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Nuclear Engineering and Design 240 (2010) 31783201

Contents lists available at ScienceDirect

Nuclear Engineering and Design

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / n u c e n g d e s

An analysis of interacting instability modes, in a phase change system

William R. Schlichting, Richard T. Lahey Jr. , Michael Z. Podowski

Center for Multiphase Research, Rensselaer Polytechnic Institute, Troy, NY 12180-3590, USA

a r t i c l e i n f o

Article history:

Received 2 March 2010

Received in revised form 22 April 2010

Accepted 25 April 2010

a b s t r a c t

This paper presents an analysis of the interaction of pressure-drop oscillations (PDO) and density-wave

oscillations (DWO) for a typical NASA type phase changesystem. A transientlumped parameter model is

developed for use in the analysis of the dynamics of this type of system. A compressible volume (e.g., an

accumulator vessel) dynamics model was also developed and PDO/DWO interactions are investigated. 2010 Elsevier B.V. All rights reserved.

1. Introduction

The purpose of this paper was to analyze interacting instability modes in phase change systems, such as those used in nuclear reactors.

This research has been motivated by the requirements of space exploration. In particular, an ambitious long range research program has

been formulated by NASA forthe human exploration anddevelopmentof space (HEDS). This program includes thepossibledevelopment of

a lunar base and subsequent manned-missions to Mars. It has been found that one of the key enabling technologies which will be required

to support the power, propulsion and life support aspects of these missions is the use of a phase change system in space ( NRC, 2000). In

particular, it appears that an on-board nuclear fission reactor, and an associated Rankine cycle energy conversion system, may be required

to supply reliable power and propulsion for NASAs missions (NRC, 2000). This, in turn, implies that one must be able to reliably predict

the performance of two-phase flows and phase change systems (e.g., boilers and condensers) in microgravity (i.e.,106 g) environments.

There is a need to develop a better understanding of the effect of gravity on many important multiphase phenomena, such as flow

regimes, critical heat flux (CHF), pressure drop and phase change system stability (NRC, 2000).

This paper is focused on the development of the analytical capability required to analyze the effect of gravity on phase change system

stability and the possible interaction of various system instability modes in typical phase change systems. In particular, we focus here on

the analysis of the interaction of density-wave oscillations (DWO) and pressure-drop oscillations (PDO) in a typical experimentwhich could

be performedboth on earth and aboard the International Space Station (ISS). These types of interactions are of concern in NASAs proposed

nuclear fission reactors, since any associated Rankine cycle energy conversion system will necessary be operating at low pressures to

minimize weight.

Fig. 1 shows a typical test loop which was designed for use by NASA. The size of this test loop is appreciably smaller than a typical

boiling system so that it can be tested aboard the ISS. Nevertheless, when properly scaled, the system instabilities are representative of

much larger phase change systems. The geometry and operating parameters of the test section shown in Fig. 1 are given in Table 1.

Various thermalhydraulic system instabilities have been observed in phase change systems. Typical instabilities are density-wave

oscillations (DWO), flow excursions (i.e., Ledinegg instabilities), and pressure-drop oscillations (PDO). Both density-wave oscillations and

pressure-drop oscillations are classified as dynamic instabilities whereas an excursive instability is normally classified as a static instability

(Lahey and Moody, 1993). Kakac and Bon (2008) have done a comprehensive review of dynamic system instabilities, including when there

may be interaction between DWO and PDO.A density-wave instability can produce oscillations in the flow, which may either increase or decay in amplitude. These oscillations

occur when a fluctuation in the subcooledinletflow creates an enthalpy perturbationin thesingle-phase regionof theheated channel, such

that the boiling boundary begins to oscillate due to these enthalpy perturbations. As a consequence, these enthalpy perturbations induce

quality and void fraction perturbations in the two-phase region of the test section. These perturbations cause, in turn, a perturbation in the

two-phasepressuredrop. If thetotal pressure drop acrossthe heatedchannel is imposed (e.g.,due toa parallel channel boundarycondition),

a perturbationin thetwo-phase pressure drop induces a feedback perturbation in the pressure drop of thesubcooled (single-phase) region,

which can either reinforce or attenuate the initial flow perturbation, and if in phase, may lead to a self-sustained DWO.

Corresponding author.

E-mail address: [email protected] (R.T. Lahey Jr.).

0029-5493/$ see front matter 2010 Elsevier B.V. All rights reserved.

doi:10.1016/j.nucengdes.2010.05.057
http://localhost/var/www/apps/conversion/tmp/scratch_4/dx.doi.org/10.1016/j.nucengdes.2010.05.057http://localhost/var/www/apps/conversion/tmp/scratch_4/dx.doi.org/10.1016/j.nucengdes.2010.05.057http://www.sciencedirect.com/science/journal/00295493http://www.elsevier.com/locate/nucengdesmailto:[email protected]://localhost/var/www/apps/conversion/tmp/scratch_4/dx.doi.org/10.1016/j.nucengdes.2010.05.057http://localhost/var/www/apps/conversion/tmp/scratch_4/dx.doi.org/10.1016/j.nucengdes.2010.05.057mailto:[email protected]://www.elsevier.com/locate/nucengdeshttp://www.sciencedirect.com/science/journal/00295493http://localhost/var/www/apps/conversion/tmp/scratch_4/dx.doi.org/10.1016/j.nucengdes.2010.05.057

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W.R. Schlichting et al. / Nuclear Engineering and Design240 (2010) 31783201 3179

Fig. 1. Typical NASA test loop.

Non-periodic flow excursions arise from the interaction between the systems impressed head flow characteristics and the hydraulic

characteristics of the heated channel. An excursive instability may occur when trying to operate where the slope of the steady-state

pressure drop versus flow curve of the heated channel is negative (e.g., region B to D in Fig. 2). We can only operate on this branch of the

systems pressure-drop curve when the flow is forced by a suitable pump (e.g., a positive displacement pump). In general, when the slopeof the pressure drop versus flow curve is negative, we may have an unstable fixed point, point A, which is a repeller ( Strogatz, 1994). For

an impressed constant pressure drop, fluctuations in the channels flow will cause the flow to be repelled from this unstable fixed point

to one of the two stable fixed points, or attractors (e.g., A or A when having the same pressure drop, as for a parallel channel boundary

condition), depending on the sign of the perturbation. If the flow reaches the single-phase attractor(e.g., the stable fixed point, A, inwhich

the flow is single-phase liquid throughout the heated section) then the flow will be stable. In contrast, if the flow is sent to the two-phase

attractor, A, then the flow may operate at this fixed point or it may oscillate about it in the form of a DWO ( Kakac and Bon, 2008; Yin et

al., 2006).

A pressure-drop instability can produce oscillations which are somewhat similar to those found in density-wave oscillations, but at a

different frequency, with the frequency based on the dynamics of the compressibility of the system. In particular, a compressible volume

will result from the use of an accumulator, such as that shown in Fig. 1. During a pressure-drop oscillation (PDO), the flow may be diverted

from entering the test section and will enter the accumulator. The increase of fluid in the accumulator volume compresses the gas in the

accumulator, which after an over-compression, will later expand (actually over-expand), pushing fluid out of the accumulator and back

into the inlet of the test section. These pressure-drop oscillations only occur for the case when, if the accumulator was coupled to the inlet

of the test section, an excursive instability can occur.Stenning et al. (1967) verified that a negative slope in the heaters pressure-drop characteristics was required for the pressure-drop

oscillations (PDOs) to occur. These oscillations were associated with the presence of a compressible volumeat the inlet of the heater. Kakac

and Bon (2008) described a PDO as a flow oscillation that roughly traces the limit cycle EBCDE in Fig. 2. Stenning and Verizoglu (1965) and

Maulbetsch and Griffith (1965) observed density-wave oscillations (DWOs) when operating on the EB branch of the systems operating

Table 1

Proposed ISS test section.

Fluid FC-72

System pressure 300 kPa

Heated channel type Annular

Hydraulic diameter 0.00127 m

Heated perimeter 0.03591 m

Channel area 2.407105 m2

Heated length 0.3048 m

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3180 W.R. Schlichting et al. / Nuclear Engineering and Design240 (2010) 31783201

Fig. 2. Pressure drop vs. mass flux curve.

curve. They also observed that PDOs occurred only on part of the negative slope region, instead of theentire region between B andD (Fig.2).

In related analyses, Ozawa et al. (1979) and Dogan et al. (1983) made the restrictive assumption that the inflow into the system, Gin, prior

to the accumulator, was constant and that the exit pressure of the heated channel was also constant. These are not valid assumptions forthe test loop shown in Fig. 1, and, as will be discussed subsequently, a different formulation is required for the analysis of this test loop.

2. Discussion

The simplest model which can describe the dynamics of a boiling channel are the one-dimensional continuity, momentum, and energy

conservation equations based on a homogenous equilibrium model (HEM), in which phasic slip and subcooled boiling are neglected. The

assumption normally made is that the system pressure is constant, which implies the pressure dependent fluid properties are given at

the system pressure. As the test section has a subcooled liquid and a two-phase region, each region has different conservation equations

associated with them.

The two-phase HEM conservation equations are, for a constant flow area and a constant, uniform, axial heat flux ( q):

t +

zG = 0 (1)

tG +

z

G2

=

zp g

f

2DH

G2

N2

n=1

Kn2

G2

(zzn) (2)

th + G

zh =

qPHAXS

(3)

and the equations of state are:

h = hf + xhfg (4a)

=1

[vf + xvfg](4b)

The mixture energy equation, Eq. (3), can be recastinto the form of a quality propagation equation by inserting theHEM state equations,

Eqs. (4a) and (4b), into Eq. (3) (Lahey and Moody, 1993) to yield:

tx + j

zx x = vf

vfg(5)

where the characteristic frequency of phase change, , and the superficial velocity of the two-phase mixture, j, are respectively:

=qPH

AXShfgvfg (6)

j =jin +(z ) (7)

In Eq. (7), jin is the inlet velocity to the heater and is the location of the boiling boundary (i.e., where bulk boiling begins).The single-phase conservation equations are a special case of the two-phase conservation equations. Due to the assumption that the

fluid properties are constant (i.e. evaluated at a constant system pressure), the single-phase fluid may be treated as an incompressible

fluid. For single-phase flow, Eqs. (1)(3) simplify to:

z

G = 0 (8)

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tG =

zp gf

f

2DH

G2

f

N1n=1

Kn2

G2

f(zzn) (9)

f

thl +G

zhl =

qPHAXS

(10)

Eq. (8) implies that the mass flux through the single-phase liquid region of the heated channel is equal to the inlet mass flux. Eq. (10)

can be used to determine the location of the boiling boundary, , where the boiling boundary is defined to be the axial position in theheater in which the liquid enthalpy is equal to the saturation enthalpy of the liquid:

hl(z= ) = hf (11)

When the heated channel is subjected to a perturbation in the flow, for an imposed pressure drop, the oscillations in the flow can either

increase in amplitude or decay back to the initial steady-state value, based on the operating conditions being perturbed. A system is stable

if the oscillations decay back to the steady-state and is unstable if the oscillations grow. A system in which small amplitude oscillations

neither grow nor decay is said to be neutrally stable. Whether the flow oscillations will increase, decrease, or remain at a very small

amplitude, can be determined without running transients in the time-domain by analyzing the stability of the linearized conservation

equations about a steady-state operating condition.

The operating conditions of the test section can be described using non-dimensional numbers. By introducing these non-dimensional

numbers, the conservation equations can be simplified by removing the majority of the fluid-dependent properties and channel geometry.

The inlet enthalpy is quantified by a subcooling number, Nsub, Eq. (12), and the phase change number, Npch, Eq. (13), which describesthe ratio of the power to the flow. The friction number, , Eq. (14) denotes the relationship between the friction factor (f), the hydraulicdiameter (DH), and the lengthof the heatedchannel (LH). For the geometry givenin Table 1, the friction numberhas a value of 2.436.Finally,

the Froude number, Fr, Eq. (15), quantifies the effect of gravity.

Nsub =hf hin

hfg

vfg

vf(12)

Npch =q0PHLH

Gin0AXShfg

vfg

vf(13)

=f

2DHLH (14)

Fr=j2

in0

gLH=

G2in0

2f

gLH(15)

It is convenient to remove the spatial dependence in the conservation equations by integrating them along the single-phase and two-

phase regions of the heated channel and by introducing appropriately spatially-averaged terms into these equations. Integrating Eq. (8)

along the subcooled length implies, as noted previously, that the mass flux throughout the single-phase region is equal to that of the inlet

mass flux. Integrating Eq. (10) along the single-phase region results in a boiling boundary () dynamics equation:

d

dt = 2

Gin0f

Gin

Gin0

0

(16)

in which the steady-state boiling boundary, 0, can be expressed as:

0 =NsubNpch

LH (17)

The average two-phase density dynamics equation is determined from the two-phase continuity equation, Eq. (1):

d

dt2 = 2

Gin00

1

2f

LH 2 (18)

where the average density two-phase is defined as:

2 =1

LH LH

dz (19)

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The final state variable equation in the nodal model is the lumped parameter momentum equation for the heated channel, which is

comprised of the combined single-phase and two-phase momentum equations: + (LH )

2f

d

dtGin = p 2Gin0

0

Ginf

+

2(LH )

1

2f

Kin2

1

G2in

f

Kout

2+ 1

Gin

2

2f

1

+ (LH )2

Ginf

+(LH )

gf

+ (LH )

2f

2Gin0f

Gin

1

2f

(LH )2

Gin

Gin0

0

+ (LH )2

Ginf

+

2(LH )

G2

in

f

LH+ 2

Ginf

+

2(LH )

21

LH

(20)

These nodal HEM conservation equations are derived in Appendix A. It should be noted that similar, but more detailed, nodal models

can be derived using a drift-flux model, and taking into account subcooled boiling (Schlichting, 2009).

3. Linear analysis

Eqs. (16), (18), and (20) can be linearized to investigate the neutral stabilityboundaryof the nodal model, fora uniformlyheated parallel

channel. The resulting linear system can be written as:d

dt =A (21)

where

- (t) 0 = [ Gin 2 ]

T(22)

and

A =

a11 a12 a13a21 a22 0

0 a32 a33

(23)

The terms in the Jacobian matrix, A, are:

a11 =

[(Kout/2)(1 (20/f)) + 1](LH 0) (Kin + Kout+ 2)(Gin0/f)

0 + (LH 0)(20/f) (24a)

a12 = (Kout/2 + 1)(u20 + uout0)20 Gin0((LH 0)/0)(20/f) + 3/2Gin0(1 (0/LH))

0 + (LH 0)20/f

(f 20)2(Gin0/f)(u20/(LH 0))(LH/0) +g

0 + (LH 0)(20/f)(24b)

a13 = Koutuout0u20 +u20(LH 0) +u20

2(1 (0/LH)) +g(LH 0)

0 + (LH 0)20/f(24c)

a21 =2

f(24d)

a22 = 20Gin0f

(24e)

a23 = 0 (24f)

a31 = 0 (24g)

a32 = 2Gin00

LH

(LH 0)2

1

20f

(24h)

a33 = 2u20

LH 0(24i)

where the average two-phase velocity and the two-phase exit velocity are, respectively:

u20 =Gin0f

+

2(LH 0) (25)

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Fig. 3. Linearized nodal model neutral stability boundary.

uout0 =Gin0

f+(LH 0) (26)

Assuming a modal solution of the form,

-= - exp(t) (27)

Eqs. (21) and (27) yield a cubic characteristic equation, where are the eigenvalues of the system:

3 2(a11 + a22 + a33) + (a22a33 + a11a33 + a11a22 a12a21) + a12a21a33 a13a21a32 a11a22a33 = 0 (28)

At least one of the eigenvalues will be real and the other two will either be real or complex conjugates. For the complex conjugate

eigenvalues, the operatingconditions thatwill generate the neutral stabilityboundary arethose in whichthe complex conjugateeigenvalues

are purely imaginary (i.e., the real part is zero). Fig. 3 shows a typical neutral stability boundary for the heated channel in the NASA test

loop (Fig. 1) using the nodal model.

A comparison between an exact analytical solution (Lahey and Moody, 1993; Lahey and Podowski, 1989; Yin et al., 2006) and the nodal

HEM models prediction of the neutral stability boundary is given in Fig. 4 for the NASA test section. For the same subcooling number,

the lumped parameter model yields a larger stability margin than that for the exact analytical solution, and thus it is somewhat non-

conservative. This discrepancy is due to the formulation of the simple, two node, lumped parameter model. Increasing the number of

nodes in the lumpedparameter model yields a neutral stabilityboundary which agrees with the exact analytical model (Garea et al., 1999).

Nevertheless, the basic phenomena are well captured by the simple two node lumped parameter model.

Fig. 5 shows that for the NASA test section shown in Fig. 1 the effect of gravity implied by the nodal model is not great. However, one

must be careful in generalizing these results since other analysis (Achard et al., 1981) has shown that the effect of gravity strongly depends

on heater geometry, orientation and operating conditions.

Fig. 4. Neutral stability model comparisons.

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Fig. 5. Effect of gravity on the neutral stability boundary.

Fig. 6. Nodal model: transient for a stable point.

Fig. 7. Nodal model: transient for operating conditions near the neutral stability boundary.

4. Non-linear analysis

Figs. 68 show typical transients for the NASA test section using the non-linear nodal model, given by Eqs. (16), (18) and (20). The

initial perturbation used in these figures was an initial mass flux perturbation which was one percent above the steady-state inlet mass

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Fig. 8. Nodal model: transient envelope for an unstable fixed point leading to a limit cycle (with an insert of the oscillations).

Fig. 9. Three transient limit cycle envelopes.

flux. The heater power was different in these three figures, as shown by the increase in the phase change number, Npch, for a constant

steady-state inlet mass flux (Gin0). Fig. 6 shows an inlet mass flux transient decaying sinusoidally to the steady-state, signifying that the

system is damped. Fig. 7 shows a system that is essentially neutrally stable, with the oscillations neither growing nor decaying (i.e., a small

amplitude limit cycle).

Fig. 8 plots the envelope (the values of the maximum and minimum of the oscillations during limit cycle oscillations) for a system that

is unstable, with the amplitude of the mass flux oscillations increasing until reaching a maximum (i.e., as the system reaches a relatively

large amplitude limit cycle). The insert in Fig. 8 shows the oscillations of this limit cycle, which has a period, TDWO, that is approximately

twice the fluids transient time in the heater (Lahey and Podowski, 1989).

It is interesting to analyze the envelopes of the limit cycles. By using the envelopes, multiple limit cycles can be conveniently viewed

in one figure. Fig. 9 shows three such inlet mass flux envelopes, the values of the maximum and minimum of the oscillations during the

limit cycles, for different phase change numbers for the NASA test section. Those transients with a higher power level, and thus a higher

phase change number (since the steady-state inlet mass flux was the same for all transients), will have larger oscillations and will reach a

limit cycle more quickly than those closer to the neutral stability boundary.

The increase in the amplitude of the oscillations for higher phase change numbers implies that our non-linear nodal model might be

experiencing a supercritical Hopf bifurcation (Achardet al., 1985). For example, consider the control parameter, , given byEq. (29) below,where Npch,C is the critical phase change number on the neutral stability boundary:

=Npch Npch,C

Npch,C(29)

For negative values of, theflow is stable, andthusit isexpected todecayto thesteady-statein response toa smallexternalperturbation.For positive values of, the flow is unstable, and for a supercritical Hopf bifurcation, the amplitude of the limit cycle oscillations will beproportional to the square root of. Fig. 10 shows that, for the NASA test section, this is indeed the case.

Fig. 11 shows a pressure-drop vs. mass flux curve for a heated channel in which the operating conditions are at point A (i.e., see Fig. 2).

Recalling the discussion ofFig.2, a perturbation off of the steady-state curve will induce an excursive (Ledinegg) instability to occur, causing

the flow to move to either point A

or A

, depending on the direction of the initial perturbation.

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Fig. 10. Hopf bifurcation diagram.

Fig. 11. Pressure drop vs. mass flux curve.

Fig. 12. Excursive (Ledinegg) instabilities leading to either a single-phase flow or a two-phase flow with density-wave oscillations (DWO).

In Fig. 11, if a parallel heated channel is operating on the negative slope region (e.g., point A) of the steady-state operating curve and the

flow is perturbed to point A , the flow will become a single-phase liquid throughout the channel and thus be stable. A flow going to point

A may experience DWO after the initial Ledinegg excursion, depending on the stability of point A . Fig. 12 shows these two transients

for a given Ledinegg instability. The dashed curve is the transient to point A

and the solid curve is the transient to point A

, which shows

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Fig. 13. Heated channel and an inline accumulator geometry for a test loop with a constant inlet mass flux.

a Ledinegg-type excursive instability that leads to a DWO. The initial steady-state was a repeller, and the flow was sent to an attractor,

which, in the case of point A , was unstable, leading to a relatively large amplitude DWO limit cycle.

5. Accumulator dynamics

By adding an accumulator at the inlet of the boiling channel the dynamics of the system can change and there is the possibility of

pressure-drop oscillations (PDOs), where the occurrence of these oscillations depends on the geometry of the accumulator which may

have a surge line connecting the accumulator to the inlet of the heated channel. The test section and type of accumulator analyzed by

Ozawa et al. (1979) and Dogan et al. (1983) is shown schematically in Fig. 13. The inlet mass flux prior to the accumulator junction

(Gin0) and the exit pressure (Pout) were specified as being constant, and the imposed pressure drop on the heated channel is between

the junction (pJ), which may vary in time, and the outlet of the heated channel; or conversely, between the accumulators gas pressure

(pgas) and the exit pressure. These are rather non-physical boundary conditions, but such a system can be investigated experimentally and

analyzed.

By introducing the accumulator dynamics, the system of conservation equations are modified. In particular, Eq. (16) is modified to use

the inlet mass flux to the heated channel, GHC, rather than Gin:

d

dt = 2

Gin0f

GHCGin0

0

(30)

and the corresponding modified momentum equation, Eq. (20), becomes: + (LH )

2f

d

dtGHC =pJpout 2Gin0

0

GHCf

+

2(LH )

1

2f

+

G2HCf

Kout

2+ 1

GHC

2

2f

1

+ (LH )2

GHCf

+ (LH )

gf

+ (LH )

2f

2Gin0f

GHC

1

2f

(LH )2

GHCGin0

0

+ (LH )2

GHCf

+

2(LH )

G2HC

f

LH+ 2

GHCf

+ 2

(LH )2

1 LH

gf

+ (LH ) 2f

(31)

where the driving pressure drop is between the so-called junction (J) and the impressed outlet pressure. The average two-phase density

dynamics equation, Eq. (18), does not have to be modified.

Let us now consider the accumulator dynamics. The continuity of the liquid in the accumulator vessel (for a positive flow rate, wacc)specifies an inflow that increases the height of the liquid level in the accumulator vessel:

fAaccd

dtHacc= wacc (32)

and the corresponding momentum equation in the accumulator vessel is:

HaccAacc

d

dtwacc=pJpgas fgHacc 8lwacc

Hacc

A2acc

Kin,acl|wacc|

2f

wacc

A2acc

(33)

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Fig. 14. Nodal model: test section with an accumulator transient inlet mass flux of the heated channel.

Fig. 15. Nodal model: test section with an accumulator transient boiling boundary.

The pressure of the gas in the accumulator is related to the height of the liquid level by, noting that the cross-sectional area of the

accumulator is constant,

pgas =pgas,i

Hacc,m Hacc,iHacc,m Hacc

n(34)

where n can either be unity, for an isothermal process, or = cp/cv=1.4, for an isentropic process. The parameter Hacc,m is the maximumheight of the liquid level (i.e., the height of the vessel), while Hacc,i is the initial liquid level height. For most purposes, the compressible

volume dynamics can be treated as an isothermal process (i.e. n = 1.0). For a constant mass flux into the system, the accumulator flow rate

is given by the continuity equation at the junction:

wacc= (Gin0 GHC)AXS (35)

Substituting Eqs. (34) and (35) into Eq. (32), and allowing the accumulator gas pressure to replace the liquid level height as a state

variable, yields:

d

dtpgas =

nAXSfAacc

pgas1+1/n

p1/n

gas,i(Hacc,m Hacc,i)

(Gin0 GHC) (36)

For an isothermal process, Eq. (36) simplifies to the state variable equation given by Gurgenci et al. (1983) and Akyuzlu et al. (1980),

neglecting liquid volume evaporation in the accumulator.

Figs. 1417 show transients for a typical heated channel/accumulator system (i.e., see Fig. 1). The heated channel operating con-

ditions are those in Fig. 11, with the initial conditions corresponding to point A. The exit pressure was imposed to be the same

pressure as given in the transients in Fig. 12. The initial and maximum heights of the accumulator vessel liquid level were given as

was the diameter of the accumulator vessel. These three geometrical values can be varied to determine their effects on system dynam-

ics.

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Fig. 16. Nodal model: test section with an accumulator transient accumulator gas pressure.

Fig. 17. pHCGHC phase plane plot of a PDO superimposed on a steady-state operating curve.

Fig. 14 shows the transient of the mass flux entering the inlet of the heated channel, in which the maximum and minimum values

over/under shoot the values of the mass flux at points A and A, respectively, thus showing that the inlet mass flux is oscillating between

the two attractors as a typical PDO.

Fig. 15 shows the corresponding transient of the non-dimensional boiling boundary in the test section, defined by:

=

LH. (37)

Forthis transient, the flowthroughout the heatedchannel becomes single-phase as theboiling boundary reaches, or exceeds, the length

of the channel (i.e., * = 1), before returning to two-phase flow at the exit.A change in the accumulator gas pressure (about a 2 kPa difference between the maximum and minimum values) is shown in Fig. 16.

Fig. 17 is a phase plot of the transient superimposed with the corresponding steady-state pressure drop vs. mass flux curve (the dashedcurve) for the pressure drop between the junction and exit pressures:

pHC =pJpout (38)

The essential difference between the junction pressure and the accumulator gas pressure is the hydrostatic head of the fluid, as the

friction and local losses are negligible compared to the gravity head. As expected, the change in the pressure drop between the exit and

junction pressures is about 2 kPa, similar to the change in pressure in accumulator gas pressure.

The pressure-drop oscillation (PDO) shown in Fig. 17 has a period ofTPDO and is centered about an initial mass flux of 1200 kg/m2 s. It

is similar in shape to the experimental limit cycles presented by Stenning et al. (1967) and Ozawa et al. (1979). Fig. 18 shows the effect of

the accumulator vessel diameter on the corresponding limit cycle. As can be seen, the smaller the accumulator diameter, the smaller the

range of mass flux variation in the PDO transient. While these phenomena are interesting, they are not typical of the transients that may

occur in the NASA test loop shown in Fig. 1.

Fig. 19 shows a schematic of the heated channel and the interconnected accumulator for the NASA test loop shown in Fig. 1. The total

pressure drop along the channel (p =pin pout) may be impressed as a parallel channel boundary condition, however, as in the previous

case, the pressure at the junction,pJ, between the heated channel andthe accumulator line, can vary with time. With that pressure varying,

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Fig. 18. pHCGHC phase plots of PDOs superimposed with the steady-state operating curve showing the effect of accumulator vessel diameter.

Fig. 19. Heated channel and accumulator geometry based on NASA test loop (Fig. 1).

the imposed pressure drop on the heatedchannel, pHC=pJpout, also varies. Part ofthe mass flux at the inlet tothe junction, Gin, can havea portion diverted into or out of the accumulator line (given by the flow rate, wacc; a positive flow defined as flow into the accumulator),while the rest of the mass flux goes into the heated channel, GHC.

The heated channel dynamics equations, Eqs. (18), (30) and (31) and the accumulator vessel continuity equation, Eq. (32) still apply for

this new accumulator model geometry. However, the previous accumulator momentum equation, Eq. (33), must be modified to include

the effect of the accumulator line:Lacl

Aacl+

HaccAacc

d

dtwacc=pJpgas

AaclAacc

1

|wacc|wacc

fA2acl

gf(Hacl +Hacc) 8f

Lacl

A2acl

+Hacc

A2acc

wacc Kacl

|wacc|wacc

2fA2acl

(39)

where

Kacl = Kin,acl + Kacl + Kout,acl (40)

We note that by setting Lacl = Hacl = 0 and Aacl =Aacc, Eq. (39) reduces to Eq. (33).

Closure comes from continuity at the junction, Eq. (41a) (where now Gin can vary with time), the pressure drop at the inlet local loss, Eq.

(41b), and the impressed constant pressure drop from the inlet to the exit pressure, Eq. (41c), for a parallel channel boundary condition:

wacc= (Gin GHC)AXS (41a)

pin =pin pJ =Kin2

G2in

f(41b)

p = pin + pHC (41c)

Combining and linearizing Eqs. (32), (34) and (39), we obtain a second-order system for the accumulators dynamics:

d2

dt2Hacc+ 2n

d

dtHacc+

2nHacc=

1

((Lacl/Aacl) + (Hacc,i/Aacc))fAaccpJ (42)

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Fig. 21. Nodal model: test section with an accumulator transient accumulator flow rate.

where the angular frequency (and thus the period of oscillation) is:

n =

n(pgas,i/(Hacc,m Hacc)) +gf

fAacc((Lacl/Aacl) + (Hacc,i/Aacc))=

2

TPDO(43a)

and the damping factor is:

=4f

n

(Lacl/A2acl

) + (Hacc,i/A2acc)

(Lacl/Aacl) + (Hacc,i/Aacc)= 4f

((Lacl/A2acl

) + (Hacc,i/A2acc))

fAacc

(n(pgas,i/Hacc,m Hacc) +gf)((Lacl/Aacl) + (Hacc,i/Aacc))

(43b)

The basic geometry of the accumulator vessel and surge line is taken to have the length of the surge line and the liquid level heightcomparable to the length of the heated channel, and for the flow area of the surge line to be on the same order of magnitude as the flow

area of the heated channel. The geometry of the accumulator vessel is that of the model given for Fig. 13, for consistency between the two

different accumulator models.

Whereas the accumulator shown in Fig. 13 oscillates about the fixed inlet mass flux, the accumulator shown in Fig. 19 will initially

undergo an excursion to either point A or A (Fig. 11). An excursive transient from point A to A is shown in Fig. 20, with the only difference

in this transient and the classical Ledinegg excursion, shown in Fig. 12, being the damped oscillations about point A, due to effect of the

damped accumulator dynamics.

The oscillations in Fig. 20 occur due to the change of the operating pressure of the gas in the accumulator vessel. When the system is

operating at a steady-state (i.e., point A in Fig. 11), the mass flux at the inlet of the system, Gin, is equal to the mass flux at the inlet to

the heated channel, GHC. An increase of the mass flux into the heated channel causes the junction pressure, pJ, to decrease. At steady-state

operating conditions, the difference between the junction pressure and the gas pressure is just the hydrostatic head in the accumulator;

therefore the pressure of the gas would also decrease during the excursive transient. To reduce the gas pressure, the height of the liquid

level must decrease, in accordance with Eq. (34). Flow out of the accumulator, a negative flow, increases the flow into the heated channel,

causing the accumulator outflow to overshoot and then oscillate to the new steady-state operating point (point A

) (Fig. 21).

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Fig. 23. *GHC phase plane plot.

A transient with an initial perturbation opposite that in Fig. 20; that is, from point A to A (see Fig. 11), is given as Fig. 22. The effect

of the accumulator dynamics adds PDO interaction to the DWO expected for such a system without an accumulator vessel (Fig. 12). As

can be seen for the case analyzed the PDO and DWO have about the same frequency and their interaction yields a beating-type periodic

oscillation. It is interesting to note that similar PDO/DWO interactions have been found using linear frequency domain techniques ( Yin et

al., 2006).

Fig. 23 is the corresponding phase plane plot of the non-dimensional boiling boundary, Eq. (37), and the mass flux into the heated

channel. This phase plot shows the results of the two interacting limit cycles. Interestingly, the flow at the exit of the heated channel never

becomes single-phase and Fig. 24 gives the dynamics of the heated channels pressure drop, pHC, which indicates the transient junctionpressure, pJ.

The corresponding phase plane plot of the inlet mass flux to the heated channel, GHC, and the heated channels pressure drop, pHC,

superimposed with the steady-state operating curve, is shown in Fig. 25. Note that we have a period two DWO, which oscillates about thetwo-phase attractor, and is superimposed with a periodic PDO.

As before (see Fig. 18), changing the accumulator vessel diameter, Dacc, will change the resulting transient. However, for

the coupled system shown in Fig. 19, such a change can completely change the ensuing transient. For instance, increasing the

accumulator diameter from 10 to 13cm causes the oscillations to increase in amplitude until the flow becomes single-phase

throughout the heated channel. Once the flow is single-phase, it overshoots the single-phase attractor (i.e., point A in Fig. 11)

and eventually reaches the steady-state operating conditions, as seen in Fig. 26. The corresponding transient is given as a phase

plane plot in Fig. 27, with the flow becoming single-phase at the exit of the channel when the inlet mass flux is sufficiently

large.

By decreasing the accumulator vessel diameter to, for example, 0.07 m, the system no longer experiences interacting oscilla-

tions, but instead exhibits a pure DWO. Fig. 28 shows such a transient, with the amplitude of the oscillations comparable to those

seen in Fig. 12, after the initial transient has decayed. However, before the initial transient disappears, the amplitude of the oscilla-

tions in Fig. 28 varies, due to an interaction of damped accumulator dynamics with the DWO associated with the heated channel.

This interaction should be physically possible by arranging the accumulator line in a helical fashion (i.e., to achieve the desired

surge line length). Note that the corresponding envelope (i.e., the dashed lines) shows the interactions of these oscillations. Even-

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Fig. 24. Nodal model: test section with an accumulator transient heated channels pressure drop.

Fig. 25. pHCGHC phase plane plot superimposed with the steady-state operating curve.


tually, the accumulator dynamics decay away, leaving just a DWO. These interactions are also readily shown in the transient of

the accumulator flow rate, Fig. 29, and in the *GHC phase plane plot, Fig. 30. We note that the theoretical approximations forthe periods of these oscillations (TPDO =4.4s and TDWO =0.88s) is consistent with the non-linear time-domain predictions shown in

Figs. 28 and 29.

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Fig. 27. Nodal model: test section with an accumulator phase plane transient.

Fig. 28. Nodal model: test section with an accumulator transient heated channel inlet mass flux with the corresponding envelope.

Fig. 29. Nodal model: test section with an accumulator transient accumulator flow rate.

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Fig. 30. *GHC phase plane plot.

Fig. 31. pHCGHC phase plane plot superimposed with the steady-state operating curve.

Also, as can be seen in Fig. 31, the pHCGHC phase plane plot, which is superimposed with the dashed steady-state operating curve,clearly shows a DWO about the two-phase attractor (i.e., at GHC=850kg/m

2 s).

Finally, it should be noted that the results using a more detailed multinode drift-flux model with subcooled boiling are qualitatively the

same as those presented herein (Schlichting, 2009). However, these models are necessarily more complicated to derive and evaluate.

6. Summary and conclusions

This paper presents an analysis the interactions of pressure-drop oscillations (PDO) and density-wave oscillations (DWO) for a typical

NASA test loop. It is shown that very interesting non-linear interactions are possible. Indeed, these type of interactions are much richer

than has been previously discussed (Kakac and Bon, 2008).

Acknowledgements

The authors would like to acknowledge the financial support given this research project by NASA-GRC. Moreover, the support given by

Rensselaer Polytechnic Institute (RPI) is gratefully acknowledged.

Appendix A. Derivation of non-linear and linear nodal models

Let us assume that the uniform axial heat flux can be treated as a time dependent variable for purposes of perturbing the system. In all

other aspects, including linearization of the model, the heat flux was treated as a constant.

Single-phase continuity equation

z

G(z, t) = 0 (A.1)

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Due to Eq. (A.1), the incompressible single-phase mass flux in a constant area duct does not have spatial dependence, thus:

G1(t) = Gin(t) (A.2)

Single-phase energy equation

Since the system pressure is assumed to be constant:

f

th(z, t) +Gin(t)

zh(z, t) =

qPHAHT

(A.3)

Integrating Eq. (A.3) from the inlet to the boiling boundary, (t):

f

(t)0

th(z, t)dz+Gin(t)

(t)0

zh(z, t)dz=

(t)0

qPHAHT

dz (A.4)

Leibnizs rule is defined in Eq. (A.5):b(t)a(t)

tf(z, t)dz=

d

dt

b(t)a(t)

f(z, t)dz+f(a(t), t)d

dta(t) f(b(t), t)

d

dtb(t) (A.5)

Applying Leibnizs rule to Eq. (A.4):

f

t

(t)0

h(z, t)dz fh((t), t)d

dt(t) +Gin(t)

(t)0

zh(z, t)dz=

(t)0

qPHAHT

dz (A.6)

f

t(t)

0

h(z, t)dz f

h((t), t)d

dt(t) +G

in(t)[h((t), t) h(0, t)] =

qPH

AHT(t) (A.7)

Noting that the inlet and boiling enthalpies are given below as Eqs. (A.8a) and (A.8b). The inlet enthalpy is assumed to be constant and

liquid saturation enthalpy is constant for a constant system pressure:

h(0, t) = hin (A.8a)

h((t), t) = hf (A.8b)

Now let:

h1(t) =1

(t)

(t)0

h(z, t)dz (A.9)

Substituting Eqs. (A.8a), (A.8b) and (A.9) into Eq. (A.7):

fd

dt

[h1(t)(t)] fhfd

dt

(t) + Gin(t)(hf hin) =qPH

AHT(t) (A.10)

f(t)d

dth1(t) + f(h1(t) hf)

d

dt(t) +Gin(t)(hf hin) =

qPHAHT

(t) (A.11)

Let the single-phase average liquid enthalpy be the numerical average of the inlet and liquid saturation enthalpies. The single-phase

average enthalpy would then be a constant:

h1(t) = h1 hf + hin

2(A.12)

Substituting Eq. (A.12) into Eq. (A.11):

f

hf + hin

2 hf

d

dt(t) + Gin(t)(hf hin) =

qPHAHT

(t) (A.13)

d

dt

(t) = 2Gin(t)

f 2

qPH

AHTf(hf hin)

(t) (A.14)

Let:

=qPHAHT

vfg

hfg(A.15a)

Nsub =hf hin

hfg

vfg

vf(A.15b)

Npch =qPHLH

Gin0AHThfg

vfg

vf(A.15c)

Substituting Eqs. (A.15a), (A.15b) and (A.15c) into Eq. (A.14):

d

dt(t) = 2

Gin0f

Gin(t)

Gin0

NpchNsub

(t)

LH (A.16)

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Note that the steady-state of Eq. (A.16) yields:

0 =NsubNpch

LH (A.17)

Substituting Eq. (17) into Eq. (16):

d

dt(t) = 2

Gin0f

Gin(t)

Gin0

(t)

0

(A.18)

Two-phase continuity equation:

t(z, t) +

zG(z, t) = 0 (A.19)

Integrating Eq. (A.19) from the boiling boundary, (t), to the exit:LH(t)

t(z, t)dz+

LH(t)

zG(z, t)dz= 0 (A.20)

LH(t)

t(z, t)dz+ G(LH, t) G((t), t) = 0 (A.21)

Again using Leibnizs rule:

tLH

(t)

(z, t)dz+ ((t), t)d

dt(t) + G(LH, t) G((t), t) = 0 (A.22)

The density and mass flux at the boiling boundary (and at the inlet) are just the single-phase values for each, Eqs. (A.23a) and (A.23b),

and the mass flux at the exit is given by Eq. (A.23c):

((t), t) = (0, t) = f (A.23a)

G((t), t) = G(0, t) = Gin(t) (A.23b)

G(LH, t) = Gout(t) (A.23c)

Let the two-phase average density be defined as:

2(t) =1

LH (t)

LH(t)

(z, t)dz (A.24)

Substituting Eqs. (A.23a), (A.23b), (A.23c) and (A.24) into Eq. (A.22):

ddt

[2(t)(LH (t))] + f ddt(t) + Gout(t) Gin(t) = 0 (A.25)

(LH (t))d

dt2(t) + (f 2(t))

d

dt(t) +Gout(t) Gin(t) = 0 (A.26)

or,

(LH (t))d

dt2(t) = Gin(t) Gout(t) (f 2(t))

d

dt(t) (A.27)

The two-phase superficial velocity (Lahey and Moody, 1993) is:

u(z, t) =Gin(t)

f+ (z (t)) (A.28a)

The average two-phase superficial velocity is thus:

u2(t) = 1LH (t)LH

(t)

Gin(t)f

+ (z (t))

dz= Gin(t)f+

2(LH (t)) (A.28b)

Similarly the exit two-phase superficial velocity:

uout(t) =Gin(t)

f+ (LH (t)) (A.28c)

The two-phase mass flux, defined as the spatial average mass flux in the two-phase region, is the product of the average two-phase

superficial velocity and the average two-phase density:

G2(t) =1

(LH (t))

LH(t)

G(z, t)dz= 2(t)u2(t) (A.28d)

The exit mass flux is thus:

Gout(t) = 2G2(t) Gin(t) = out(t)uout(t) (A.28e)

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Substituting Eqs. (A.18), (A.28b), (A.28d), and (A.28e) into Eq. (A.27):

d

dt2(t) = 2

Gin00

1

2(t)

f

(t)

LH (t) 2(t) (A.29)

The momentum equation:

tG(z, t) +

zG2(z, t)

(z, t) = zp(z, t) f2DH

G2(z, t)

(z, t)

g(z, t) KinG2(z, t)

2(z, t)(z) Kout

G2(z, t)

2(z, t)(z LH)

(A.30)

Integrating Eq. (A.30) from the inlet to the exit of the heated channel:LH0

tG(z, t)dz+

LH0

z

G2(z, t)

(z, t)

dz=

LH0

zp(z, t)dz

LH0

f

2DH

G2(z, t)

(z, t)dz

LH0

g(z, t)dz

LH0

KinG2(z, t)

2(z, t)(z)dz

LH0

KoutG2(z, t)

2(z, t)(z LH)dz

(A.31)

The interval union property of integrals is:

ba

f(z)dz=c

af(z)dz+

bc

f(z)dz if a < c < b (A.32)

Applying this property of integrals to Eq. (A.31):(t)0

tG(z, t)dz+

LH(t)

tG(z, t)dz+

LH0

z

G2(z, t)

(z, t)

dz=

LH0

zp(z, t)dz

f

2DH

(t)0

G2(z, t)

(z, t)dz

f

2DH

LH(t)

G2(z, t)

(z, t)dzg

(t)0

(z, t)dzg

LH(t)

(z, t)dz

LH

0

KinG2(z, t)

2(z, t)(z)dz

LH

0

KoutG2(z, t)

2(z, t)(z LH)dz

(A.33)

Substituting in the single-phase values for the density and the mass flux:(t)0

d

dtGin(t)dz+

LH(t)

tG(z, t)dz+

LH0

z

G2(z, t)

(z, t)

dz=

LH0

zp(z, t)dz

f

2DH

G2in

(t)

f

(t)0

dzf

2DH

LH(t)

G2(z, t)

(z, t)dzgf

(t)0

dzg

LH(t)

(z, t)dz

LH0

KinG2(z, t)

2(z, t)(z)dz

LH0

KoutG2(z, t)

2(z, t)(z LH)dz

(A.34)

Using Leibnizs rule on Eq. (A.34):

d

dt

(t)0

d

dtGin(t)dzGin(t)

d

dt(t) + G((t), t)

d

dt(t) +

d

dt

LH(t)

G(z, t)dz+

LH0

z

G2(z, t)

(z, t)

dz=

LH0

zp(z, t)dz

f

2DH

G2in

(t)

f

(t)0

dzf

2DH

LH(t)

G2(z, t)

(z, t)dzgf

(t)0

dzg

LH(t)

(z, t)dz

LH0

KinG2(z, t)

2(z, t)(z)dz

LH0

KoutG2(z, t)

2(z, t)(z LH)dz

(A.35)

The friction number is defined as:

=f

2DH

LH (A.36)

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Substituting Eqs. (A.23b) and (A.36) into Eq. (A.35) and simplifying:

d

dtGin(t)

(t)0

dz+d

dt

LH(t)

G(z, t)dz+

LH0

z

G2(z, t)

(z, t)

dz=

LH0

zp(z, t)dz

LH

G2in

(t)

f

(t)0

dz

LH

LH(t)

G2(z, t)

(z, t)dzgf

(t)0

dzg

LH(t)

(z, t)dz

LH

0

KinG2(z, t)

2(z, t)(z)dz

LH

0

KoutG2(z, t)

2(z, t)(z LH)dz

(A.37)

Let the average two-phase dynamic pressure be defined as:

G22

(t)

2(t)=

1

LH (t)

LH(t)

G2(z, t)

(z, t)dz (A.38)

Substituting Eqs. (A.24), (A.28d) and (A.36) into Eq. (A.37):

d

dtGin(t)

(t)0

dz+d

dt[G2(t)(LH (t))] +

LH0

z

G2(z, t)

(z, t)

dz=

LH0

zp(z, t)dz

LH

G2

in(t)f

(t)0

dz G2

2

(t)2(t)

1 (t)

LH

gf

(t)0

dzg2(t)(LH (t))

LH0

KinG2(z, t)

2(z, t)(z)dz

LH0

KoutG2(z, t)

2(z, t)(z LH)dz

(A.39)

d

dt[Gin(t)(t)] +

d

dt[G2(t)(LH (t))] +

G2(LH, t)

(LH, t)

G2(0, t)

(0, t)=p(0, t) p(LH, t)

Gin

2(t)

f

(t)

LH

G22

(t)

2(t)

1

(t)

LH

gf(t) g2(t)(LH (t))

KinG2(0, t)

2(0, t)

KoutG2(LH, t)

2(LH, t)

(A.40)

For a parallel channel boundary condition, a constant pressure drop is imposed across the heated channel:

p =p(0) p(LH) =p(0, t) p(LH, t) (A.41a)

Let:

out(t) = (LH, t) (A.41b)

Substituting Eqs. (A.23a), (A.23b), (A.23c), (A.41a) and (A.41b) into Eq. (A.40):

d

dt[Gin(t)(t)] +

d

dt[G2(t)(LH (t))] = p

Kin2

1

G2in

(t)

f

Kout

2+ 1

G2out(t)

out(t)

Gin2(t)

f

(t)

LH +

G22

(t)

2(t)

1 (t)

LH

g(f(t) + 2(t)(LH (t)))

(A.42)

or,

(t)d

dtGin(t) + (Gin(t) G2(t))

d

dt(t) + (LH (t))

d

dtG2(t) = p

Kin2

1

G2in

(t)

f

Kout

2+ 1

G2out(t)

out(t)

G2

in(t)

f

(t)

LH+

G22

(t)

2(t)

1

(t)

LH

g(f(t) + 2(t)(LH (t)))

(A.43)

Substituting Eq. (A.28b) into Eq. (A.28d):

G2(t) = Gin(t)2(t)

f+

2(L (t))2(t) (A.44)

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Substituting Eqs. (A.28d), (A.28e) and (A.44) into Eq. (A.43):

(t)d

dtGin(t) + (Gin(t) G2(t))

d

dt(t) + (LH (t))

d

dt

Gin(t)

2(t)

f+

2(L (t))2(t)

= p

Kin2

1

G2in

(t)

f

Kout

2+ 1

Gout(t)uout(t)

Gin

2(t)

f

(t)

LH+

G22

(t)

2(t)

1

(t)

LH

g(f(t) + 2(t)(LH (t)))

(A.45)

or, (t) + (LH (t))

2(t)

f

d

dtGin(t) +

Gin(t) G2(t)

2(LH (t))2(t)

d

dt(t)

+(LH (t))u2(t)d

dt2(t) = p

Kin2

1

G2in

(t)

f

Kout

2+ 1

Gout(t)uout(t)

G2

in(t)

f

(t)

LH+

G22

(t)

2(t)

1

(t)

LH

gf

(t) + (LH (t))

2(t)

f

(A.46)

Substituting Eqs. (A.18), (A.28d) and (A.29) into Eq. (A.46):(t) + (LH (t))

2(t)

f

d

dtGin(t) = p 2Gin0u2(t)

(t)

0

1

2(t)

f

+ (LH (t))G2(t)

2Gin0f

Gin(t) G2(t)

2(LH (t))2(t)

Gin(t)

Gin0

(t)

0

gf

(t) + (LH (t))

2(t)

f

Kin2

1

G2in

(t)

f

Kout

2+ 1

Gout(t)uout(t)

G2

in(t)

f

(t)

LH+G2(t)u2(t)

1

(t)

LH

(A.47)

Finally, substituting Eqs. (A.28b), (A.28c), (A.28d), and (A.28e) into Eq. (A.47) we obtain:

(t) + (LH (t))2(t)

f d

dtGin(t) = p 2Gin0

(t)

0 Gin(t)

f+

2(LH (t))1

2(t)

f

Kin2

1

G2in

(t)

f

Kout

2+ 1

Gin(t)

2

2(t)

f 1

+ (LH (t))2(t)

Gin(t)

f+ (LH (t))

gf

(t) + (LH (t))

2(t)

f

2

Gin0f

Gin(t)

1

2(t)

f

(LH (t))2(t)

Gin(t)

Gin0

(t)

0

+(LH (t))2(t)

Gin(t)

f+

2(LH (t))

G2

in(t)

f

(t)

LH+ 2(t)

Gin(t)

f+

2(LH (t))

21

(t)

LH

(A.48)

Using a similar approach more involved multimode models can be derived based on a drift-flux model with subcooled boiling

(Schlichting, 2009), however due to the complexity of these models they have not been presented in this paper.

References

Achard, J.L., Drew, D.A., Lahey Jr., R.T., 1985. The analysis of nonlinear density-wave oscillations in boiling channels. J. Fluid Mech. 155, 213232.Achard, J.L., Drew, D.A., Lahey Jr., R.T., 1981. The effect of gravity and friction on the stability of boiling flow in a channel. Chem. Eng. Commun. 11, 5979.Akyuzlu,K., Veziroglu, T.N.,Kakac, S., Dogan, T., 1980.Finite differenceanalysisof two-phase flowpressure-drop and density-waveoscillations.Warme-und Stoffubertragung

26, 365376.Dogan, T., Kakac, S., Vezirglu, T.N., 1983. Analysis of forced-convection boiling flow instabilities in a single-channel upflow system. Int. J. Heat Fluid Flow 4, 145

156.Garea, V.B., Drew, D.A., Lahey Jr., R.T., 1999. A moving-boundary nodal model for the analysis of the stability of boiling channels. Int. J. Heat Mass Transfer 42 (19), 3575

3584.Gurgenci, H., Veziroglu, T.N., Kakac, S., 1983. Simplified nonlinear descriptions of two-phase flow instabilities in vertical boiling channel. Int. J. Heat Mass Transfer 26, 671

679.Kakac, S., Bon, B., 2008. A review of two-phase flow dynamic instabilities in tube boiling systems. Int. J. Heat Mass Transfer 51, 399433.Lahey Jr., R.T., Podowski, M.Z., 1989. On the analysis of various instabilities in two-phase flows. Multiphase Science & Technology, vol. 4. Hemisphere Press, pp. 183370.Lahey Jr., R.T., Moody, F.J., 1993. The thermal-hydraulics of a boiling water nuclear reactor. ANS Monogr..Maulbetsch, J.S., Griffith, P., 1965. A study of system-induced instabilities in forced-convection flows with subcooled boiling. MIT Eng. Lab. Rep., 53355382.NRC Space Studies Board, 2000. Microgravity research in support of technologies for the human exploration and development of space and planetary bodies. Natl. Res.

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Ozawa, M., Nakanishi, S., Ishigai, S., Mizuta, Y., Tarui, H., 1979. Flow instabilities in boiling channels. Bull. JSME 22, 11131117.Schlichting, W.R., 2009. An analysis of the effect of gravity on interacting DWO/PDO instability modes. PhD Thesis. Rensselaer Polytechnic Institute.Stenning, A.H., Verizoglu, T.N., 1965. Flow oscillation modes in forced-convection boiling. In: Proceedings of the Heat Transfer & Fluid Mechanics Institute, pp. 301

316.Stenning, A.H., Verizoglu, T.N., Callahan, G.M., 1967. Pressure-drop oscillations in forced convection flow with boiling. In: Proceedings of the Symposium on Two-Phase Flow

Dynamics, Eindhoven, pp. 405427.Strogatz, S.H., 1994. Nonlinear Dynamics and Chaos, with Applications to Physics, Biology, Chemistry, and Engineering. Perseus Books Publishing.Yin, J., Lahey Jr., R.T., Podowski, M.Z., Jensen, M.K., 2006. An analysis of interacting stability modes. J. Multiphase Sci. Technol. 18 (4), 359385.

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