flux delayed variations throughout the channel. Nsub ¼ 8 and Q¼ 133 kW.

pressure drops (by testing several correlations for the two-phase friction factor multiplier F2lo ). In this respect, a comparison of the considered friction models is provided in Table 3: Homogeneous equilibrium pressure drop model (HEM), Lockhart-Martinelli multiplier (Todreas and Kazimi, 1993), Jones expression of Martinelli-Nelson method (Todreas and Kazimi, 1993) and Friedel correlation (Friedel, 1979) are selected, respectively, for the analysis. The various formulations are indeed shown in Table 4, where F2lo ¼ DP tp =DP lo . It is worth noticing that in this case the main contribution to channel total pressure drops is given by the two-phase terms, both frictional and in particular concentrated losses at channel exit (nearly 40– 50%). The fractional distribution of the pressure drops along the channel plays an important role in determining the stability of the system. Concentration of pressure drops near the channel exit is such to render the system prone to instability: hence, DWOs triggered at low qualities may be expected with the analysed system. The effects of two-phase frictions on the instability threshold (discussed also by Furutera (1986) and Dokhane et al. (2005)) are evident from the stability maps shown in Fig. 10. The higher are the two-phase friction characteristics of the system (that is, with

8.05

8

7.95 247

248

249

250 251 Time [s]

252

253

254

Fig. 8. Total pressure drop oscillation showing ‘‘shark-fin’’ shape. Nsub ¼2 and Q¼ 103 kW.

Lockhart-Martinelli and Jones models), the most unstable results the channel (being the instability induced at lower thermodynamic quality values). Moreover, RELAP5 calculations about DWO occurrence in the same system (Colombo et al., 2012) are reported as well (see Section 4). In these conditions, Friedel correlation for two-phase multiplier is the preferred one. The influence of the two-phase friction losses on the system stability (via the channel pressure drop distribution) is made apparent also in terms of eigenvalue computation. Fig. 11 reports the results of the linear stability analysis corresponding to the four cases depicted in Table 3. With reference to the outlined conditions, ‘‘high friction’’ models (Lockhart-Martinelli and Jones ones) turn out into unstable operating mode (with complex conjugate eigenvalues showing positive real part), whereas according to ‘‘low friction’’ models (HEM and Friedel ones) the same void generation is still not enough to induce the instability. Moreover, linear stability results are presented reporting in Fig. 12 the neutral stability boundaries obtained with respect to two different frictional models. In particular, Jones correlation (as ‘‘high friction’’ model) and HEM assumption (as ‘‘low friction’’

128


200 190

Total Channel Δp [kPa]

180 170 160 150 140 130 120 110 100 0

10

20

30 t [s]

40

50

60

Fig. 9. Experimental recording of total pressure drop oscillation showing ‘‘sharkfin’’ shape (collected at SIET labs). P¼ 80 bar and Nsub ¼5.1.

Table 3 Fractional contributions to total channel pressure drop (at steady-state conditions). HEM Term

DP (kPa)

DPgrav DPacc DPin DPfrict,1f DPfrict,2f DPex

12.82 10.24 15.35 0.96 10.61 24.06

DPtot

74.03

Lockhart-Martinelli % of total 17.31 13.84 20.74 1.29 14.33 32.50 100

Jones Term

DPgrav DPacc DPin DPfrict,1f DPfrict,2f DPex DPtot

DP (kPa) 12.82 10.24 15.35 0.96 23.54 54.07 116.97

DP (kPa) 12.82 10.24 15.35 0.96 39.84 81.73 160.94

% of total 7.96 6.36 9.54 0.59 24.75 50.79 100

Friedel % of total 10.96 8.76 13.12 0.82 20.12 46.22 100

DP (kPa) 12.82 10.24 15.35 0.96 14.97 33.36 87.69

% of total 14.62 11.68 17.51 1.09 17.07 38.04 100

as summarized also by Goswami and Paruya (2011). Fig. 12 shows indeed the comparison with the distributed-parameter model of Ambrosini et al. (2000), based on HEM formulation of two-phase flow and subsequent linearization of the discretized equations. Though the different behaviour at low subcooling (explainable by the distributed parameter features of the two-phase zone in the referenced model), the agreement is excellent in terms of instability quality line prediction. Finally, the frictional model is effective besides on the predicted oscillation period (Fig. 13). The period increases when a larger fraction of the total pressure drops is concentrated near the channel exit (that is, with Lockhart-Martinelli and Jones models), as more delayed and slowed down feedbacks are supposed to trigger the instability. On the whole, the calculations of the oscillation period are in good agreement with literature results (Ambrosini et al., 2000; Ambrosini and Ferreri, 2006), as the period (nearly twice the mixture transit time according to the two density perturbations at the basis of the phenomenon) grows with the subcooling number. This is confirmed by the comparison with RELAP5 calculations (Colombo et al., 2012), though a slight underestimation of the oscillation period can be inferred with the analytical model, in particular when ‘‘low friction’’ is considered. Some evident discrepancies are depicted just at very low inlet subcooling (Nsub o1), where T/t drops considerably. This simple two-node lumped parameter model sounds less accurate when the second node (two-phase region) is too large. Better simulations of the oscillation period may be expected by increasing the number of nodes in the two-phase zone. Mixture transit time in the heated channel has been calculated with classical homogeneous flow theory (Masini et al., 1968; Yadigaroglu, 1981).

4. Numerical modelling with computer codes In this section several advances reached when applying timedomain numerical codes to predict the inception of DWOs in boiling systems are discussed. Dimensions and operating conditions listed in Table 2, referring to classical BWR core subchannel, are considered in the analysis. Both, utilization of the RELAP5 thermal-hydraulic code (U.S. NRC, 2001) – largely adopted in the nuclear field of light water reactors (LWRs) – and multipurpose numerical tool COMSOL Multiphysicss (COMSOL, Inc., 2008) have been accounted for. According to the theoretical tone of this work, mostly the mathematical bases of the developed numerical models are addressed in the followings. A final benchmark, aimed at validating the theoretical analytical model predictions presented in Section 3, is at last presented.

Test case: G ¼ 0.12 kg/s; Tin ¼239.2 1C; Q¼ 100 kW (xex ¼0.40).

4.1. Numerical modelling by means of RELAP5 code model) are considered. The effect of two-phase frictions on the instability threshold is confirmed. Again, the higher are the twophase frictions (that is, with Jones model), the smaller is the stable region. Considering the calculated eigenvalues in response to thermal power variations, it is noticed that, within the stable operating region, the imaginary part reduces as the power is decreased (hence, the amplitude of the oscillations during transients). By progressively reducing the thermal power, the imaginary part vanishes, resulting in five purely real negative eigenvalues (with no damped flow oscillations). Further reduction of thermal power may lead (in particular at high subcooling) to positive real part appearance. This behaviour, occurring at low qualities in the high subcooling zone of the stability plane, is ascribed to the prediction of Ledinegg unstable operating points (Ledinegg, 1938). Confirmations of this outcome can be found in the works of Ambrosini et al. (2000) and Ambrosini and Ferreri (2006), where similar ‘‘nose’’ of the stability boundary is depicted,

4.1.1. RELAP5 models and numerical settings The RELAP5 code (U.S. NRC, 2001) is a transient analysis code for complex systems, developed within the nuclear engineering research field, widely used for safety analysis application and mainly designed as the reference thermal-hydraulic code for calculations with water–steam mixture in 1D ‘‘pipe-oriented’’ systems. Specific applications of RELAP5 include best-estimate transient simulation of LWR coolant systems during postulated accidents, such as loss of coolant accidents (LOCAs), anticipated transients without scram (ATWS), and operational transients such as loss of offsite power, loss of feedwater, station blackout, and turbine trip (U.S. NRC, 2001). However, RELAP5 is a highly generic code that can be used also for simulation of a wide variety of hydraulic and thermal transients in both nuclear and non-nuclear systems, involving mixtures of steam, water and non-condensable gas. RELAP5 is based on a non-homogeneous non-equilibrium set


129

Table 4 List of the two-phase friction factor multipliers considered in the analysis. Multiplier formulaa

Friction model

Homogeneous (HEM) (Todreas and Kazimi, 1993)

vf g vf

F2lo ¼ 1 þ x

"

Computational notes

1þx

mf g mf

!#n

– n ¼0.25 for Re r 2 104 (Blasius correlation) – n ¼0.2 for Re 42 104 (McAdams correlation)

!

"

Jones (Martinelli-Nelson method) (Todreas and Kazimi, 1993)

F2lo ¼ OðP,GÞ 1:2

#

rf 1 x0:824 þ 1:0 rg

– P is expressed in psia – G is expressed in lb/(ft2h)

8 > < 1:36 þ 0:0005P þ 0:1 10G6 0:000714P 10G6 if 10G6 r 0:7 6 6 OðP,GÞ ¼ > : 1:260:0004P þ 0:119 10G þ 0:00028P 10G if G6 40:7 10

Lochkart-Martinelli (Todreas and Kazimi, 1993)

F2l ¼ 1 þ

20 1 þ X tt X 2tt

– ‘‘Only-liquid’’ multiplier formula – To pass to ‘‘liquid-only’’ mode: (Todreas and Kazimi, 1993)

F2lo ¼ F2l ð1xÞ2n 3:23FH Fr 0:045 We0:035 rf f go E ¼ ð1xÞ2 þ x2 F ¼ x0:78 ð1xÞ0:224 rg f lo !0:91 !0:19 !0:7

Friedel (Friedel, 1979)

F2lo ¼ E þ

rf rg

H¼ Fr ¼

G2 gDr2H

mg mf

We ¼

1

mg mf

rH ¼

rf rg xrf þ ð1xÞrg

G2 D

rH s

Conversely, when ‘‘l’’ subscript is applied, ‘‘only-liquid’’ approach is considered. That is, the liquid phase is assumed to flow alone at its actual flow rate. a

When ‘‘lo’’ subscript is added to the friction multiplier, ‘‘liquid-only’’ approach is considered. That is, the liquid phase is assumed to flow alone with total flow rate.

5

9 RELAP5

8

Model -HEM

Model -HEM

STABLE

6

4

Model -Friedel

Model -Friedel

7

Model -Lockhart-Martinelli


Model -Jones

Model -Jones

3

x = 0.3

4

x = 0.4

2 Imaginary Axis

Nsub

x = 0.2

5

x = 0.5

3

-45

2

UNSTABLE

1

-35

-25

-15

-5

0

5

-1

1 -2

0 0

5

10

15

20 Npch

25

30

35

40

-3 -4

Fig. 10. Sensitivity on two-phase friction factor multiplier in terms of stability maps (Npch Nsub stability plane).

-5 Real Axis of 6 partial derivative balance equations for the steam and liquid phase (sharing the same pressure). A non-condensable component in the steam phase and a non-volatile component (e.g. boron) in the liquid phase can be treated. A fast partially implicit numerical scheme (so named semi-implicit numerical scheme) is used to solve the flow conservation equations inside control volumes connected by junctions (in 1D sense). Starting from the RELAP5 studies available in the open literature on DWO instability concern, which is generally addressed by

Fig. 11. Sensitivity on two-phase friction factor multiplier in terms of system eigenvalues. Test case: G ¼0.12 kg/s; Tin ¼ 239.2 1C; Q ¼100 kW (xex ¼ 0.40).

working with the impressed and constant DP boundary condition on the boiling channel (Ambrosini et al., 2000; Ambrosini and Ferreri, 2006; Rizwan-Uddin, 1994), the attempt to reproduce more realistic experimental apparatus for DWO investigation has been pursued.

130


20 Ambrosini et al. (2000)

18

Single Junction

Model eigenvalues -HEM Model eigenvalues -Jones

16 14

x = 0.3

Heated Channel

x = 0.5

Nsub

12

Time-dependent Volume

Upper Branch Bypass

10

Heat Structure

8

Lower Branch

6 Time-dependent Volume

4

Time-dependent Junction

2 x=0

0 0

5

10

15 Npch

20

25

30

Fig. 14. RELAP5 nodalization concept for single heated channel with bypass layout.

Fig. 12. Neutral stability boundary as predicted by linear stability analysis with different two-phase friction factor models.

Single Junction 3.5

Heated Channel A

3.0

Heat Structures

2.5

Time-dependent Volume Upper Branch Heated Channel B Lower Branch

T/τ

2.0

Time-dependent Volume

1.5

Time-dependent Junction

Model -HEM

1.0

Model -Friedel

Fig. 15. RELAP5 nodalization concept for two parallel channels layout.


0.5

Model -Jones RELAP5

0.0 0

1

2

3

4

5

6 7 Nsub

8

9

10 11 12 13

Fig. 13. Sensitivity on two-phase friction factor multiplier in terms of ratio between period of oscillations and transit time.

In case of single boiling channel studies, as a matter of fact, the imposition of constant DP across the heated tube is not a realistic assumption in the respect of typical experimental setup, where the mass flow rate is forced by an external feedwater pump instead of being freely driven according to the supplied power level. A large bypass tube is instead usually connected to the heated channel, such to respect the constant-pressure-drop boundary condition required to excite DWOs. When dealing on the other hand with parallel channel case, the common connection provided by the lower and upper headers is sufficient to maintain the mentioned boundary condition (strictly speaking, equal-pressure-drop condition between the two channels), no matter if the flow rate is forced at inlet by the respective pump. In this framework, both the experimental configurations of interest for density wave studies (i.e., single boiling channel with bypass and two parallel channels) have been object of the RELAP5 modelling. Aim of the analysis has been first to assess the capability of the code in detecting the onset of instability with the selected layout, and then to obtain validation results for the predictions given by the analytical lumped parameter model (Section 3).

The nodalization concepts adopted in this work, in which MOD3.3 version of the RELAP5 code is used, are depicted in Figs. 14 and 15, as concerns single channel with bypass and two parallel channels respectively (Colombo et al., 2012). The bypass layout (Fig. 14) is realized by connecting two pipe components of different diameter by means of two branches. The mass flow rate is provided by a time-dependent junction, connected directly to the lower branch. Inlet pressure and temperature are set using a time-dependent volume. Outlet pressure is imposed by another time-dependent volume connected to the upper branch by means of a single junction. Pressure drop across the channel is imposed by the exit pressure, the mass flow rate and the characteristics of the channel. The heated channel is subdivided for these studies in 48 nodes. Imposed heat flux condition is adopted on the heated channel. The heat structures (wherein power generation is accounted for) are assumed very thin and present high thermal conductivity and low heat capacity to avoid distortions in the imposed thermal flux condition, and to neglect tube wall dynamic behaviour as well. As far as the bypass ratio Rby (ratio of bypass cross sectional area to heated channel cross sectional area) is concerned, the initial value of the bypass diameter is assumed equal to 10 times the diameter of the heated channel (i.e., Rby ¼100), in order to ensure that the suited boundary condition is preserved (Collins and Gacesa, 1969; Colombo et al., 2012). Two channels layout (Fig. 15) is based on similar features, with now two pipe components of the same diameter connected by means of two branches, which represent lower and upper header of the real experimental apparatus. Mass flow rate is again provided by a time-dependent junction, connected to the lower


branch. Each pipe is thermally bounded by the respective heat structure, imposing the same uniform heat flux. Excessive numerical damping, roughly dealt by RELAP5 code to achieve stable numerical solutions, may be detrimental in stability analyses. To reduce as much as possible the amount of numerical diffusion introduced, specific numerical settings have been adopted and the time step is always forced to be as close as possible to the Courant limit (Colombo et al., 2012). The other choices are consistent with the numerical assumptions of Ambrosini and Ferreri (2006). 4.1.2. Results and discussion To establish the instability occurrence for the different operating conditions, the following procedure has been adopted. Specific values of exit pressure, inlet mass flow rate and inlet temperature are selected as initial conditions, at the beginning of each run. Flow circulation in the system starts at zero power, then power generation within the heat structures is increased gradually till the instability is reached. Thermal power increase rate is higher at the beginning of the transient, to quickly approach the unstable region, then it is lowered to guarantee an easier detection of the instability onset. Comprehensive discussion of the results of the RELAP5 analyses is provided in Colombo et al. (2012), studying both single boiling channel case (parametric studies included, above all the influence on stability of the bypass ratio) and twin parallel channel case. Comparison between the two different experimental layouts is discussed in more details in Section 5, as one of the principal achievements of this paper. It is just noticed here that the instability threshold is detected by the appearance of growing oscillations in the mass flow rate through the heated channel. In particular, in this work the system was considered unstable when mass flow rate oscillation amplitude reached 100% of the steady-state value. Typical transient is shown in Fig. 16: mass flow rate decreases with time as the supplied power is increased, until it starts to oscillate when the instability threshold is approached.

mentioned that this approach is globally different from the previous one discussed (i.e., the RELAP5 code), which indeed considers finite volume discretization of the governing equations, and of course from the simple analytical treatment thoroughly discussed in Sections 2 and 3. Modelling equations, derived from conservation of mass, energy and momentum, have been implemented separately with respect to single-phase region and two-phase region. In single-phase region, the mass, momentum and energy balance equations take the following form: @r @G ¼0 þ @z @t ! @G @ G2 @P @P þ ¼ rg siny @t @z r @z @z f rict @ @t

rhP þ

0.25 0.20

Г [kg/s]

0.10 0.40 0.20

0.00

0.00

-0.05

-0.20 250

-0.10

260

ð61Þ The physical system represented in Eqs.(59)–(61) is made by a straight tube of length H, circular cross-section of diameter D and slope angle y (between tube axis and the horizontal direction). Of course, siny ¼ 1 applies in order to simulate the straight vertical BWR subchannel under investigation (Table 2). A simpler enthalpy balance equation can replace Eq. (61), by subtracting a convenient form of Eq. (60) from Eq. (61). One obtains: " # @ @

q00 Z G @P @P þ rhP þ ðGhÞ ¼ þ ð62Þ @t @z A r @z f rict @z In two-phase region, the mass, momentum and energy balance equations read: @rm @G ¼0 þ @z @t ! TP @G @ G2 @P @P þ r g sin y ¼ m þ @t @z rm @z @z f rict @ @t

rm hm P þ

! þ

@ G3 þ Ghm þ þ2 @z 2rm

! ¼

ð63Þ

ð64Þ

q00 Z G @P TP þ Gg siny A rm @z f rict

ð65Þ Alike the single-phase region equations, an enthalpy balance equation is more convenient than Eq. (65). Hence, the kinetic energy terms within Eq. (65) can be eliminated by subtracting Eq. (64). One finally obtains: " # @ þ q00 Z @

G @P TP @P Ghm ¼ þ rm hm P þ þ ð66Þ @t @z A rm @z f rict @z

"

50

G2 þ 2rm

rm ¼ arg þ ð1aÞrf

270

-0.15 0

ð60Þ

Though fundamentally similar to single-phase region equations, two-phase flow conservation equations require proper definition of the two-phase flow structure. In particular, suited expressions for static and dynamic two-phase density and enthalpy, as well as for two-phase frictional term (via the two-phase friction multiplier F2lo concept), must be considered. General expressions for the terms representing a two-phase mixture are given by

0.15

0.05

ð59Þ

! ! G2 @ G3 q00 Z G @P Ghþ þ Gg siny þ ¼ @z A 2r r @z f rict 2r2

4.2. Numerical modelling by means of COMSOL code 4.2.1. Mathematical formulation and model development COMSOL Multiphysicss (COMSOL, Inc., 2008) is a numerical code that is gaining importance in the recent years, based on its possibility to solve different numerical problems by implementing directly the system of equations in PDE form. PDEs are then solved numerically by means of finite element techniques. It is

131

100

150 t [s]

200

250

300

Fig. 16. Mass flow rate behaviour in the heated channel at instability inception (Colombo et al., 2012). Simulation performed with Nsub ¼4.

ð67Þ 2 #1

1w w2 r ¼ þ arg ð1aÞrf þ m

hm ¼

arg hg þ ð1aÞrf hf rm

ð68Þ

ð69Þ

132


þ hm ¼ whg þ 1w hf

ð70Þ

The widespread S–a–w relation is used to relate the previous quantities: " #1 1w rg a ¼ 1þ S ð71Þ

w rf

The value of the slip ratio S depends on the choice of the twophase flow model adopted. In this section, both a simple homogeneous flow model (S ¼1, as the assumption on which basis the analytical lumped parameter model has been derived, see Section 2.1.4), and a more accurate drift-flux model (DFM) have been considered. The latter accounts for the effect of the relative velocity between the two phases (the so named slip), by defining a suitable slip ratio S as follows: ðC 0 1Þwrf V vj rf S ¼ C0 þ

þ

1w rg 1w G

ð72Þ

The quantities C0 and Vvj indicate respectively the concentration parameter (void distribution parameter) and the effective drift-flux velocity. The first represents the global effect due to non-uniform void distribution and velocity profiles in the channel section, whereas the latter represents the local relative velocity effect and generally depends on the flow regime. Several correlations exist for the two parameters. Various combinations of drift-flux models, together with different twophase friction multiplier expressions, have been tested. Comprehensive results can be found in the Master Thesis of Giorgi (2010). For the aims of this paper, just the simple DFM based on a constant value of the concentration parameter (C0 ¼1.13) (Kakacand Bon, 2008) and Lahey and Moody correlation for the drift-flux velocity (Todreas and Kazimi, 1993) are considered. HEM model is implemented with homogeneous F2lo , whereas DFM is implemented with Jones correlation for F2lo (both already utilized in the analytical model in Section 3.3). Implementation of the modelling equations set has been carried out using the PDE General Form module of COMSOL Multiphysicss. The obtained 1D thermal-hydraulic simulator works out switching between single-phase and two-phase regions according to a selection logic that is based on the following check variable vc: þ

vc ¼

hm hf hg hf

ð73Þ

The value of vc (basically representing the thermodynamic quality x) is evaluated in each node. Single-phase zone equations are considered if vc o0 or vc 4 1, whereas two-phase zone equations are considered if 0 r vc r1. Similar computational procedure is followed, for instance, in the work of Colorado et al. (2011).

4.2.2. Results and discussion Steady-state predictions from the described modelling with COMSOL code have been first validated before any attempt to apply this numerical modelling tool for the study of density wave instabilities. Experimental test-case for the 1D simulator built-up with the COMSOL code has been represented by the thorough pressure drop database obtained by Santini (2008) and Santini et al. (2008), who worked on the helical-coiled tube test section located at SIET labs. COMSOL thermal-hydraulic simulator has been adapted to the peculiar referenced geometry, on the account of specific modifications including the approximation with a straight inclined

Fig. 17. Comparison of numerical results in COMSOL with Santini’s experimental pressure drop data (Giorgi, 2010). (a) Pin ¼ 20 bar – red curve: G¼ 200 kg/m2s; q00 ¼ 43.7 kW/m2, blue curve: G ¼ 400 kg/m2s; q00 ¼ 111.1 kW/m2; (b) Pin ¼ 40 bar – red curve: G¼600 kg/m2s; q00 ¼ 142.4 kW/m2, blue curve: G¼ 800 kg/m2s; q00 ¼191.6 kW/m2; (c) Pin ¼ 60 bar – red curve: G¼400 kg/m2s; q00 ¼ 85.2 kW/m2, blue curve: G¼ 600 kg/m2s; q00 ¼117.4 kW/m2. (For interpretation of the references to colour in the legend of this figure, the reader is referred to the web version of this article.)

channel, with same length and inclinational angle y of the facility helix (Papini et al., 2011b). The comparison between the numerical predictions and the experimental pressure drop data on the helical coil facility is depicted in Fig. 17 (respectively for three different pressure levels). The agreement is satisfactory, with maximum error less than 10%. As final step, the water–steam thermal-hydraulic simulator has been applied to boiling channel stability analysis, via linearization of the equation system (Eqs.(59)–(66)) and computation of the eigenvalues, on the basis of the experience collected in Section 2.3. Just the single boiling channel case is hereby presented. COMSOL Eigenvalue Solver has been used, yielding the linearization of the equations about an equilibrium point. The linearization tool acts by substituting each temporal derivative


operator in the following way:

9

@ -l @t

ð74Þ

8

where l is an eigenvalue. Neglecting tedious computational details about the linearization and consequent linear stability analysis on the boiling channel, one is directed to the following section for the comparison of the COMSOL results with RELAP5 and analytical model results.

7

4.3. Validation benchmark of analytical and numerical studies on DWOs A comprehensive comparison of all tools developed in this work to predict the inception of DWOs in simple vertical tube geometry is shown in Fig. 18, where the noteworthy work of Ambrosini et al. (2000) is reported as well. Stability maps drawn in the Npch Nsub stability plane are compared. RELAP5 results rely on UVUT model selection, i.e. non-homogeneous non-equilibrium model. Theoretical model uses HEM two-phase flow model, respectively with homogeneous and Friedel friction factor multiplier. Finally, both HEM and DFM are considered as concerns COMSOL results. It is immediate to notice that all the results with this simple vertical tube geometry confirm the classical DWO theory expectations. The exit quality (proportional to the ratio between thermal power and mass flow rate) turns out as key parameter for the stability boundary definition. At high subcoolings, the stability boundary follows roughly a constant-exit-quality line; at medium-low subcoolings, the so named ‘‘L shape’’ appears (Zhang et al., 2009). On the whole, parametric effect of an increase of the inlet subcooling is stabilizing at high subcooling and destabilizing at low subcooling (Papini et al., 2011b; Yadigaroglu, 1981). The results of the proposed combined approach to boiling channel stability analysis – based on both system codes (RELAP5) and the theoretical nodal model developed – have been then assessed by reproducing ‘‘classical’’ stability experimental results. In particular, the experimental data of Solberg (1966), studying density wave instabilities with a simple bypass layout, have been considered (Fig. 19). Respective operational and geometrical conditions are listed in Table 5. It is just noticed that the same experimental work was among the referred ones by Ishii and Zuber (1970) to validate the predictions of their own analytical

9 x = 0.5 8 7 Model -Friedel

Nsub

6

Model -HEM Ambrosini et al. (2000)

5

RELAP5

4

COMSOL -HEM COMSOL -DFM

3 2 1

x = 0.3

0 0

5

10

133

15 Npch

20

25

30

Fig. 18. Validation benchmark between analytical model and numerical models with RELAP5 and COMSOL codes.

x = 0.8

Nsub

6 5 4 3

Model -Friedel Model -Lockhart-Martinelli RELAP5 Solberg's data

2 1 x = 0.4

x = 0.6

0 0

5

10

15 Npch

20

25

30

Fig. 19. Comparison between analytical model, RELAP5 results and the experimental data of Solberg (1966) (available from Ishii and Zuber, 1970).

Table 5 Dimensions and operating conditions of the Solberg’s experiment (available from Ishii and Zuber, 1970). Heated channel Diameter (m) Length (m)

0.00525 2.9

Operating parameters Pressure (bar) Inlet temperature (1C) kin kex

81.1 139.6–291.8 17.8 0.03

model. A rather good validation is obtained for the RELAP5 results, as well as a good envelope set of the experimental findings is offered by the results of the analytical model according to the different friction options available (Friedel model as for ‘‘low friction’’ family and Lockhart-Martinelli model as for ‘‘high friction’’ family). The classical ‘‘L shape’’ of the stability boundary is hardly visible from the experimental data, on which cannot be neglected the influence of uncertainties such the ones related to the procedure and the criterion followed to record the instability. All things considered, it can be concluded that the Friedel friction model might be more accurate at high subcoolings (predicting the constant-exit-quality line of about 0.6), whereas the LockhartMartinelli friction model performs better at medium-low subcooling conditions. The accuracy of a HEM thermal-hydraulic model (as assumed in Van Bragt and Van Der Hagen (1998) and ˜ oz-Cobo et al. (2002)) can be reasonably demonstrated. Mun However, the applicability of the HEM for calculation of stability threshold is still debatable (Goswami and Paruya, 2011; Nayak et al., 2007). An alternative to the two-fluid approach (complicated by bunch of constitutive relations for interfacial laws) is the DFM, based on a sole momentum equation for the mixture plus non-linear constitutive law for the relative velocity, which should provide more accurate predictions. Nevertheless, it is known that the choice of both concentration parameter and drift-flux velocity significantly affects the stability, since their increase or decrease modifies the void fraction, hence mixture density and transportation time lags (Dokhane et al., 2005; Nayak et al., 2007; Rizwan-Uddin and Dorning, 1986).


The drift-flux model implemented by COMSOL simulator shows the least agreement in the benchmark of Fig. 18, but this discrepancy might be indeed ascribed to the friction factor selected (Jones one, i.e. ‘‘high friction’’ model yielding less stable regions) rather than to the particular DFM parameters adopted. As the correct prediction of the instability threshold depends highly on the effective frictional characteristics of the reproduced channel (see Section 3.3), the possibility of implementing most various kinds of two-phase flow models (DFM kind, with different void fraction expressions) makes in principle the developed COMSOL model suitable to apply for most different heated channel systems. DFM parameter adjustment will be required separately case by case.

9

x = 0.4

8 7 6 Model -single channel (DP = 80 kPa)

Nsub

134

5

Model -single channel (DP = 110 kPa)

4

Model -twin channels

3 2 1

5. Main achievements

As concerns the instability mechanism, the key role played by

the void propagation time delay in the two-phase region is well known. At sufficiently large values of void fraction (i.e., exit thermodynamic quality), any small fluctuation in the inlet velocity may lead to large fluctuation of the two-phase frictional pressure losses. Multiple feedback effects are triggered by the mandatory constant-pressure-drop boundary condition. Enthalpy and void transportation lags throughout the channel are evident in a non-sinusoidal wave for flow rate and pressure drop oscillations (Rizwan-Uddin, 1994). ‘‘Shark-fin’’ shape of oscillating total pressure drops is our final outcome, corroborated with some experimental evidence. When simulating a simple geometry, such as a vertical channel, classical DWO theory is respected. Instability boundary shows the ‘‘L-shape’’ inclination in the Npch Nsub plane, with inlet subcooling increase that is stabilizing at high subcooling and destabilizing at low subcooling. The period of oscillations is nearly twice the mixture transit time, and grows with the inlet subcooling. Fractional distribution of the pressure drops along the channel plays an important role in determining the stability of the system (Rizwan-Uddin, 1994). In this respect, a proper simulation of the two-phase frictional pressure losses – prior to proper representation of the pressure drop distribution within the channel – is well indentified as the most critical concern for an accurate prediction of the instability threshold. The homogenous friction factor model shows the most stable system (Nayak et al., 2007). Higher friction models (in ascending order: Friedel, Lockhart-Martinelli and Jones) reduce the system stability. Two different system layouts are dealt with for instability investigation: a single heated channel and two parallel channels. The equivalence of the two approaches is found in terms of stability maps, both with analytical calculations (Fig. 20: single heated channel is studied with two different values of

0 0

5

10

15 Npch

20

25

30

Fig. 20. Stability maps for single heated channel and two parallel channels cases, as calculated with analytical lumped parameter model (Friedel friction model).

9 x = 0.5

8

x = 0.6

RELAP5 -imposed DP

7 RELAP5 -bypass (Rby = 200)

6 Nsub

Time-domain linear and non-linear studies presented in this work permitted to understand more in depth the basic phenomena at the basis of DWOs, ranging from development of a selfsustained flow rate oscillation and identification of critical twophase flow model parameters for accurate prediction of instability threshold, to assessment of experimental apparatus layouts suited for instability detection. State-of-the-art advances reached by the paper results can be summarized as in the followings. Similar discussions have been found in the work of Nayak et al. (2007), who investigated various two-phase friction factor multiplier models, DFM parameters and several geometrical parameters in case of two-phase natural circulation system instabilities.

RELAP5 -twin channels

5 4 3 2 1 0 0

5

10

15 Npch

20

25

30

Fig. 21. Stability maps for single heated channel and two parallel channels cases, as calculated with RELAP5 code. Single heated channel simulations performed with impressed constant DP (Ambrosini and Ferreri, 2006) and with bypass tube nodalization (Rby ¼200) (Colombo et al., 2012).

imposed DP) and RELAP5 simulations (Fig. 21: single heated channel is studied with constant DP fictitiously impressed and with the bypass nodalization). Such equivalence (strictly valid for the analysed vertical tube case (Yadigaroglu, 1981)) is important because, if one has to study the instability boundary of a multi-channel system (i.e., one single channel working rigorously under constant DP), the experimental apparatus might be designed just with two parallel tubes connected by two headers, thus without the need for a complex parallel large bypass tube. Several sensitivity studies can be provided with the developed models. The effect of tube diameter on boiling channel stability may be relevant when using scaled facilities in simulation of the instability behaviour of prototype systems (Nayak et al., 2007). Sensitivity analysis with the analytical model (single channel, Friedel friction model), reported in Fig. 22, reveals that the stability boundary is independent on tube diameter at high subcooling conditions (where constantexit-quality line is roughly respected). On the contrary, the


9 x = 0.4

Model -nominal A

8

Model -2*A Model -4*A

7

Model -1/4*A

Nsub

6 5 4 3 2 1 0 0

5

10

15 Npch

20

25

30

Fig. 22. Effect of channel cross-sectional area on DWO instability threshold, as calculated with analytical lumped parameter model (single channel case, Friedel friction model).

system results less stable as the channel area is reduced at medium-low subcooling conditions. Decrease in tube diameter has a destabilizing effect (Nayak et al., 2007), as a reduction of tube area increases the frictional contribution within total pressure drop (but this increase is such to destabilize the system only when the two-phase region is dominant, i.e. at medium-low subcooling). 6. Conclusions Density wave instability phenomena have been described in this work, featured as topic of interest in the nuclear area, both to the design of BWR fuel channels and the development of the steam generators with reference to the new generation reactors. Theoretical studies based on analytical and numerical modelling have been presented, aimed at gaining insight into the distinctive features of DWOs as well as predicting instability onset conditions. An analytical lumped parameter model has been first developed. Non-linear features of the modelling equations have permitted to describe the complex interactions between the variables triggering the instability. Enthalpy and void transportation lags throughout the channel, at the basis of the inception of the instability, turn out in non-sinusoidal wave for flow rate and pressure drop oscillations. ‘‘Shark-fin’’ shape of oscillating total pressure drops is our final outcome. Moreover, a proper simulation of the two-phase frictional pressure drops – prior to proper representation of the pressure drop distribution within the channel – has been depicted as the most critical concern for an accurate prediction of the instability threshold. The higher are the two-phase friction characteristics of the system, the less stable results the channel. Dealing with the simple and acknowledged case of vertical tube geometry, theoretical predictions from the analytical model have been validated with numerical results obtained with the RELAP5 and COMSOL codes, which have proved to successfully predict the DWO onset. Moreover, a fair assessment of the analytical model predictions has been obtained thanks to available stability experimental results. RELAP5 code has permitted to analyse two experiment-like configurations for stability studies, i.e. a single heated channel

135

connected to a large bypass tube and two parallel channels. The equivalence of the two approaches for DWO investigation has been confirmed. Whereas homogeneous flow assumption was considered in the analytical model, HEM and DFM options have been compared with the 1D thermal-hydraulic simulator built-up with COMSOL code. Improvement of the DFM formulation is still required. Systematic and comprehensive study on DWOs has been completed by linearizing the modelling equations and computing the system eigenvalues. Linear stability analysis has pointed out the ‘‘nose’’ feature of the stability boundary, with prediction of Ledinegg instability at high subcooling, as well as quickly highlighted the effect of different two-phase friction models. The whole work is preparatory to the study of DWO instabilities in the complex geometry represented by two helical coil tubes for simulation of helically coiled steam generators (envisaged e.g. within the Generation III þ Small Modular Reactors – SMRs – with integral layout). All things considered, a proper representation of the pressure drop distribution (with correct prediction of two-phase frictional term) in a helical channel will be fundamental. Modelling efforts will have to address the definition of most appropriate correlations for two-phase friction multiplier and void fraction. With suitable modifications to comply with the helical coil geometry, the developed two-node analytical model as well as the COMSOL model (thanks to the high potentialities offered by the DFM formulation implemented for two-phase flow) might be a valuable contribution to the study. Nomenclature A C0 c D f Fr G g H h j k M Npch Nsub n P Q Q 000 q00 Rby Re S T t Vch Vvj

n vc We Xtt

tube cross-sectional area (m2) concentration parameter (DFM) (dimensionless) specific heat (J/kg 1C) tube diameter (m) single-phase friction factor (dimensionless) Froude number (G2/gDr2H) (dimensionless) mass flux (kg/m2 s) acceleration of gravity (m/s2) tube length (heated zone) (m) specific enthalpy (J/kg), heat transfer coefficient in Eqs. (8) and (9) (W/m2 1C) volumetric flux ((x/rg þ(1 x)/rf)G2f) (m/s) concentrated loss coefficient (dimensionless) tube mass (kg) phase change number (Q/(Ghfg) vfg/vf) (dimensionless) subcooling number (Dhin/hfg vfg/vf) (dimensionless) exponent of Reynolds number in single-phase friction factor formulas (dimensionless) pressure (bar) thermal power (W) thermal power per unit volume (Q/Vch) (W/m3) heat flux (W/m2) bypass ratio (dimensionless) Reynolds number (GD/m) (dimensionless) slip ratio (dimensionless), heat transfer surface in Eqs. (8) and (9) (m2) period of oscillations (s), temperature in Eqs. (8) and (9) (1C) time (s) heated channel volume (m3) drift-flux velocity (DFM) (m/s) specific volume (m3/kg) COMSOL model check variable (dimensionless) Weber number (G2D/rHs) (dimensionless) Lockhart-Martinelli parameter (((1 x)/x)0.9(rg/rf)0.5(mf/mg)0.1) (dimensionless)

136

x Z z


thermodynamic quality (dimensionless) heated perimeter (m) tube abscissa (m)

Greek symbols

a G

g DP

Z y l

m r s t F2

w O

void fraction (dimensionless) mass flow rate (kg/s) density ratio (rf/rg) (dimensionless) pressure drops (Pa) state variable channel inclination angle (with horizontal direction) (deg) system eigenvalue (dimensionless) dynamic viscosity (Pa s) density (kg/m3) surface tension (N/m) mixture transit time (s) two-phase friction factor multiplier (dimensionless) two-phase flow quality (DFM) (Gg/Gtot) (dimensionless) reaction frequency (Q/(AH) vfg/hfg) (1/s)

Subscripts

introduced within the resulting linear system are listed. Calculation of the eigenvalues of the linearized system is at last discussed. A.1. Mass-energy conservation equation in the single-phase region The initial equation is represented by the dynamics of the boiling boundary zBB(t). By considering the definition of the dynamic coefficients of the non-linear model (see Table 1), this equation reads: dzBB 2Gin 2Q zBB ¼

dt rf hf hin V ch rf

ðA:1Þ

First-order perturbation of the state variables zBB(t) and Gin(t) about the respective steady-state conditions is introduced in Eq. (A.1), in the form of zBB ðtÞ ¼ z0BB þ dzBB elt

ðA:2Þ

Gin ðtÞ ¼ G0in þ dGin elt

ðA:3Þ

One obtains dz0BB 2Q z0BB 2G0in 2dGin lt 2Q dzBB þ e

elt þ dzBB lelt ¼ dt rf rf hf hin V ch rf hf hin V ch rf

ðA:4Þ acc av BB ex f fl frict g go grav H h in l lo m tot tp

accelerative average boiling boundary exit saturated liquid fluid bulk frictional saturated vapour gas-only (gas phase with total flow rate) gravitational homogeneous model heated wall inlet only-liquid (liquid phase at its actual flow rate) liquid-only (liquid phase with total flow rate) mixture total two-phase

Superscripts 0 1f 2f þ

steady-state value single-phase region two-phase region dynamic quantity (referred to two-phase density and enthalpy)

Steady-state conditions, from Eq. (A.1), are defined as 0¼

2G0in

rf

2Q z0BB

hf hin V ch rf

ðA:5Þ

By introducing Eq. (A.5) within Eq. (A.4), and simplifying the exponential perturbation terms, one gets

dzBB l ¼

2dGin

rf

2Q dzBB

hf hin V ch rf

ðA:6Þ

Eq. (A.6) can be finally rearranged pointing out the system variables (in terms of small variations about steady-state values): ! 2Q 2 lþ

dzBB þ 0 dxex dGin ¼ 0 ðA:7Þ rf hf hin V ch rf

A.2. Mass-energy conservation equation in the two-phase region The dynamics of the exit quality (see Section. 2.1.4) is considered. This equation – Eq. (33) – is linearized cancelling the derivative terms of steady-state conditions, and neglecting the second-order infinitesimal terms: 0 dzBB dxex dx0 ¼ b9 ¼ ex þ dxex lelt ¼ b6 þ b7 þ dzBB lelt ðA:8Þ dt dt dt Linearized forms of the coefficients b6 and b7 must be introduced. One can write, respectively: 0

b6 ¼ b6 þD3 elt dzBB þ D4 elt dxex þD5 elt dGin Acknowledgements The contribution of Gianluca Giorgi and Giovanna Malatesta to the development of the water-steam thermal-hydraulics 1D simulator using COMSOL is gratefully acknowledged.

Appendix A The computational procedure leading to the linearization of the ordinary differential equations of the analytical model (see Section 2.3) is presented in this Appendix. All the coefficients

ðA:9Þ

0

b7 ¼ b7 þD1 elt dzBB þ D2 elt dxex

ðA:10Þ 0 b6

It is simple (see Table 1) to demonstrate that ¼ 0. The introduction of Eqs. (A.9) and (A.10) into Eq. (A.8) gives:

lelt dxex ¼ D3 elt dzBB þ D4 elt dxex þD5 elt dGin

0 þ b7 þ D1 elt dzBB þ D2 elt dxex lelt dzBB

ðA:11Þ

which yields, after simplification of second-order terms and exponential small perturbations: 0 b7 l þD3 dzBB þ ðl þ D4 Þdxex þD5 dGin ¼ 0 ðA:12Þ


137

The linearization of Eq. (A.9) returns the coefficients D3, D4 and D5, which are:

within Eqs. (A.28) and (A.29) – is hereby considered. The selected form for the two-phase multiplier is:

D3 ¼ p5 O

ðA:13Þ

F2lo ¼ F2 Oj ¼ ð1þ C F xÞð1:9 þ C O GÞ

D4 ¼ p5 G0in vf g

ðA:14Þ

D5 ¼ p5 x0ex vf g

ðA:15Þ

where 1 p5 ¼

0 Hz0BB vf þ vf g x0ex rf g b5

ðA:16Þ

where C F ¼ 30 and C O ¼ 5 104 at the nominal operating pressure of 70 bar. The linearization coefficients read:

g 1þ g1 x0ex g g1 x0ex M1 ¼ ðA:31Þ

2 1þ g1 x0ex " #

0 1

g1

ln 1 þ g 1 x 2 ex 1 þ g1 x0ex g1 x0ex x0ex

g

M2 ¼

A.3. Momentum conservation equation

M3 ¼

Momentum conservation equation, written down for the whole channel, is strongly non-linear and rather complex. One can start from Eq. (24): ðzBB þb3 Þ

dGin dzBB dxex þ ðGin þ b1 Þ þ b2 ¼ DPDP acc DPf rict DPgrav dt dt dt ðA:17Þ

Left-hand-side term can be linearized following the rules introduced in the previous sections (cancelling steady-state derivative terms and neglecting second order terms), obtaining: 0 0 0 z0BB þb3 lelt dGin þ G0in þb1 lelt dzBB þ b2 lelt dxex ¼ DPDP acc DPf rict DPgrav

DP

¼ DP 0acc þ DP 0f rict þ DP 0grav

ðA:19Þ

The linearized relations between the following system parameters must be then calculated:

daex ¼ M1 dxex

ðA:20Þ

da ¼ M2 dxex

ðA:21Þ

dGex ¼ M3 dzBB þM4 dxex þ M5 dGin

d G2f ¼ M6 dzBB þ M7 dxex þ M8 dGin

ðA:22Þ

DP acc ¼ DP 0acc þ D6 elt dzBB þD7 elt dxex þ D8 elt dGin

ðA:24Þ

DP grav ¼ DP 0grav þD9 elt dzBB þ D10 elt dxex

ðA:25Þ

DP f rict,in ¼ DP0f rict,in þD11 elt dGin

ðA:26Þ

DP f rict,1f ¼ DP 0f rict,1f þ D12 elt dzBB þ D13 elt dGin

ðA:27Þ

DP f rict,2f ¼ DP 0f rict,2f þ D14 elt dzBB þ D15 elt dxex þ D16 elt dGin

ðA:28Þ

DP f rict,ex ¼ DP0f rict,ex þ D17 elt dzBB þ D18 elt dxex þ D19 elt dGin

ðA:29Þ

Detailed linearization calculations are left to a willing reader. In the following, the complete list of the above defined linearization coefficients is reported. Note that the March-Leuba approximation of Jones two-phase friction factor multiplier (March-Leuba, 1984, õz-Cobo et al., 2002) – which plays a fundamental role 1986; Mun

ðA:35Þ

vf vf g 0 ln 1þ x 1 vf ex vf g x0ex

vf g x0ex

O Hz0BB

M7 ¼

ðA:36Þ

2 vf g x0ex

vf vf g 0 ln 1 þ x 1 vf ex vf g x0ex

vf vf g 0 ln 1 þ xex 0 vf vf g xex

M8 ¼

ðA:34Þ

O

M6 ¼

ðA:32Þ

ðA:33Þ

vf vf þvf g x0ex

M5 ¼

ðA:37Þ

ðA:38Þ

D6 ¼ 2G0ex p7 M 3

ðA:39Þ

2 D7 ¼ 2G0ex p7 M 4 þ G0ex p8

ðA:40Þ

D8 ¼ 2G0ex p7 M 5

p7 ¼

ðA:23Þ

which are useful in linearizing each pressure drop term (accelerative, gravitational, and concentrated and distributed frictional terms as well). Resulting (refer to Section 2.1.2 for pressure drops computation):

O vf þvf g x0ex

h

i vf g 0 0 M4 ¼

2 O HzBB þ vf Gin vf þ vf g x0ex

ðA:18Þ

Right-hand-side term (in which pressure drop calculation is accomplished) comprehends most non-linear features of the model. Obviously, the total DP is constant, and equal to the stationary value of pressure drops, such that:

ðA:30Þ

p8 ¼

0 2 xex

2G0in

ðA:41Þ

rf

2 1x0ex 1a0ex rf

þ

a0ex rg

ðA:42Þ

2

2 1 2x0ex a0ex x0ex M 1 1 2 1x0ex 1a0ex þ 1x0ex M 1 þ

2

2 rg rf a0ex 1a0ex ðA:43Þ

D9 ¼ g rf g

h

1a0 rf þ a0 rg

D10 ¼ Hz0BB gM2 rg rf D11 ¼

D12 ¼

kin G0in

rf 2 f G0in 2Drf

D13 ¼

f z0BB G0in Drf

D14 ¼

f 2Drf

i

ðA:44Þ ðA:45Þ

ðA:46Þ

ðA:47Þ

ðA:48Þ

E D E2 D CF 0 xex 1:9 þ C O G0in 1þ Hz0BB 2 G02f M 6 G02f 2

ðA:49Þ

138


Table A1 List of the coefficients Eii, which constitute the matrix of dynamics of the linearized system. 2Q E11 ¼ l þ

hf hin V ch rf 0

E13 ¼

0

E31 ¼ l G0in þ b1 þ D6 þ D9 þ

0 E33 ¼ l z0BB þ b3 þ D8 þ D11 þ

0

E32 ¼ lb2 þ D7 þ D10 þ D15 þ D18

D12 þ D14 þ D17

D15 ¼

2

rf

E23 ¼ D5

E22 ¼ l þ D4

E21 ¼ b7 l þ D3

E12 ¼ 0

D13 þ D16 þ D19

D E f

CF 0 C F D 0 E2 1þ Hz0BB 1:9 þ C O G0in xex 2 G02f M 7 þ G2f 2Drf 2 2

ðA:50Þ D E D E2 f

CF 0 D16 ¼ Hz0BB 1 þ 1:9 þ C O G0in 2 G02f M 8 þ C O G02f xex 2Drf 2

ðA:51Þ D17 ¼

kex

1 þ C F x0ex 1:9þ C O G0in 2G0ex M 3 2rf

ðA:52Þ

D18 ¼

2 kex 1þ C F x0ex 2G0ex M 4 þ C F G0ex 1:9 þ C O G0in 2rf

ðA:53Þ

D19 ¼

2 kex

1:9þ C O G0in 2G0ex M 5 þ C O G0ex 1 þ C F x0ex 2rf

ðA:54Þ

By introducing Eqs. (A.31)–(A.54) into Eq. (A.18), on the account of Eq. (A.19), one finally obtains the linearized version of the momentum conservation equation. After some algebra: h i l G0in þ b01 þ D6 þ D9 þD12 þD14 þ D17 dzBB h i 0 þ lb2 þD7 þ D10 þD15 þD18 dxex h i 0 þ l z0BB þ b3 þD8 þ D11 þD13 þD16 þ D19 dGin ¼ 0 ðA:55Þ A.4. System eigenvalue computation

System eigenvalues are computed by solving Eq. (A.56), which, all things considered, reads: l þD20 0 D21 0 D4 l D5 3 b7 l þD ¼0 ðA:62Þ 0 0 l G0 þb0 þ D 0 þ D l b þ D l z þ b 22 23 24 in 1 2 3 BB The better way to solve the determinant of Eq. (A.62) is to consider the elements of the first row (first equation of the linear system), and the respective complementary minors. That yields: D4 l D5 ðl þ D20 Þ 0 0 0 þ D l b þ D l z þ b 23 24 2 3 BB 0 b7 l þD3 þ D21 0 0 l Gin þb1 þ D22

¼0 0 lb2 þ D23 D4 l

ðA:63Þ

After some calculus, the following cubic equation is finally obtained: 3

2

l R3 þ l R2 þ lR1 þ R0 ¼ 0

ðA:64Þ

where 0

R3 ¼ z0BB þb3

ðA:65Þ

0 0 0 0 0 R2 ¼ ðD4 D20 Þ z0BB þ b3 D24 b2 D5 þ b2 b7 D21 þ D21 G0in þ b1 ðA:66Þ

With respect to the single boiling channel linearized model, three are the eigenvalues that govern the system dynamics (by neglecting the heated wall model). The calculation of the three eigenvalues relies on solving the characteristic equation, based on the matrix of dynamics: E11 E12 E13 E ðA:56Þ 21 E22 E23 ¼ 0 E31 E32 E33 Theoretical expressions of the terms Eii are recapitulated in Table A1, with explicit reference to all the coefficients defined above. Obviously, the choice of the coefficients to group the system variables is totally arbitrary. For the sake of brevity, the following further coefficients are introduced: 2Q D20 ¼

hf hin V ch rf D21 ¼

2

rf

ðA:57Þ

ðA:58Þ

D22 ¼ D6 þ D9 þ D12 þ D14 þD17

ðA:59Þ

D23 ¼ D7 þ D10 þD15 þ D18

ðA:60Þ

D24 ¼ D8 þ D11 þD13 þ D16 þ D19

ðA:61Þ

0 0 R1 ¼ D4 D24 D5 D23 þD4 D20 z0BB þ b3 D20 D24 b2 D5 D20 0 0 0 þ D21 b7 D23 þ b2 D3 D4 D21 G0in þ b1 þ D21 D22

ðA:67Þ

R0 ¼ D4 D20 D24 D5 D20 D23 þ D3 D21 D23 D4 D21 D22

ðA:68Þ

The roots of Eq. (A.64) are, at the end, the eigenvalues of the linearized model. References Abdalla, M.A., 1994. A four-region, moving-boundary model of a once-through, helical-coil steam generator. Ann. Nucl. Energy 21 (9), 541–562. Ambrosini, W., Di Marco, P., Ferreri, J.C., 2000. Linear and nonlinear analysis of density wave instability phenomena. Int. J. Heat Technol. 18 (1), 27–36. Ambrosini, W., Ferreri, J.C., 2006. Analysis of basic phenomena in boiling channel instabilities with different flow models and numerical schemes. In: Proceedings of 14th International Conference on Nuclear Engineering (ICONE 14), Miami, FL, USA, July 17–20, 2006. Boure´, J.A., Bergles, A.E., Tong, L.S., 1973. Review of two-phase flow instability. Nucl. Eng. Des. 25, 165–192. Cinotti, L., Bruzzone, M., Meda, N., Corsini, G., Lombardi, C., Ricotti, M., Conway, L., 2002. Steam generator of the international reactor innovative and secure. In: Proceedings of 10th International Conference on Nuclear Engineering (ICONE 10), Arlington, VA, USA, April 14–18, 2002. Collins, D.B., Gacesa, M., 1969. Hydrodynamic instability in a full scale simulated reactor channel. In: Proceedings of the Symposium on Two Phase Flow Systems, Leeds, Institute of Mechanical Engineers, London, UK, pp. 117–128.


Colombo, M., Cammi, A., Papini, D., Ricotti, M.E., 2012. RELAP5/MOD3.3 study on density wave instabilities in single channel and two parallel channels. Prog. Nucl. Energy 56, 15–23. Colorado, D., Papini, D., Hernańdez, J.A., Santini, L., Ricotti, M.E., 2011. Development and experimental validation of a computational model for a helically coiled steam generator. Int. J. Therm. Sci. 50, 569–580. Comsol, Inc., 2008. COMSOL Multiphysicss User’s Guide, Version 3.5a. Dokhane, A., Hennig, D., Chawla, R., Rizwan-Uddin, 2005. Semi-analytical bifurcation analysis of two-phase flow in a heated channel. Int. J. Bifurcation Chaos 15 (8), 2395–2409. Dokhane, A., Hennig, D., Rizwan-Uddin, Chawla, R., 2007. Interpretation of inphase and out-of-phase BWR oscillations using an extended reduced order model and semi-analytical bifurcation analysis. Ann. Nucl. Energy 34 (4), 271–287. Farawila, Y.M., Pruitt, D.W., 2006. A study of nonlinear oscillation and limit cycles in boiling water reactors, Part I: the global mode, Part II: the regional mode. Nucl. Sci. Eng. 154 (3), 302–327. Friedel, L., 1979. Improved friction pressure drop correlation for horizontal and vertical two-phase pipe flow. In: European Two-Phase Flow Group Meeting, Ispra, Italy, June 5–8, 1979, Paper E2. Furutera, M., 1986. Validity of homogeneous flow model for instability analysis. Nucl. Eng. Des. 95, 65–77. Giorgi, G., 2010. Single and parallel channel stability analysis by means of multiphysics modeling. M.Sc. Thesis, Politecnico di Milano, Department of Energy, Milan, Italy, October 2010. Goswami, N., Paruya, S., 2011. Advances on the research on nonlinear phenomena in boiling natural circulation loop. Prog. Nucl. Energy 53 (6), 673–697. Guo Yun, Qiu, S.Z., Su, G.H., Jia, D.N., 2008. Theoretical investigations on two-phase flow instability in parallel multichannel system. Ann. Nucl. Energy 35, 665–676. Hennig, D., 1999. A study on boiling water reactor stability behavior. Nucl. Technol. 126 (1), 10–31. Incropera, F.P., Dewitt, D.P., Bergman, T.L., Lavine, A.L., 2007. Fundamentals of Heat and Mass Transfer, sixth ed. John Wiley & Sons, Hoboken. Ishii, M., Zuber, N., 1970. Thermally induced flow instabilities in two-phase mixtures. In: Proceedings of 4th International Heat Transfer Conference, Paris, France, August 31–September 5, 1970, vol. 5, paper B5.11. Jensen, J.M., Tummescheit, H., 2002. Moving boundary models for dynamic simulations of two-phase flows. In: Proceedings of 2nd International Modelica Conference, Oberpfaffenhofen, Germany, March 18–19, 2002, pp. 235–244. Kakac- , S., Bon, B., 2008. A review of two-phase flow dynamic instabilities in tube boiling systems. Int. J. Heat Mass Transfer 51, 399–433. Lahey Jr., R.T., Moody, F.J., 1977. The Thermal-hydraulics of a Boiling Water Nuclear Reactor. American Nuclear Society. Lange, C., Hennig, D., Hurtado, A., 2011. An advanced reduced order model for BWR stability analysis. Prog. Nucl. Energy 53, 139–160. Ledinegg, M., 1938. Instability of flow during natural and forced circulation. Die ¨ Warme 61 (48), 891–898. Li, H., Huang, X., Zhang, L., 2008. A lumped parameter dynamic model of the helical coiled once-through steam generator with movable boundaries. Nucl. Eng. Des. 238, 1657–1663. March-Leuba, J., 1984. Dynamic behaviour of boiling water reactors. Ph.D. Thesis, University of Tennesse, Knoxville, TN, USA. March-Leuba, J., 1986. A reduced-order model of boiling water reactor linear dynamics. Nucl. Technol. 75, 15–22. Masini, G., Possa, G., Tacconi, F.A., 1968. Flow instability thresholds in parallel heated channels. Energ. Nucl. 15 (12), 777–786.

139

˜ oz-Cobo, J.L., Podowski, M.Z., Chiva, S., 2002. Parallel channel instabilities in Mun boiling water reactor systems: boundary conditions for out of phase oscillations. Ann. Nucl. Energy 29, 1891–1917. Nayak, A.K., Dubey, P., Chavan, D.N., Vijayan, P.K., 2007. Study on the stability behaviour of two-phase natural circulation systems using a four-equation drift flux model. Nucl. Eng. Des. 237, 386–398. ¨ Paladino, D., Huggenberger, M., Schafer, F., 2008. Natural circulation characteristics at low-pressure conditions through PANDA experiments and ATHLET simulations. Sci. Technol. Nucl. Install., article ID 874969, 14 pp. Papini, D., 2011. Modelling and experimental investigation of helical coil steam generator for IRIS small-medium modular reactor. Ph.D. Thesis, Politecnico di Milano, Department of Energy, cycle XXIII, Milan, Italy, January 2011. Papini, D., Colombo, M., Cammi, A., Ricotti, M.E., Colorado, D., Greco, M., Tortora, G., 2011a. Experimental characterization of two-phase flow instability thresholds in helically coiled parallel channels. In: Proceedings of the International Congress on Advances in Nuclear Power Plants (ICAPP 2011), Nice, France, May 2–6, 2011. Papini, D., Cammi, A., Colombo, M., Ricotti, M.E., 2011b. On density wave instability phenomena: modelling and experimental investigation. In: Ahsan, A. (Ed.), Two Phase Flow, Phase Change and Numerical Modeling. InTech Publisher, Rijeka, ISBN: 978-953-307-584-6, pp. 257–284. Rizwan-Uddin, Dorning, J.J., 1986. Some nonlinear dynamics of a heated channel. Nucl. Eng. Des. 93, 1–14. Rizwan-Uddin, 1994. On density-wave oscillations in two-phase flows. Int. J. Multiphase Flow 20 (4), 721–737. Rizwan-Uddin, 2006. Turning points and sub- and supercritical bifurcations in a simple BWR model. Nucl. Eng. Des. 236, 267–283. Saha, P., Ishii, M., Zuber, N., 1976. An experimental investigation of the thermally induced flow oscillations in two-phase systems. J. Heat Transfer—Trans. ASME 98, 616–622. Santini, L., 2008. Thermalhydraulic issues of IRIS nuclear reactor helically coiled steam generator and emergency heat removal system. Ph.D. Thesis, Politecnico di Milano, Department of Energy, cycle XX, Milan, Italy. Santini, L., Cioncolini, A., Lombardi, C., Ricotti, M., 2008. Two-phase pressure drops in a helically coiled steam generator. Int. J. Heat Mass Transfer 51, 4926–4939. Schlichting, W.R., Lahey Jr., R.T., Podowski, M.Z., Ortega Go´mez, T.A., 2007. Stability analysis of a boiling loop in space. In: Proceedings of the COMSOL Conference 2007, Boston, MA, USA, October 4–6, 2007. Schlichting, W.R., Lahey Jr., R.T., Podowski, M.Z., 2010. An analysis of interacting instability modes, in a phase change system. Nucl. Eng. Des. 240, 3178–3201. Solberg, K.O., 1966. Resultats des Essais d’Instabilites sur la Boucle ‘‘Caline’’ et Comparaisons avec un Code de Calcul. Centre d’Etudes Nucleáires de Grenoble (CENG), Note 225. The Math Works, Inc., 2005. SIMULINKs software. Todreas, N.E., Kazimi, M.S., 1993. Nuclear Systems I: Thermal Hydraulic Fundamentals, second printing. Taylor & Francis, Washington. U.S. NRC Nuclear Safety Analysis Division, 2001. RELAP5/MOD3.3 Code Manual, NUREG/CR-5535/Rev 1. Van Bragt, D.D.B., Van Der Hagen, T.H.J.J., 1998. Stability of natural circulation boiling water reactors: Part I—description stability model and theoretical analysis in terms of dimensionless groups. Nucl. Technol. 121, 40–51. Wallis, G.B., 1969. One Dimensional Two-phase Flow. McGraw-Hill, New York. Yadigaroglu, G., 1981. Two-phase flow instabilities and propagation phenomena. In: Delhaye, J.M., Giot, M., Riethmuller, M.L. (Eds.), Thermohydraulics of TwoPhase Systems for Industrial Design and Nuclear Engineering. Hemisphere Publishing Corporation, Washington, pp. 353–396. Zhang, Y.J., Su, G.H., Yang, X.B., Qiu, S.Z., 2009. Theoretical research on two-phase flow instability in parallel channels. Nucl. Eng. Des. 239, 1294–1303.

Time-domain linear and non-linear studies on density

Time-domain linear and non-linear studies on density

Suggest Documents

Linear and Nonlinear Programming

Linear and nonlinear waves

Linear and Nonlinear Programming

LINEAR AND NONLINEAR PROGRAMMING

Meltblowing: II-linear and nonlinear waves on

Studies of Linear and Nonlinear Photoelectric Emission for Advanced ...

Linear and Nonlinear Lagrange Relaxation

Linear and Nonlinear Programming.pdf - WordPress.com

Controlled Linear and Nonlinear Optical

LINEAR AND NONLINEAR BEHAVIOR OF

LINEAR AND NONLINEAR OPTICAL PROPERTIES

Linear and Nonlinear Nanophotonic Devices Based on Silicon-on

Ab initio studies on the mechanism for linear and nonlinear optical

Linear and Nonlinear Synchronization Analysis and

Relativistic Density Functional Studies on Ligated ... - CiteSeerX

Theoretical studies on high energetic density polynitroimidazopyridines

On the Boundary between Nonlinear Jump Phenomenon and Linear

Some results on controllability for linear and nonlinear heat equations

On the solution of linear and nonlinear partial differential equations

Lectures on Linear and Nonlinear Dispersive ... - Semantic Scholar

Recent results on conservative linear and nonlinear models for a

Attribute Selection Impact on Linear and Nonlinear Regression ...

On the extraction of linear and nonlinear physical parameters in ...

On linear and nonlinear instability of the ... - Archivo Digital UPM