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Nuclear Engineering and Design 250 (2012) 6067

Contents lists available at SciVerse ScienceDirect

Nuclear Engineering and Design

j ournal homepage: www.elsevier .com/ locate /nucengdes

Inertia and compressibility effects on density waves and Ledinegg phenomena intwo-phase flow systems

L.C. Ruspini

Department of Energy andProcess Engineering, Faculty of Engineering Science andTechnology, NorwegianUniversity of Science andTechnology, Kolbjrn Hejes vei1B, Glshaugen,

N-7491 Trondheim,Norway

h i g h l i g h t s

The stability influence ofpiping fluid inertia on two-phase instabilities is studied. Inlet inertia stabilizes the system while outlet inertia destabilizes it. High-order modes oscillations are found and analyzed. The effect ofcompressible volumes in the system is studied. Inlet compressibility destabilizes the system while outlet comp. stabilizes it.

a r t i c l e i n f o

Article history:

Received 28 November 2011

Received in revised form 19 May 2012

Accepted 25 May 2012

Keywords:

Density waves

LedineggThermal-hydraulic

Instabilities

Two-phase flows

Boiling

High-order modes

a b s t r a c t

The most common kind ofstatic and dynamic two-phase flow instabilities namely Ledinegg and density

wave oscillations are studied. A new model to study two-phase flow instabilities taking into account

general parameters from real systems is proposed. The stability influence ofexternal parameters such as

the fluid inertia and the presence ofcompressible gases in the system is analyzed. High-order oscillation

modes are found to be related with the fluid inertia of external piping. The occurrence of high-order

modes in experimental works is analyzed with focus on the results presented in this work. Moreover,

both inertia and compressibility are proven to have a high impact on the stability limits ofthe systems.

The performed study is done by modeling the boiling channel using a one dimensional equilibrium model.

An incompressible transient model describes the evolution ofthe flow and pressure in the non-heated

regions and an ideal gas model is used to simulate the compressible volumes in the system. The use of

wavelet decomposition analysis is proven to be an efficient tool in stability analysis ofseveral frequencies

oscillations.

2012 Elsevier B.V. All rights reserved.

1. Introduction

The occurrence of oscillations and instabilities maycausesevere

damages in many industrial systems, such as heat exchangers,

nuclear reactors, re-boilers, steam generators, thermal-siphons,

etc. These phenomena induced in boiling flows are of relevance for

the design and operation of two-phase systems. Consequently thestability in thermal-hydraulic variables such as mass flux, pressure

and temperature should be studied in detail to better understand

and characterize the conditions for the occurrence of these phe-

nomena. Several types of thermal-hydraulic instabilities can be

found in two-phase flow systems as shown in Bour et al. (1973).

Ledinegg instability, introduced by Ledinegg (1938), is consid-

ered the most common type of static instability. The occurrence

Tel.: +47 73593985; fax: +47 73593491.E-mail address: [email protected]

of this instability is related to the slope of the pressure drop vs.

flow characteristic curve of the system. Several works described

the experimental occurrence of this phenomenon in several kinds

of systems (see Padki et al., 1992; Zhang et al., 2009; Hamidouche

et al., 2009).

The phenomenon called density wave oscillations (DWO), or

thermally induced two-phase flow instability, is the most commontype of dynamic instability occurring in real systems. There exist

several experimental works describing the occurrence of this phe-

nomenon (Ishii and Zuber, 1970; Yadigaroglu and Bergles, 1972;

Saha et al., 1976; Yuncu, 1990; Wanget al., 1994; Dinget al., 1995).

The main contradiction between those works is the description of

high-order modes. While in Yadigaroglus work (Yadigaroglu and

Bergles, 1972) higher-order modes are experimentally observed,

in Saha et al.s (1976) investigation no higher modes are reported,

even if special focus is made on searching for these modes. In

addition, regarding the experimental study performed in the latter

work, it is necessary to remark the fact that in Sahas experiment

0029-5493/$ seefront matter 2012 Elsevier B.V. All rights reserved.

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L.C. Ruspini / Nuclear Engineering andDesign250 (2012) 6067 61

Nomenclature

Lowercase

h specific enthalpy

f Darcy friction factor

t time coordinate

v specific volume

x themodynamic quality

z space coordinate density

Uppercase

Axs cross section area

DH hydraulic diameter

T temperature

TP two-phase

G mass flux

K valve constant

L pipe length

P pressure

PH wet perimeter

Q constant heat source

Subscripts

HS heated section

in inlet

out outlet

l,f liquid

g gas

the heated section is fixed in a by-pass configuration to assure a

constant pressure drop condition (parallel channel condition). In

this case all the pipes are 50mm inside diameter and the heater

is 10mm inside diameter. Moreover, from the given information,

the section before the heater (where the pre-heater, turbine flow-

meter and flex-joint are situated) is at least 2 m long. It means thatthe inlet external inertia is higher than the inertia terms in the

heated section. As it is shown later, the fluid inertia is playing an

important role in the occurrence of high order modes. In the rest

of the experimental works no focus is made regarding high-order

modes. Actually by the data reported in Yuncu (1990), Wang et al.

(1994), Ding et al.(1995) where theprincipal focus is given to pres-

suredrop oscillations(slow oscillations), it is possible to see thatthe

acquisition systemwas notable to samplefast enoughto described

these high-order phenomena.

Regarding the modeling and theoretical background of den-

sity waves oscillations, the investigations carried out in Ishii and

Zuber (1970), Ishii (1971) constitute the theoretical basis in the

understanding of density wave phenomenon. In these works a

thermal equilibrium model is used to describe the system in aone-dimensional model. The IshiiZuber stability maps are also

introduced in those works. In Saha et al. (1976), the use of a non-

equilibrium model is proposed. For low sub-cooling this model

seems to fit better to the experimental data. Nevertheless, in the

high sub-cooling cases the equilibrium model fits better the exper-

imental data. In Furutera (1986), the validity of the homogeneous

model is discussed. In the latter work several pressure, sub-cooling

and heat capacity models are compared with experimental data. It

is proven that in general terms the best approximation is made

with no sub-cooling model and heat capacity of the wall when

that mechanism could be important (massive tubes). In Fukuda

and Kobori (1979) a classification of the different types of den-

sity waves according to the most significant effects occurring in

the system (inertia, gravity, friction) is presented. More recently,

in Rizwan (1994b,a) a homogeneous equilibrium (no sub-cooling)

model is used to study the phenomenon. Several aspects of the

classical theoretical description of DWO are critically discussed.

The introduction of non-uniform heating is discussed in Narayanan

et al. (1997), Rizwan (1994a). The useof commercial codes andsim-

plified lumped methods based on the homogeneous equilibrium

models are described in Achard et al. (1985), Lahey and Podoski

(1989), Ambrosini et al. (2000), Ambrosini (2007). Nevertheless,

none of these theoretical works predict the occurrence of high-

order modes in the sense of Yadigaroglus work. In most of these

works linearization techniques are used to analyze the stability of

the systems. In addition, none of these works takes into account

parameters of the external thermal-hydraulic loop as compress-

ibility (gases) and external fluid inertia (piping). Just in some cases

of vertical tubes a non-heated riser section is modeled butthe main

focus is given to the gravity and pressure drop influence.

The purpose of this work is to analyze the influence of the

fluid inertia and compressibility volumes in different parts of the

thermal-hydraulic loop. A new general model that includes those

external parameters (inertia, compressibility, pump response) is

introduced. A non-dimensional stability analysis of different cases

is presented.

2. General model for instability analysis

In this section a new general model to study stability of two-

phase flow systems is presented, as shown in Fig. 1. This model

consists of: a constant pressure tank, Pout; a variable pressure tank

in order to take into account the pump response and the pump

evolution, Pin(G1, t); a heated section; two different surge tanks

to simulate the effects of compressible volumes (non-condensable

gas), VSi and VSo; four incompressible pipe lines (inertia effects),

Li; and finally four localized pressure drops in each section, Ki. In

this model the pressure difference between both tanks acts as the

driving force and, according to the valves opening, the external

characteristic (P vs. G) results in a quadratic decreasing curve.The implemented model is based on the following assumptions,

One-dimensional model. Two-phase homogeneous model. Thermodynamic equilibrium conditions. Colebrook pressure drop correlation in the single phase region

and two-phase Mller-Steinhagen and Heck pressure drop cor-

relation for two-phase flow region (Thome, 2006).

The mathematical description of the external system (surge

tanks and piping) corresponds to the conservation of momentum,

since an unheated incompressible model is assumed. For the case

of the surge tanks an ideal isothermal gas model is assumed. The

equations of the external system can be expressed as

Psi = P2si

Psi0Vsi0Axsl

(G2 G1) (1)

Pso =P2so

Pso0Vso0

Axsl

(G4 G3) (2)

G1 =Pin(G1, t) Psi K1

G1|G1|2l

1

L1(3)

G2 =Psi P2 (K2 + 1)

G2|G2|2l

1

L2(4)

G3 =P3 Pso (K3 1)

G3|G3|2out

1

L3(5)

G4 = Pso Pout K4G4|G4|2

out

1

L4

(6)

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62 L.C. Ruspini / Nuclear Engineering andDesign250 (2012) 6067

Fig. 1. Scheme of the implemented model.

The mathematical model used to describe the evolution of the

heated section is based on the mass, momentum and energy con-

servation. It can be expressed as

t+ G

z= 0 (7)

G

t+

z

G2

+P

z+ f

DH

G|G|2

= 0 (8)

h

t+ Gh

z= QPH

Ax(9)

As shown in Eqs. (1)(6) the pressure drop in the valves is cal-

culated using a homogeneous local pressure drop model, Ki, for

each valve. Friction losses are neglected in the energy equation

and the friction factor in Eq. (8) is given by the known Colebrook

correlation for the single phase regions (liquid or gas) and by

Mller-Steinhagen and Heck correlation for the two-phase region

(Thome, 2006). The main differences of the heated section model

used in this work and the models used in previous works is that all

the fluid properties are updated in each point as a function of the

local enthalpy and pressure instead of being consider constant.

Even if not all the introduced parameters are analyzed in thiswork, their description is necessary for the sake of describing a

general model to study two-phase instabilities that includes the

main effects present in real industrial cases. None of the previous

mentioned worksanalyze directly these external effects. In Ruspini

et al. (2011b) a complementaryanalysisto the one presentedin this

work but with focus on pressure drop oscillations is presented.

2.1. Stability criteria

The stability to these phenomena is normally analyzed by the

construction of non-dimensional stability maps. In this work the

sub-cooling and Zuber (phase-change) numbers are used following

(Ishii and Zuber, 1970). They correspond to

Nsub = hf hinhfg

fgg

NZu = Npch = QGAxshfgfgg

(10)

The numerical stability limits obtained in this work are com-

pared to existing stability limit correlations. One the most used

limits to analyze the DWO stability is the Ishis simplified stability

criterion (Ishii and Zuber, 1970). It is obtained by the develop-

ment of a thermal equilibrium homogeneous model and it is only

applicable for high subcooling numbers (Nsub >). It is expressedas

Nsub = NZu 2(Kin + Kout+fTP)

1 + (1/2(2Kout+fTP))(11)

whereKin and Koutrepresent the constant inlet and outlet pressure

losses andfTP is the two-phase mixture friction factor.

Another more recent stability limit for DWO is Guido et al.s

(1991) criterion. This stability limit is obtained using a simple

lumped model assuming all the pressure losses represented by an

inlet and outlet valves. The representativevalues of these valves are

assumed constant, represented with * in Eq (13). The expression of

this limit is given by

Npch = 2

1 + 2Nsub

5

2+

2

1 + 2

Nsub

5

2

2+ 1

2 +Nsub(12)

where the value of is given by

= 2Kin +KoutKout +1

(13)

The latter limit for DWO is valid for all the range of subcooling

numbers. As this criterion does not represent distributed friction

losses it is only valid in the cases where the local pressure losses in

the valves are significant respect to the total pressure drop in the

system. In the analysis presented by Guido it is also obtained the

classical stability limit for Ledinegg phenomenon as

Nsub = NZu + 2 . (14)

3. Numerical approximation

The introduction of a significant diffusion by low-order meth-

ods can affect the numerical solution, describing inaccurately the

modeled problem. High-order discretization reduces the numeri-

cal diffusion. The necessity of solving thermal-hydraulic problems

with high accuracy is analyzedin Ambrosini and Ferreri (1998) and

Ruspini et al. (2011a). Thenumericalsolver used inthis work (based

on Least Squares spectral element method) is introduced and

explainedin Ruspini et al. (2010, 2011b,c). An extensive description

of the mathematical and numerical aspects of the implementation

of this kind of methods can be found in Deville et al. (2002). Inthe next section a summary of the mathematical basis needed to

develop the current solver is presented.

3.1. Least squares formulation

In a general case the least squares formulation is based on the

minimization of a norm-equivalent functional. For simplicity, the

system of equations can be represented as

Lu = g in (15)

Bu = u on (16)

withL a linear partial differential operator andBthe trace operator.

We assume that the system is well-posed and the operator (L,B)

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is a continuous mapping between the function spaceX() onto the

space Y()Y().The norm equivalent functional becomes

J(u) 12Lu g2

Y()+ 1

2Bu u2Y() (17)

Based on variational analysis, the minimization statement is

equivalent to:

lim0

dd

J(u+ v) = 0 u X() (18)

Hence, the necessary condition for the minimization ofJ is

equivalent to:

FindfX() such that

A(u,v) = F(v) vX() (19)with

A(u,v) = Lu,LvY() + Bu,BvY() (20)

F(v) = g,LvY() + u,BvY() (21)

A(vh,vh) = F(vh) vh Xh(h) (22)whereA :X

XR is a symmetric, continuous bi-linearform, and

F :X R a continuous linear form.The introduction of the boundary residual allows the use of

spacesX() that are not constrained to satisfy the boundary con-ditions. The boundary terms can be omitted and the boundary

conditions must be enforced strongly in the definition of the space

X(). Finally, the searching space is restricted to a finite dimen-

sional space such that uh Xh()X().

3.2. Numerical description of the internal problem

From Eqs. (7)(9) it is possible to see that the system is non-

linear (quasi-linear). For that reason it is necessary to find a linear

form for this set of equations in order to use Least Square Spectral

Method (LSSM). Non-linear effects and the couplingbetween inter-

nal and the external systems are considered by implementing an

iterative Picard loop. The linearization of the system described in

Eqs. (7)(9) results in

Lint=

z

0 0

t

z

0

0 0 t

+ G z

(23)

gint=

t

zG2

fDH

G|G|2

QPHAx

(24)

uint=

G

P

h

(25)

where G* and * correspond to the old values of flow and densityrespectively. The operatorBint is the matrix where the correspond-

inginitialandboundarynodesaresettooneandtheu corresponds

to the initial values for flow, pressure and enthalpy. In x = 0 the

boundary conditions for enthalpy, flow are added to the vectoru,

while the boundary condition for the pressure is set atx = 1.

3.3. Numerical description of the external problem

In contrast with the internal system, the external system is

solved just as a function of the time (t). The operator description of

this system, Eqs. (1)(6)

Lext=

t

0 0 0 0 0

0 t

0 0 0 0

0 0 t

0 0 0

0 0 0 t

0 0

0 0 0 0 t

0

0 0 0 0 0 t

(26)

gext=

P2si

Psi0Vsi0Axsl

(G2 G1)

P2soPso0Vso0

Axsl

(G4 G3)Pin Psi K1

G1|G1|2l

1

L1Psi P2 (K2 + 1)

G2|G2|2l

1

L2P3 Pso (K3 1)

G3|G

3|

2out

1

L3

Pso Pout K4G4|G4|2out

1L4

(27)

uext=

Psi

Pso

G1

G2

G3

G4

(28)

where the * values correspond to the values of the variables in the

previous non-linear step.

3.4. Spectral element approximation

The computational domain is dividedintoNe non-overlappingsub-domainse of diameterhe, called spectral elements, such that

=Ne

e=1e, e l = , e /= l (29)

The global approximation in , uh, is constructed by gluing thelocal approximations ue

h, i.e.

uh

=Ne

e=1ueh (30)

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Fig. 2. Density wave oscillations and wavelet decomposition. Two different modes

can be observed. This case corresponds to the example ofSection 4.1.3. The pair

(Npch,Nsub) for this caseis (17, 7.5).

The local approximation solution ueh

can be expressed like

ueh(x, t) =N1i=0

N2j=0

ueiji(x)j(t), with ueij = u(xi, tj) (31)

where i(x) and j(t) are the one-dimensional basis functions.These basis functions consist of Lagrangian interpolants polynomi-

als through the GaussLobattoLegendre (GLL) collocation points.

For example the polynomial j(x) defined in the reference domain

= [1,1] is given by

j(x) =(x2 1)(dLN2 (x))/(dx)N2(N2 + 1)LN2 (x)(xxj)

(32)

where the (N2 + 1) GLL-points,xj, are the roots of the first deriva-

tive of the Legendre polynomial of degree N2, extended with the

boundary nodes (Deville et al., 2002).

3.5. Wavelet stability analysis

To evaluate the stability of the numerical solution the evolution

of theinletflow is analyzed usingwavelet decomposition (Addison,

2002). In particular the Mexican hat wavelets family is used. This

analyzing techniqueis useful notjust to analyzethe frequencyspec-

trumbut alsoto obtain theevolution of thespectrumcorresponding

to the signal. For a given function u(t), the Mexican hat wavelet

transform is written as

u(, t) =

u(t)(t t, )dt (33)

where the function is the negative normalized second derivative

of a Gaussian function,

(, t) =d2F

d2t =2

21/4 1 t2

2e

(

t2)/(22)

(34)

Finally the variable is transformed to frequency (Hz)and the signal spectrum is analyzed. In Fig. 2 the wavelet

decomposition of a simulated case is presented. The color-bar

corresponds to the percentage of energy for each coefficient of

the wavelet decomposition. In this case the decomposition is

characterized by two peaks at different frequencies. The lower

frequency component behaves in convergent fashion, while the

high frequency component evolves with a divergent behavior.

The peak position for each characteristic frequency is detected

and an exponential curve is fitted to the maximum values

of the wavelet decomposition evolution at this particular fre-

quency. Then according to the obtained exponential function,

f(t) =Aet

, the stability criterion for each mode corresponds

Fig. 3. Stability map for L1234 =0. This stability map corresponds to the normalDWO mode.Green and black linesare, respectively, Ishiis simplified criterion (Ishii

and Zuber, 1970) and Guido et al.s (1991) correlation. Pink and red lines show

the numerical stability limit and Guidos criterion for the occurrence of Ledinegg

instability. (For interpretation of the references to color in this figure legend, the

readeris referred to theweb version of the article.)

toEq. (35). In this way the use of wavelet analysis allowsto analyze

independently the evolution of different frequency modes.

> 0 stable < 0 unstable (35)

As it is shown in the introduction several mathematical and

modeling techniques could be used to simplify this problem (lin-

earization, Laplace transforms, etc.). Nevertheless the main idea of

studying the system stability in this way (simulating each case) is

to use the same analysis as it used for industrial systems where

those mathematical tools and modeling simplifications cannot be

applied.

4. Numerical results

4.1. Inertia analysis

In the followingexamples the influence of the fluid inertia in theexternal pipes is discussed. The presence of compressible gases in

the system is assumed to be negligible. All the simulations in this

section are done in a system with the following characteristics:

Fluid: R134a LHS=1m, DH=5mm Pout= 8105 Pa, Pin =Pstationary(Gin =500[kg/m2 s]) K1 =10, K2 = 0, K3 = 0, K4 = 1 Vsi =Vso = 0

The numerical order of approximation of time and space is 4.

The number of elements in which the space is discretized is Ne = 50

and the time step is t = 102 sec for all the cases.

4.1.1. No external inertia

In this example a no external inertia case is analyzed,

L1234 = 0. The corresponding stability map is presented in Fig. 3.A total of 204 simulations have been used for the construction of

this map. Most of these simulations are localized close to the sta-

bility limit to assure an accurate description of the system in that

region. Ishiis simplified stability criterion and Guidos criterion are

plotted in this figure. Guidos prediction does notdescribe thelimit

of stability in an accurate way. The difference is probably due to

the simple lumped model used by Guido to obtain that limit. As

described above this limit does not take into account distributed

pressure losses. In contrast,in the case of Ishiis criterionthe stabil-

ity limit is predicted accurately. Nevertheless, for high sub-cooling

this simplified criterion and the present numerical limit does not

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Fig. 4. Stability map for L2 = 1m , L134 =0. This stability map corresponds to thenormal DWO mode. Green and blacklines are,respectively,Ishiisand Guidos crite-

ria. Pink and red lines show, respectively, the numerical stability limit and Guidos

correlation for the occurrenceof Ledinegg instability. (For interpretation of the ref-

erences tocolorin this figurelegend, thereaderis referredto theweb version of the

article.)

follow the same trend. The main differences of the model used in

this work and other works is the two-phase pressure drop model

(Mller) and the factthat all the properties (alsosingle-phase prop-erties) are updated as a function of local the enthalpy and pressure

profiles.

Moreover, the present numerical limit and Guidos correlation

for the occurrence of Ledinegg excursions are also plotted (pink

and red line). The predicted limit differs significantly from the

present numerical limit. Therefore, the usage of Guidos limit for

both Ledinegg and DWO to real cases should be very carefully

analyzed, since these criteria do not seem to reflect the nature of

the involved phenomena. In contrast Ishiis limit seems to predict

properly and conservatively the occurrence of DWO.No high-order

modes are observed in the analyzed simulations for this case.

4.1.2. Inlet inertia

The same case as the previous example is analyzed but in thiscase usinga one meter inlet pipe, L2 =1m, L134 = 0. 226numericalsimulations have been used to construct the map ofFig. 4. Same as

beforethe stabilitylimit criteria forDWO and Ledinegg instabilities

are plotted. None of the stability limits seems to predict accurately

the DWO stability limit, since the models used to obtain these lim-

its does not reflect the influence of external parameters (pipe line

lengths). Moreover the system behaves in a more stable way when

inertia is introduced at the inlet of the heated pipe. No high-order

modes are observed in this case.

4.1.3. Outlet inertia

For this case an outlet one meter pipe is considered, L124 = 0,L3 =1m. A total number of 225 cases have been simulated to con-

struct thestability mapofFig.5(a). Ishiis andGuidosstability limitsfor DWO and Ledinegg are also plotted. None of these criteria seem

to reflect the stability limit correctly, since none of them take into

account external parameters such as the inertia of the fluid in non-

heated pipes. In contrast with the other two examples, in this case

high-order DWO appear for high sub-cooling numbers. The lim-

its for the normal mode and the high-order modes are plotted in

Fig. 5(a) and (b). For Nsub numbers above 5 the high-order modes

become unstable evenwhen thenaturalDWO modeis stable. More-

over for Nsub higher than 10 the higher-order oscillations are not

a pure frequency oscillation but conversely they correspond to the

sum of different frequencies, as shown in the simulation in Fig. 6.

This last fact is completely in accordance with the experimental

data presentedin (Yadigaroglu andBergles, 1972), where for higher

sub-cooling the superposition of different higher-order modes is

Fig. 5. Stability maps for L123 = 0, L4 = 1m . Normal(a) andhigh-order (b)stabilitymaps are shown. Green and black lines represent, respectively, Ishiis and Guidos

criteria.Pink andred linesshow thenumericalstability limitand Guidos correlation

for the occurrence of Ledinegg instability. (For interpretation of the references to

color in this figure legend, thereader is referred to the web version of thearticle.)

observed. The ratio between high-order and normal modes fre-

quencies goes from 2.5 times (Npch =14, Nsub = 5) to approximately

10 times (Npch =21, Nsub = 12).

4.1.4. Discussion

In Fig.7 all the stabilitylimits forthe inertia analysis areplotted.

As it is possible to see, the effects of inlet external inertia stabilize

the system. In contrast, the outlet inertia not only destabilizes the

system but also causes the occurrence of high-order DWO. In con-

clusion, as an important design rule for two-phase systems, it is

possible to say that the outlet pipes (two-phase outlet) should be

shortenedas muchas possible in orderto stabilizenormalDWO and

do not induce high-order modes. Moreover, the occurrence of high-

order modes can affect more strongly the control systems since

Fig.6. Densitywave oscillationsand wavelet decomposition.(Npch,Nsub) forthiscase

are (19, 10.5). In this case thehigh-order modes are thesum of differentfrequency

components.

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Fig.7. Comparisonof thestabilitylimitfor thethreeanalyzedexamples. Noexternal

inertia (blueline), inletinertia (greenline) andoutlet inertia (red line).In thecaseof

outlet inertia higher-modes are observed in the system. The higher-mode stability

limit is also plotted in red line. (For interpretation of thereferences to color in this

figure legend, thereader is referred to theweb version of thearticle.)

their frequency is higher than a normal DWO. The period of the

normal oscillations does not seem to change significantly within

these three different examples. In addition, none of the existing

stability criteria for DWO predicts the change on the stability limitdue to these inertia effects.

These stability limits are in accordance with the experimental

data presented in Yadigaroglu and Bergles (1972). These results

seem to be the link between Sahass and Yadigaroglus experimen-

talworks. Inthosecases itis just necessaryto remarkthatthe outlet

pipe used in Yadigaroglus experiment has had a strong influence

over the system and it seems to induce the described high-order

modes. Onthe other hand, as describedin theintroduction, in Sahas

experiment the used by-pass configuration and the long inlet sec-

tion played a stabilization role and as a result no high-order mode

oscillations are observed.

4.2. Compressibility analysis

In this section theinfluence of compressiblevolumes on the sta-

bility of the system is studied. The same methodology as described

in the inertia cases is applied. The numerical order of approxima-

tion of time and space is 4. The number of elements in which the

space is discretized is Ne = 50and the timestep ist= 102 s for all

the cases. All the simulations in this section are done in a system

with the following characteristics:

Fluid: R134a LHS=1m, DH=5mm

Pout= 8105

Pa, Pin =Pstationary(Gin = 500 [kg/m2

s]) K1 =10, K2 = 0, K3 = 0, K4 = 4 L1 =1m, L2 = L3 = 0, L4 = 1 m

Three different cases are presented: no compressibility, inlet

compressibility and outlet compressibility. In the compressibil-

ity cases a 10 liters volume up-stream or down-stream of the

heated section, respectively, is modeled. The compressible volume

is assumed to behave as an ideal gas, as described in Section 2. As

shownabovea onemeter pipes aresimulated in theinlet andoutlet

of the system. In Fig. 8 a comparison between the stability limits

forthe three differentcases is shown. Significant differences on the

stability limits are observed. For all the cases the oscillations are

mainly dominated by the main mode. Only in the no compress-

ibility case high-order modes are observed for high sub-cooling

numbers. The high-order modes are observed as a wave in the sta-

bility limit for sub-cooling number larger than 7. No high-order

modes are observed in the cases where the compressible volume is

taken into account. For the construction of the maps a total of 227

(no comp.), 260 (outlet comp.) and 280 (inlet comp.) simulations

have been analyzed. Thesoft undulations in the boundaries aredue

to the numerical approximation of the stability surface and do not

represent any physical effect.

4.2.1. Discussion

In Fig. 8 the limit for both Ledinegg and DWO are presented

for the three different cases. For the inlet compressibility case the

stability limit of both Ledinegg and DWO are highly influenced.

These two limitsare more unstable comparedto thecaseof nocom-

pressibility. The destabilization of the Ledinegg limit is explained

by the decoupling between K1 section and the heated section

that the compressible tank produces. So in this case the external

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 2032

1

0

1

2

3

4

5

6

7

8

9

10

11

12

Npch

Ns

ub

Led Guido Limit

DWO Guido Limit

DWOV

so= 10 l

DWONo tankDWO

Vsi

= 10 l

DWO Ishii Limit

LedineggNo tank

LedineggV

so= 10 l

LedineggV

si= 10 l

Fig. 8. Comparison of the stability limit for the case of inlet and outlet compressible volume. For DWO the different cases are: no compressibility case (blue lines), inlet

compressibility(green lines) and outlet compressibility(red lines). Ishiis and Guidos stability limits are plotted in dashed lines. (Forinterpretationof the references to color

in this figure legend, thereader is referred to theweb version of thearticle.)

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characteristic pressure drop vs. flow curve, affecting the heated

section, is not following the valve behavior. Moreover, the stability

limit for density wave oscillations is changing its shape. It seems

that in this case the limit is influenced by the Ledinegg stability

limit. So the influence of the compressible volume is not just mov-

ing the stability limit to the left (destabilizing) but also adding a

region close to the Ledinegg limit as unstable zone.

In contrast when a compressible volume is added in the outlet

of the heatedsection then thelimitfor DWO is stabilized (moved to

the right) for every sub-cooling number. It also stabilizes the high-

order modes, discussed on previous sections. The stabilization of

high-order modes seems to be the consequence of the decoupling

between the heated section and the outlet pipe. In that sense it

cannot be directly concluded that a compressible volume stabilizes

high order modes. This effect is a secondary consequence of fixing

the volume just right after the heater. In any case it can be used as

a valid way of stabilizing the system. As analyzed in Ruspini et al.

(2011b) theinfluence of the compressibletanks will be related with

the distance respect to the heated section. So for both cases (inlet

and outlet) the influence will be stronger as much as the heated

section is closer to the compressible volume.

In conclusion the insertion of a compressible volume after the

heatedsectioncanresultin a more stablesystem.Moreoverin most

of the two-phase systems the tanks where vapor can produce theeffect of compressible volume are located down-stream the heated

section. In addition, special attention has to be paid to the use of

pressurizers andexpansion volumes whentheyare fixedup-stream

of the heated section.

5. Conclusions

A general model that takes into account several external param-

eters (external to the heated section) such as fluid inertia in pipes,

compressible gasesand pumpresponse is introduced.Densitywave

and Ledinegg phenomena are analyzed with focus on the influ-

ence of these external parameters not taken into account in other

works.

The inertia influence of the connecting pipes is analyzed. It isfound that the inlet inertia (longer inlet pipes) increases the sta-

bility of the system. On the contrary for the increase of outlet

inertia (longer outlet pipes) the stability of the system is not just

decreased but also high-order oscillations are induced. The occur-

rence of high-order oscillation modes, reported experimentally, is

described. The frequency of these high-order oscillations is found

to be between 2.5 and 10 times the frequency of the normal mode,

according to the (Npch,Nsub) region. Different modes are induced

in different parameter region as observed experimentally. Wavelet

decomposition is proven to be an efficient analysis technique when

the evolution of different modes is needed.

The effect of compressible gases in the system is also studied.

It is found that when a compressible volume is up-stream of the

heated section, then the system becomes highly more unstable forboth Ledinegg and density wave phenomena. In contrast,when the

compressible volume is fixed down-stream of the heated section

the system becomes more stable in the density wave sense.

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