# Flow Instabilities in Boiling Two-Phase Natural Circulation Systems: A Review

Authors: A. K. Nayak; P. K. Vijayan
Journal: Science and Technology of Nuclear Installations
 (2008)

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## Abstract

Several decades have been spent on the study of flow instabilities
in boiling two-phase natural circulation systems. It is felt to have a review
and summarize the state-of-the-art research carried out in this area, which
would be quite useful to the design and safety of current and future light water
reactors with natural circulation core cooling. With that purpose, a review of flow
instabilities in boiling natural circulation systems has been carried out. An attempt
has been made to classify the instabilities occurring in natural circulation systems
similar to that in forced convection boiling systems. The mechanism of instabilities
occurring in two-phase natural circulation systems have been explained based on
these classifications. The characteristics of different instabilities as well as the effects
of different operating and geometric parameters on them have been reviewed.


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## Body

## 1. Introduction

Natural circulation (NC) systems are susceptible to several kinds of
instabilities.Although instabilities are common to both forced and natural
circulation systems, the latter is inherently more unstable than forced
circulation systems due to more nonlinearity of the NC process and its low
driving force. Because of this, any disturbance in the driving force affects
the flow which in turn influences the driving force leading to an oscillatory
behavior even in cases where eventually a steady state is expected. In other
words, a regenerative feedback is inherent in the mechanism causing NC flow due
to the strong coupling between the flow and the driving force. Even among
two-phase systems, the NC systems are more unstable than forced circulation
systems due to the above reasons.Before we proceed further, let us define
the term “instability”. Following a perturbation, if the system returns
back to the original steady state, then the system is considered to be stable.
If on the other hand, the system continues to oscillate with the same
amplitude, then the system is neutrally stable. If the system stabilizes to a
new steady state or oscillates with increasing amplitude, then the system is
considered as unstable. It may be noted that the amplitude of oscillations
cannot go on increasing indefinitely even for unstable flow. Instead for almost
all cases of instability, the amplitude is limited by nonlinearities of the
system and limit cycle oscillations (which may be chaotic or periodic) are
eventually established. The time series of the limit cycle oscillations may
exhibit characteristics similar to the neutrally stable condition. Further,
even in steady state case, especially for two-phase systems with slug flow,
small amplitude oscillations are visible. Thus, for identification purposes
especially during experiments, often it becomes necessary to quantify the
amplitude of oscillations as a certain percentage of the steady state value.
Amplitudes more than± 10% of the mean value is often considered as an
indication of instability. However, some authors recommend the use of ±
30% as the cutoff value [1].Instability is undesirable as sustained
flow oscillations may cause forced mechanical vibration of components. Further,
premature CHF (critical heat flux) occurrence can be induced by flow
oscillations as well as other undesirable secondary effects like power
oscillations in BWRs. Instability can also disturb control systems and cause
operational problems in nuclear reactors. Over the years, several kinds of
instabilities have been observed in natural circulation systems excited by
different mechanisms. Differences also exist in their transport mechanism,
oscillatory mode, and analysis methods.In addition,effects of loop geometry
and secondary parameters also cause complications in the observed
instabilities. Under the circumstances, it looks relevant to classify
instabilities into various categories which will help in improving our
understanding and hence control of these instabilities.


## 2. Instability Classification

Mathematically, the
fundamental cause of all instabilities is due to the existence of competing
multiple solutions so that the system is not able to settle down to anyone of
them permanently. Instead, the system swings from one solution to the
other. An essential characteristic of
the unstable oscillating NC systems is that as it tries to settle down to one
of the solutions, a self-generated feedback appears making another solution
more attractive causing the system to swing toward it. Again, during the
process of settling down on this solution, another feedback of opposite sign
favoring the original solution is self-generated and the system swings back to
it. The process repeats itself resulting in perpetual oscillatory behavior if
the operating conditions are maintained constant. Although this is a general
characteristic it hardly distinguishes the different types of instabilities
found to occur in various systems. In general, instabilities can be classified
according to various bases as follows:(a)
analysis method;(b)
propagation
method;(c)
number of
unstable zones;(d)
nature of
the oscillations;(e)
loop
geometry;(f)
disturbances
or perturbations.


### 2.1. Based on the Analysis Method (or Governing Equations Used)

In some cases, the occurrence of multiple solutions
and the instability threshold itself can be predicted from the steady-state
equations governing the process (pure or fundamental static instability). The
Ledinegg-type instability is one such example occurring in boiling two-phase NC
systems. The occurrence of this type of instability can be ascertained by
investigating the steady-state behavior alone. The criterion for this type of
instability is given by(1)∂Δpint∂w−∂Δpdν∂w≤0, where Δpint is the internal pressure loss in the system
and Δpdν is the driving head due to buoyancy. The
internal pressure loss of the system includes the losses due to friction,
elevation, acceleration and local in the heated portion, the riser pipes and
the steam drum, and all the losses except the elevation loss in the downcomers.
The driving head is basically the gravitational head available from the steam
drum to the bottom of the heated section. Figures 1(a) and 1(b) show an
example of occurrence of Ledinegg-type instability at different powers [2] in a boiling two-phase NC system. The instability is found to occur
when the power is more than 285 MWth and less than 460 MWth if the operating
pressure is 0.1 MPa and the subcooling is 30 K. When the power is in between
the above specified range, the internal pressure loss curve intersects the
driving buoyancy curve at three points (i.e., three operating points at a given
power level) which makes the system unstable. Thus at 30 K subcooling, the
system can have two threshold points of instability.Typical stable, unstable and neutrally stable behaviour for Ledinegg type instability.
(a)(b)Like the Ledinegg instability, the flow pattern transition
instability is another static instability caused by the excursion of flow due
to differences in the pressure drop characteristics of different flow patterns.
To analyze this type instability, it is required to predict the pressure drop
characteristics of the system against the flow rate similar to the Ledinegg-type
instability [3]. Figure 2 shows an example of the steady-state
pressure drop characteristics of the system for analysis of flow pattern
transition instability. The gravitational head, which depends on the density of
the single-phase fluid, remains constant at a particular core inlet
temperature. The different flow patterns in the vertical and horizontal
portions of the riser pipes are shown in the two-phase region at the operating
conditions. It can be observed from Figure 3 that there can be multiple steady-state
flow rates (point at which the driving head intersects the internal loss curve)
at this operating condition. The number of flow excursions is seen to be five,
unlike that of the Ledinegg-type instability. The type of flow excursion in
different flow regimes are observed to be as follows: there can be one flow
excursion in the annular region itself due to reduction of pressure drop with
reduction in quality as in the Ledinegg-type instability. The next flow
excursion occurs due to rise in pressure drop when the flow pattern changes
from annular to slug flow in the vertical portion of the riser pipes. The other
flow excursion occurs when the flow pattern changes from the annular to
dispersed bubbly flow in the horizontal portion of riser pipes due to reduction
of pressure drop with flow rate. The last flow excursion occurs when the flow
becomes single phase and the pressure drop increases with increase in flow
rate. Thus, there can be five different flow rates for a particular operating
condition of power and subcooling as indicated in Figure 2 by points A–E. The existence
of multiple flow rates as a particular operating power and subcooling makes the
system unstable. For example, if the system is initially operating at point C,
any slight disturbance causing the flow to increase will shift the flow rate to
point D and the to point E. Similarly, any slight disturbance causing the flow
rate to decrease will shift the operating point to B and then to point A. Thus,
the flow rate can jump from one value to the other even though the operating
power and pressure are constant. This makes the system unstable.Figure 2
Typical flow pattern transition instability in boiling natural circulation systems.(a)
Typical low power and high power unstable zones for two-phase NC flow(b)
Typical stability map for two-phase density wave instability [2]However, there are many situations
with multiple steady-state solutions where the threshold of instability cannot
be predicted from the steady-state laws alone (or the predicted threshold is
modified by other effects). In this case, feedback effects are important in
predicting the threshold (compound static instability). Besides, many NCSs with
only a unique steady-state solution can also become unstable during the
approach to the steady state due to the appearance of competing multiple
solutions due to the inertia and feedback effects (pure dynamic instability).Neither the cause nor the threshold of instability of such systems can be
predicted purely from the steady state equations alone. Instead, it requires
the full transient governing equations to be considered for explaining the
cause and predicting the threshold. In addition, in many oscillatory
conditions, secondary phenomena get excited and they modify significantly the
characteristics of the fundamental instability. In such cases, even the
prediction of the instability threshold may require consideration of the
secondary effect (compound dynamic instability). A typical case is the
neutronic feedback responding to the void fluctuations resulting in both flow
and power oscillations in a BWR. In this case, in addition to the equations
governing the thermalhydraulics, the equations for the neutron kinetics and
fuel thermal response also need to be considered.Thus we find that the analysis to
arrive at the instability threshold can be based on different sets of governing
equations for different instabilities.Boure et al. [4]
classified instabilities into four basic types as follows:(a)
pure static instability;(b)
compound static instability (It may be noted that
Boure et al. [4] named this instability as compound relaxation
instability);(c)
pure dynamic instability;(d)
compound dynamic instability.


### 2.2. Based on the Propagation Method

This classification is actually
restricted to only the dynamic instabilities. According to Boure et al. [4],
the mechanism of dynamic instability involves the propagation or transport of
disturbances. In two-phase flow, the disturbances can be transported by two
different kinds of waves: pressure (acoustic waves) and void (or density)
waves. In any two-phase system, both types of waves are present, however, their
velocities differ by one or two orders of magnitude allowing us to distinguish
between the two.


#### 2.2.1. Acoustic Instability

Acoustic instability is considered
to be caused by the resonance of pressure waves. Acoustic oscillations are also
observed during blowdown experiments with pressurized hot-water systems
possibly due to multiple wave reflections. Acoustic oscillations are
characterized by high frequencies of the order of 10–100 Hz related to
the pressure wave propagation time [4]. Acoustic oscillations
have been observed in subcooled boiling, bulk boiling, and film boiling. The
thermal response of the vapor film to passing pressure wave is suggested as a
mechanism for the oscillations during film boiling. For example, when a
compression (pressure wave consists of compression and rarefaction) wave passes,
the vapor film is compressed enhancing its thermal conductance resulting in
increased vapor generation. On the other hand when a rarefaction wave passes,
the vapor film expands reducing its thermal conductance resulting in decreased
vapor generation. The process repeats itself.


#### 2.2.2. Density-Wave Instability (DWI)

A
density-wave instability is the typical dynamic instability which may occur due
to the multiple regenerative feedback between the flow rate, enthalpy, density,
and pressure drop in the boiling system. The occurrence of the instability
depends on the perturbed pressure drop in the two-phase and single-phase
regions of the system and the propagation time delay of the void fraction or
density in the system. Such an instability can occur at very low-power and at
high-power conditions. This depends on the relative importance of the
respective components of pressure drop such as gravity or frictional losses in
the system. Fukuda and Kobori [5] have classified the
density-wave instability as type I and type II for the low power and high-power
instabilities, respectively. The mechanisms can be explained as follows [6].
Type I Instability
For
this type of instability to occur, the presence of a long riser plays an
important role such as in a boiling two-phase natural circulation loop. Under
low quality conditions, a slight change in quality due to any disturbance can
cause a large change in void fraction and consequently in the driving head. Therefore, the flow can
oscillate at such low-power conditions. But as the power increases, the flow
quality increases where the slope of the void fraction versus quality reduces.
This can suppress the fluctuation of the driving head for a small change in
quality. Hence, the flow stabilises at higher power (Figures3(a) and 3(b)).Type II Instability
Unlike
the type I instability, the type II instability occurs at high-power
conditions. This instability is driven by the interaction between the single
and two-phase frictional component of pressure losses, mass flow, void
formation, and propagation in the two-phase region. At high power, the flow
quality or void fraction in the system is very large. Hence, the two-phase
frictional pressure loss may be high owing to the smaller two-phase mixture
density. Having a large void fraction will increase the void propagation time
delay in the two-phase region of the system. Under these conditions, any small
fluctuation in flow can cause a larger fluctuation of the two-phase frictional
pressure loss due to fluctuation of density and flow, which propagates slowly
in the two-phase region. On the other hand, the fluctuation of the pressure
drop in the single-phase region occurs due to fluctuation of flow alone since
the fluctuation of the density is negligible. The pressure drop fluctuation in
this region travels much faster due to incompressibility of single-phase
region. If the two-phase pressure drop fluctuation is equal in magnitude but
opposite in phase with that of the single-phase region, the fluctuation or
oscillation is sustained in the system since there are no attenuating mechanisms.
Divergent oscillations can occur depending on the magnitude of the pressure-loss
fluctuation in the two-phase and single-phase regions and the propagation time
delay.
Because of the importance of void fraction and its effect on
the flow as explained above, this instability is sometimes referred to as
flow-void feedback instability in two-phase systems. Since transportation time
delays (related to the spacing between the light and heavy packets of fluid as
explained above) are crucial to this instability, it is also known as “time-delay
oscillations”. Density-wave instability (DWI) or density-wave oscillations
(DWO), first used by Stenning and Veziroğlu [7], is the most common term
used for the above described phenomenon as it appears that a density wave with
light and heavy fluid packets is traveling through the loop.


### 2.3. Based on the Number of Unstable Zones

Fukuda and Kobori [5] gave a
further classification of density-wave instability based on the number of
unstable zones. Usually, there exists a low-power and a high-power unstable
zone for density wave instability in forced as well as NC two-phase flows (Figure 3(a)). For the two-phase flow density-wave instability, the unstable region
below the lower threshold occurs at a low power and hence at low quality and is
named as type I instability by Fukuda and Kobori [5]. Similarly, the
unstable region beyond the upper threshold occurs at a high power and hence at
high qualities and is named as type II instability. However, in certain cases
depending on the geometry and operating conditions, islands of instability have
been observed to occur [8–10]. In these cases, more
than two zones of instability were oserved. Chen et al. [11] also observed
hysteresis in a two-phase loop. As an unstable single-phase system progresses
through single-phase NC to boiling inception and then to fully-developed
two-phase NC with power change, it can encounter several unstable zones. In
view of the existence of more than two unstable zones, this method of
classification could be confusing at times.


### 2.4. Based on the Nature of the Oscillations

All instabilities eventually lead
to some kind of oscillations. The oscillations can be labeled as flow
excursions, pressure drop oscillations, power oscillations, temperature
excursions or thermal oscillations, and so on. Besides, classifications based
on the oscillatory characteristics are sometimes reported for dynamic
instability. For example, based on the periodicity sometimes oscillations are
characterized as periodic and chaotic. Based on the oscillatory mode, the
oscillations are characterized as fundamental mode or higher harmonic modes
[12]. In boiling NC systems with multiple parallel channels,
inphase and out of phase modes are present depending on the geometry of the channels
and heating conditions. Sometimes, dual oscillations also are possible. In
natural circulation loops, flow direction can also change during oscillations.
Based on the direction of flow, the oscillations can be characterized as
unidirectional, bidirectional, or it can switch between the two. Such switching
is often accompanied by period doubling, tripling, or n-tupling.


### 2.5. Based on the Loop Geometry

Certain instabilities are
characteristic of the loop geometry. Examples are the instabilities observed in open U-loops,
symmetric closed loops, and asymmetric-closed loops. In addition, pressure-drop
oscillations and the parallel-channel instability are also characteristic of
the loop geometry. Another type instability which can occur in systems with a
compressible volume (a pressurizer for example) at the inlet of the heated
channel is the pressure-drop-type instability. Similarly, interaction among
parallel channels can also lead to various complex instabilities as discussed
above.


### 2.6. Based on the Disturbances

Certain two-phase flow phenomena
can cause a major disturbance and can lead to instability or modify the
instability characteristics significantly. Typical examples are boiling
inception, flashing, flow pattern transition, or the occurrence of CHF. Cold
water injection can also cause a major disturbance and instability in natural
circulation systems.


## 2.1. Based on the Analysis Method (or Governing Equations Used)

In some cases, the occurrence of multiple solutions
and the instability threshold itself can be predicted from the steady-state
equations governing the process (pure or fundamental static instability). The
Ledinegg-type instability is one such example occurring in boiling two-phase NC
systems. The occurrence of this type of instability can be ascertained by
investigating the steady-state behavior alone. The criterion for this type of
instability is given by(1)∂Δpint∂w−∂Δpdν∂w≤0, where Δpint is the internal pressure loss in the system
and Δpdν is the driving head due to buoyancy. The
internal pressure loss of the system includes the losses due to friction,
elevation, acceleration and local in the heated portion, the riser pipes and
the steam drum, and all the losses except the elevation loss in the downcomers.
The driving head is basically the gravitational head available from the steam
drum to the bottom of the heated section. Figures 1(a) and 1(b) show an
example of occurrence of Ledinegg-type instability at different powers [2] in a boiling two-phase NC system. The instability is found to occur
when the power is more than 285 MWth and less than 460 MWth if the operating
pressure is 0.1 MPa and the subcooling is 30 K. When the power is in between
the above specified range, the internal pressure loss curve intersects the
driving buoyancy curve at three points (i.e., three operating points at a given
power level) which makes the system unstable. Thus at 30 K subcooling, the
system can have two threshold points of instability.Typical stable, unstable and neutrally stable behaviour for Ledinegg type instability.
(a)(b)Like the Ledinegg instability, the flow pattern transition
instability is another static instability caused by the excursion of flow due
to differences in the pressure drop characteristics of different flow patterns.
To analyze this type instability, it is required to predict the pressure drop
characteristics of the system against the flow rate similar to the Ledinegg-type
instability [3]. Figure 2 shows an example of the steady-state
pressure drop characteristics of the system for analysis of flow pattern
transition instability. The gravitational head, which depends on the density of
the single-phase fluid, remains constant at a particular core inlet
temperature. The different flow patterns in the vertical and horizontal
portions of the riser pipes are shown in the two-phase region at the operating
conditions. It can be observed from Figure 3 that there can be multiple steady-state
flow rates (point at which the driving head intersects the internal loss curve)
at this operating condition. The number of flow excursions is seen to be five,
unlike that of the Ledinegg-type instability. The type of flow excursion in
different flow regimes are observed to be as follows: there can be one flow
excursion in the annular region itself due to reduction of pressure drop with
reduction in quality as in the Ledinegg-type instability. The next flow
excursion occurs due to rise in pressure drop when the flow pattern changes
from annular to slug flow in the vertical portion of the riser pipes. The other
flow excursion occurs when the flow pattern changes from the annular to
dispersed bubbly flow in the horizontal portion of riser pipes due to reduction
of pressure drop with flow rate. The last flow excursion occurs when the flow
becomes single phase and the pressure drop increases with increase in flow
rate. Thus, there can be five different flow rates for a particular operating
condition of power and subcooling as indicated in Figure 2 by points A–E. The existence
of multiple flow rates as a particular operating power and subcooling makes the
system unstable. For example, if the system is initially operating at point C,
any slight disturbance causing the flow to increase will shift the flow rate to
point D and the to point E. Similarly, any slight disturbance causing the flow
rate to decrease will shift the operating point to B and then to point A. Thus,
the flow rate can jump from one value to the other even though the operating
power and pressure are constant. This makes the system unstable.Figure 2
Typical flow pattern transition instability in boiling natural circulation systems.(a)
Typical low power and high power unstable zones for two-phase NC flow(b)
Typical stability map for two-phase density wave instability [2]However, there are many situations
with multiple steady-state solutions where the threshold of instability cannot
be predicted from the steady-state laws alone (or the predicted threshold is
modified by other effects). In this case, feedback effects are important in
predicting the threshold (compound static instability). Besides, many NCSs with
only a unique steady-state solution can also become unstable during the
approach to the steady state due to the appearance of competing multiple
solutions due to the inertia and feedback effects (pure dynamic instability).Neither the cause nor the threshold of instability of such systems can be
predicted purely from the steady state equations alone. Instead, it requires
the full transient governing equations to be considered for explaining the
cause and predicting the threshold. In addition, in many oscillatory
conditions, secondary phenomena get excited and they modify significantly the
characteristics of the fundamental instability. In such cases, even the
prediction of the instability threshold may require consideration of the
secondary effect (compound dynamic instability). A typical case is the
neutronic feedback responding to the void fluctuations resulting in both flow
and power oscillations in a BWR. In this case, in addition to the equations
governing the thermalhydraulics, the equations for the neutron kinetics and
fuel thermal response also need to be considered.Thus we find that the analysis to
arrive at the instability threshold can be based on different sets of governing
equations for different instabilities.Boure et al. [4]
classified instabilities into four basic types as follows:(a)
pure static instability;(b)
compound static instability (It may be noted that
Boure et al. [4] named this instability as compound relaxation
instability);(c)
pure dynamic instability;(d)
compound dynamic instability.


## 2.2. Based on the Propagation Method

This classification is actually
restricted to only the dynamic instabilities. According to Boure et al. [4],
the mechanism of dynamic instability involves the propagation or transport of
disturbances. In two-phase flow, the disturbances can be transported by two
different kinds of waves: pressure (acoustic waves) and void (or density)
waves. In any two-phase system, both types of waves are present, however, their
velocities differ by one or two orders of magnitude allowing us to distinguish
between the two.


### 2.2.1. Acoustic Instability

Acoustic instability is considered
to be caused by the resonance of pressure waves. Acoustic oscillations are also
observed during blowdown experiments with pressurized hot-water systems
possibly due to multiple wave reflections. Acoustic oscillations are
characterized by high frequencies of the order of 10–100 Hz related to
the pressure wave propagation time [4]. Acoustic oscillations
have been observed in subcooled boiling, bulk boiling, and film boiling. The
thermal response of the vapor film to passing pressure wave is suggested as a
mechanism for the oscillations during film boiling. For example, when a
compression (pressure wave consists of compression and rarefaction) wave passes,
the vapor film is compressed enhancing its thermal conductance resulting in
increased vapor generation. On the other hand when a rarefaction wave passes,
the vapor film expands reducing its thermal conductance resulting in decreased
vapor generation. The process repeats itself.


### 2.2.2. Density-Wave Instability (DWI)

A
density-wave instability is the typical dynamic instability which may occur due
to the multiple regenerative feedback between the flow rate, enthalpy, density,
and pressure drop in the boiling system. The occurrence of the instability
depends on the perturbed pressure drop in the two-phase and single-phase
regions of the system and the propagation time delay of the void fraction or
density in the system. Such an instability can occur at very low-power and at
high-power conditions. This depends on the relative importance of the
respective components of pressure drop such as gravity or frictional losses in
the system. Fukuda and Kobori [5] have classified the
density-wave instability as type I and type II for the low power and high-power
instabilities, respectively. The mechanisms can be explained as follows [6].
Type I Instability
For
this type of instability to occur, the presence of a long riser plays an
important role such as in a boiling two-phase natural circulation loop. Under
low quality conditions, a slight change in quality due to any disturbance can
cause a large change in void fraction and consequently in the driving head. Therefore, the flow can
oscillate at such low-power conditions. But as the power increases, the flow
quality increases where the slope of the void fraction versus quality reduces.
This can suppress the fluctuation of the driving head for a small change in
quality. Hence, the flow stabilises at higher power (Figures3(a) and 3(b)).Type II Instability
Unlike
the type I instability, the type II instability occurs at high-power
conditions. This instability is driven by the interaction between the single
and two-phase frictional component of pressure losses, mass flow, void
formation, and propagation in the two-phase region. At high power, the flow
quality or void fraction in the system is very large. Hence, the two-phase
frictional pressure loss may be high owing to the smaller two-phase mixture
density. Having a large void fraction will increase the void propagation time
delay in the two-phase region of the system. Under these conditions, any small
fluctuation in flow can cause a larger fluctuation of the two-phase frictional
pressure loss due to fluctuation of density and flow, which propagates slowly
in the two-phase region. On the other hand, the fluctuation of the pressure
drop in the single-phase region occurs due to fluctuation of flow alone since
the fluctuation of the density is negligible. The pressure drop fluctuation in
this region travels much faster due to incompressibility of single-phase
region. If the two-phase pressure drop fluctuation is equal in magnitude but
opposite in phase with that of the single-phase region, the fluctuation or
oscillation is sustained in the system since there are no attenuating mechanisms.
Divergent oscillations can occur depending on the magnitude of the pressure-loss
fluctuation in the two-phase and single-phase regions and the propagation time
delay.
Because of the importance of void fraction and its effect on
the flow as explained above, this instability is sometimes referred to as
flow-void feedback instability in two-phase systems. Since transportation time
delays (related to the spacing between the light and heavy packets of fluid as
explained above) are crucial to this instability, it is also known as “time-delay
oscillations”. Density-wave instability (DWI) or density-wave oscillations
(DWO), first used by Stenning and Veziroğlu [7], is the most common term
used for the above described phenomenon as it appears that a density wave with
light and heavy fluid packets is traveling through the loop.


## 2.2.1. Acoustic Instability

Acoustic instability is considered
to be caused by the resonance of pressure waves. Acoustic oscillations are also
observed during blowdown experiments with pressurized hot-water systems
possibly due to multiple wave reflections. Acoustic oscillations are
characterized by high frequencies of the order of 10–100 Hz related to
the pressure wave propagation time [4]. Acoustic oscillations
have been observed in subcooled boiling, bulk boiling, and film boiling. The
thermal response of the vapor film to passing pressure wave is suggested as a
mechanism for the oscillations during film boiling. For example, when a
compression (pressure wave consists of compression and rarefaction) wave passes,
the vapor film is compressed enhancing its thermal conductance resulting in
increased vapor generation. On the other hand when a rarefaction wave passes,
the vapor film expands reducing its thermal conductance resulting in decreased
vapor generation. The process repeats itself.


## 2.2.2. Density-Wave Instability (DWI)

A
density-wave instability is the typical dynamic instability which may occur due
to the multiple regenerative feedback between the flow rate, enthalpy, density,
and pressure drop in the boiling system. The occurrence of the instability
depends on the perturbed pressure drop in the two-phase and single-phase
regions of the system and the propagation time delay of the void fraction or
density in the system. Such an instability can occur at very low-power and at
high-power conditions. This depends on the relative importance of the
respective components of pressure drop such as gravity or frictional losses in
the system. Fukuda and Kobori [5] have classified the
density-wave instability as type I and type II for the low power and high-power
instabilities, respectively. The mechanisms can be explained as follows [6].
Type I Instability
For
this type of instability to occur, the presence of a long riser plays an
important role such as in a boiling two-phase natural circulation loop. Under
low quality conditions, a slight change in quality due to any disturbance can
cause a large change in void fraction and consequently in the driving head. Therefore, the flow can
oscillate at such low-power conditions. But as the power increases, the flow
quality increases where the slope of the void fraction versus quality reduces.
This can suppress the fluctuation of the driving head for a small change in
quality. Hence, the flow stabilises at higher power (Figures3(a) and 3(b)).Type II Instability
Unlike
the type I instability, the type II instability occurs at high-power
conditions. This instability is driven by the interaction between the single
and two-phase frictional component of pressure losses, mass flow, void
formation, and propagation in the two-phase region. At high power, the flow
quality or void fraction in the system is very large. Hence, the two-phase
frictional pressure loss may be high owing to the smaller two-phase mixture
density. Having a large void fraction will increase the void propagation time
delay in the two-phase region of the system. Under these conditions, any small
fluctuation in flow can cause a larger fluctuation of the two-phase frictional
pressure loss due to fluctuation of density and flow, which propagates slowly
in the two-phase region. On the other hand, the fluctuation of the pressure
drop in the single-phase region occurs due to fluctuation of flow alone since
the fluctuation of the density is negligible. The pressure drop fluctuation in
this region travels much faster due to incompressibility of single-phase
region. If the two-phase pressure drop fluctuation is equal in magnitude but
opposite in phase with that of the single-phase region, the fluctuation or
oscillation is sustained in the system since there are no attenuating mechanisms.
Divergent oscillations can occur depending on the magnitude of the pressure-loss
fluctuation in the two-phase and single-phase regions and the propagation time
delay.
Because of the importance of void fraction and its effect on
the flow as explained above, this instability is sometimes referred to as
flow-void feedback instability in two-phase systems. Since transportation time
delays (related to the spacing between the light and heavy packets of fluid as
explained above) are crucial to this instability, it is also known as “time-delay
oscillations”. Density-wave instability (DWI) or density-wave oscillations
(DWO), first used by Stenning and Veziroğlu [7], is the most common term
used for the above described phenomenon as it appears that a density wave with
light and heavy fluid packets is traveling through the loop.


## 2.3. Based on the Number of Unstable Zones

Fukuda and Kobori [5] gave a
further classification of density-wave instability based on the number of
unstable zones. Usually, there exists a low-power and a high-power unstable
zone for density wave instability in forced as well as NC two-phase flows (Figure 3(a)). For the two-phase flow density-wave instability, the unstable region
below the lower threshold occurs at a low power and hence at low quality and is
named as type I instability by Fukuda and Kobori [5]. Similarly, the
unstable region beyond the upper threshold occurs at a high power and hence at
high qualities and is named as type II instability. However, in certain cases
depending on the geometry and operating conditions, islands of instability have
been observed to occur [8–10]. In these cases, more
than two zones of instability were oserved. Chen et al. [11] also observed
hysteresis in a two-phase loop. As an unstable single-phase system progresses
through single-phase NC to boiling inception and then to fully-developed
two-phase NC with power change, it can encounter several unstable zones. In
view of the existence of more than two unstable zones, this method of
classification could be confusing at times.


## 2.4. Based on the Nature of the Oscillations

All instabilities eventually lead
to some kind of oscillations. The oscillations can be labeled as flow
excursions, pressure drop oscillations, power oscillations, temperature
excursions or thermal oscillations, and so on. Besides, classifications based
on the oscillatory characteristics are sometimes reported for dynamic
instability. For example, based on the periodicity sometimes oscillations are
characterized as periodic and chaotic. Based on the oscillatory mode, the
oscillations are characterized as fundamental mode or higher harmonic modes
[12]. In boiling NC systems with multiple parallel channels,
inphase and out of phase modes are present depending on the geometry of the channels
and heating conditions. Sometimes, dual oscillations also are possible. In
natural circulation loops, flow direction can also change during oscillations.
Based on the direction of flow, the oscillations can be characterized as
unidirectional, bidirectional, or it can switch between the two. Such switching
is often accompanied by period doubling, tripling, or n-tupling.


## 2.5. Based on the Loop Geometry

Certain instabilities are
characteristic of the loop geometry. Examples are the instabilities observed in open U-loops,
symmetric closed loops, and asymmetric-closed loops. In addition, pressure-drop
oscillations and the parallel-channel instability are also characteristic of
the loop geometry. Another type instability which can occur in systems with a
compressible volume (a pressurizer for example) at the inlet of the heated
channel is the pressure-drop-type instability. Similarly, interaction among
parallel channels can also lead to various complex instabilities as discussed
above.


## 2.6. Based on the Disturbances

Certain two-phase flow phenomena
can cause a major disturbance and can lead to instability or modify the
instability characteristics significantly. Typical examples are boiling
inception, flashing, flow pattern transition, or the occurrence of CHF. Cold
water injection can also cause a major disturbance and instability in natural
circulation systems.


## 3. Characteristics of Instabilities

Several types of flow regimes can
be associated in a natural circulation loop as heating proceeds. Some of these
flow regimes are stable while others are unstable. For example, [13]
observed seven different types of flow modes in a boiling two-phase natural
circulation loop with increase in heater power such as (i) surface evaporation;
(ii) a static instability characterized by periodic exit large bubble
formation; (iii) a steady flow with continuous exit of small bubbles; (iv) a
static instability characterized by periodic exit of small bubbles; (v) another
static instability characterized by periodic extensive small bubble formation;
(vi) a steady natural circulation; and (vii) the density-wave oscillation
(dynamic instability). The static instabilities observed in their loop are due
to the high heat flux and subcooled boiling occurring in the heated section,
which are ideal for the cause of chugging-type instability. The characteristics
of the instabilities are different from one to the other due to the differences
in the physical mechanism associated with their initiations.


### 3.1. Characteristics of Instabilities Associated with Boiling  Inception

Boiling inception is a large enough disturbance that can
bring about significant change in the density and hence the buoyancy driving
force in an NCS. A stable single-phase NCS can become unstable with the
inception of boiling. Boiling inception is a static phenomenon that can lead to
instability in low-pressure systems. However, feedback effects also are
paramount in the phenomena. Hence the instability belongs to the class of
compound-static instability. In this case, however, the instability continues
with limit cycle oscillations. The oscillatory mode during boiling inception
can also be significantly affected by the presence of parallel channels.


#### 3.1.1. Effect of Boiling Inception on Unstable Single-Phase NC

With increase in
power, subcooled boiling begins in an unstable single-phase system leading to
the switching of flow between single-phase and two-phase regimes. Experiments
in a rectangular loop showed that subcooled boiling occurs first during the low
flow part of the oscillation cycle [14]. The bubbles formed
at the top horizontal-heated wall flows along the wall into the vertical limb
leading to an increase in flow rate. The increased flow suppresses boiling
leading to single-phase flow. Several regimes of unstable flow with subcooled
boiling can be observed depending on the test section power such as (a)
instability with sporadic boiling (boiling does not occur in every cycle); (b)
instability with subcooled boiling once in every cycle; (c) instability with
subcooled boiling twice in every cycle; and (d) instability with fully
developed boiling. The change in power required from the first to the last
stage is quite significant and it may not be reached in low-power loops.U-tube manometer-type instabilities have been observed in
boiling NC systems at reduced downcomer level [15] when the
loop is heated from single-phase condition. The flow is found to reverse even
before boiling is initiated (Figure 4). However, with initiation of boiling, no
flow reversal is observed (Figure 5). The
characteristics of oscillation were similar as previous cases (i.e., periodic
large amplitude oscillation with few small amplitude oscillations in between).
However, regular flow stagnation is observed, which is of concern for the
safety of nuclear reactors. Also the amplitude of oscillation was found to be
larger than that under single-phase conditions.Figure 4
Typical flow instability behavior at low power under single-phase condition (power 10 kW).Figure 5
Typical flow instability behavior at low power under two-phase condition (initiation of boiling, power 12 kW).


#### 3.1.2. Effect of Boiling Inception on Steady Single-Phase NC

A common characteristic
of the instabilities associated with boiling inception is that single-phase
conditions occur during part of the oscillation cycle. With the bubbles
entering the vertical tubes, the buoyancy force is increased which increases
the flow. As the flow is increased, the exit enthalpy is reduced leading to
suppression of boiling. This reduces the buoyancy force and the flow,
increasing the exit enthalpy resulting in boiling and leading to the repetition
of the process. Krishnan and
Gulshani [16] observed such instability in a figure-of-eight loop. They found
that the single-phase circulation was stable. However, with power increase, the
flow became unstable as soon as boiling was initiated in the heated section. Other
examples of instabilities associated with boiling inception in stable single-phase
NCs are:(a) Flashing Instability
Flashing instability is
expected to occur in NCSs with tall, unheated risers. The fundamental cause of
this instability is that the hot liquid from the heater outlet experiences
static pressure decrease as it flows up and may reach its saturation value in
the riser causing it to vaporize. The increased driving force generated by the
vaporization, increases the flow rate leading to reduced exit temperature and
suppression of flashing. This in turn reduces the driving force and flow
causing the exit temperature to increase once again leading to the repetition
of the process. The necessary condition for flashing is that the fluid
temperature at the inlet of the riser is greater than the saturated one at the
exit [17]. The instability is characterized by oscillatory
behavior and gets suppressed with rise in pressure [18, 19]. Furuya et al. [18] did
systematic analyses to characterize the difference between flashing
instability from other flow instabilities in boiling systems such as geysering,
natural circulation instabilities like a DWI, and flow pattern transition
instability. Their main observations were that the oscillation period in
flashing instability correlates well with the passing time of single-phase
liquid in the chimney section regardless of the system pressure, the heat flux,
the inlet subcooling, and the waveform. Out of phase flashing, oscillations were observed in
the parallel channels of the CIRCUS facility by Marcel et al. [20].(b) Geysering
Geysering was identified by both Boure et al. [4] and
Aritomi et al. [21] as an oscillatory phenomenon which is not necessarily
periodic. The proposed mechanism by both the investigators differ somewhat.
However, a common requirement for geysering is again a tall riser at the exit
of the heated section. When the heat flux is such that boiling is initiated at
the heater exit and as the bubbles begin to move up the riser they experience
sudden enlargement due to the decrease in static pressure and the accompanying
vapor generation, eventually resulting in vapor expulsion from the channel. The
liquid then returns, the subcooled nonboiling condition is restored, and the
cycle starts once again. The main difference with flashing instability is that
the vapor is produced first in the heated section in case of geysering, whereas
in flashing the vapor is formed by the decrease of the hydrostatic head as
water flows up.
The mechanism as proposed by Aritomi et al. [21]
considers condensation effects in the riser. According to him, geysering is
expected during subcooled boiling when the slug bubble detaches from the
surface and enters the riser (where the water is subcooled), where bubble
growth due to static-pressure decrease and condensation can take place. The sudden
condensation results in depressurization causing the liquid water to rush in
and occupy the space vacated by the condensed bubble. The large increase in the
flow rate causes the heated section to be filled with subcooled water
suppressing the subcooled boiling, and reducing the driving force. The reduced
driving force reduces the flow rate. Increasing the exit enthalpy and
eventually leading to subcooled boiling again and repetition of the process.
Geysering involves bubble formation during subcooled conditions, bubble
detachment, bubble growth, and condensation. Geysering is a thermal
nonequilibrium phenomenon. On the other hand, during flashing instability, the
vapor is in thermal equilibrium with the surrounding water and they do not
condense during the process of oscillation. Both these instabilities are
observed during low-pressure conditions only.
Instability due to
boiling inception usually disappears with increase in system pressure due to
the strong influence of pressure on the void fraction and hence the density
(Figure6).Effect of pressure on the void fraction.
(a)
For the entire range of qualities(b)
For low qualities


### 3.2. Characteristics of Two-Phase Static Instability

Static instability can lead either
to a different steady state or to a periodic behavior. Commonly observed,
static instabilities are flow excursion and boiling crisis.


#### 3.2.1. Flow Excursion Orexcursive Instability

The characteristics of the flow excursion instability
or Ledinegg type instability depend very much on the geometry as well as the
system pressure, power, and channel inlet subcooling [22].
Figure 7 shows an example of the stability maps for Ledinegg type instability
at different pressures for a natural circulation boiling water reactor [2]. The Ledinegg-type instability decreases with an increase in
pressure. This may be due to the fact that with an increase in pressure, the
void fraction decreases with quality significantly in the two-phase region,
which can reduce the S-shaped variation of the irreversible losses (i.e., ∂Δpint/∂w) responsible for the occurrence of the
Ledinegg-type instability. Similar to the type I and type II density-wave
oscillations, two types of Ledinegg instabilities are observed at any
subcooling depending on the operating power. With increase in pressure, the
threshold power for the lower instability boundary moves to much higher power
and the upper threshold boundary does not change significantly. The interesting
thing which can be observed from the figure is that this instability almost
vanishes when the operating pressure is more than 0.7 MPa.Figure 7
Characteristics of Ledinegg type instability in NCS.


#### 3.2.2. Flow Pattern Transition Instability

While there are
several experimental and analytical studies to understand the characteristics
of Ledinegg-type instability, there are not many studies on flow pattern
transition instability. Nayak et al. [3] were probably the first to clarify
some characteristics of this type instability theoretically. They compared the
stability maps between the Ledinegg and the flow pattern transition instability
(Figure 8). The Ledinegg-type instability is found to occur at a lower
power as compared to the flow-pattern transition instability at any subcooling.
However, both instabilities increase with rise in subcooling. More experimental
and theoretical studies are required to further clarify this instability.Figure 8
Comparison between Ledinegg and flow pattern transition instability maps for NC.The problems
associated with static instability is that the amplitude of oscillations can be
very high and sometimes the static instability can initiate the dynamic
oscillations in the system [17].
There are limited studies on the excursive instability behavior of a parallel
downward flow system (Babelli and Ishii (2001)).
While the mechanism of instability is same for upward- and downward-flow systems,
however, one important finding is that the flow excursion can be the dominant
mode of instability as compared to the density-wave instability in boiling NCs.


#### 3.2.3. Boiling Crisis

Following the occurrence of the critical heat flux, a
region of transition boiling, may be observed in many situations as in pool
boiling (see Figure9(a)). During transition boiling a film of vapor can
prevent the liquid from coming in direct contact with the heating surface
resulting in steep temperature rise and even failure. The film itself is not
stable causing repetitive wetting and dewetting of the heating surface
resulting in an oscillatory surface temperature. The instability is
characterized by sudden rise of wall temperature followed by an almost
simultaneous occurrence of flow oscillations. This will not be confused with
the premature occurrence of CHF during an oscillating flow, in which case the
oscillations occur first followed by CHF (see Figure 9(b)).Instability due to boiling crisis.
(a)
Pool boiling heat transfer regimes(b)
CHF occurrence during unstable oscillations


### 3.3. Characteristics of Two-Phase Dynamic Instabilities

Unlike the static instabilities,
there are several investigations in the area of dynamic instabilities,
particularly the density-wave oscillations. In fact, numerous experiments and
analytical studies are found in literature to clarify the characteristics of
the density-wave oscillations.


#### 3.3.1. Experimental Investigations

Experimental
investigations in two-phase natural circulation loops having single boiling
heated channel have been carried out by [13, 17, 24–27]. They observed density-wave instability in their experiments, which
was found to increase with increase in channel exit restriction and inlet
subcooling. Also low water level in downcomer and low system pressure increases
the density-wave instability in NCs. Such behavior was also observed in
parallel heated channels of a two-phase natural circulation loop by Mathisen
[28]. Lee and Ishii [25] found that the nonequilibrium between the phases
created flow instability in the loop. Kyung and Lee [26] investigated the
flow characteristics in an open two-phase natural circulation loop using
Freon-113 as test fluid.They observed three different modes of oscillation
with increase in heat flux such as (a) periodic oscillation (A) characterized
by flow oscillations with an incubation period. The mean circulation rate and
void fraction at the riser section were found to increase with heat flux; (b)
continuous circulation which is maintained with the churn/wispy-annular flow
pattern. This was found to be a stable operation mode in which the flow was
found to increase with heat flux first and then decrease with increase in heat
flux; and (c) periodic circulation (B) characterized by flow oscillations with
continuous boiling inside the heater section (i.e., there is no incubation
period) and void fraction fluctuates from 0.6 to 1.0 regularly. In this mode,
mean circulation rate was found to decrease with increase in heat flux,
although the mean void fraction kept on increasing. Jiang et al. [17]
observed three different kinds of flow instability such as geysering, flashing,
and density-wave oscillations during startup of the natural circulation loop.
Wu et. al. [27] observed that the flow oscialltory behavior was dependent on
the heating power and inlet subcooling. Depending on the operating conditions,
the oscillations can be periodic or chaotic. Fukuda and Kobori [5] observed
two modes of oscillations in a natural circulation loop with parallel heated
channels. One was the U-tube oscillation characterized by channel flows
oscillating with 1800 phase difference, and the other was the
inphase mode oscillations in which the channel flow oscillated alongwith the
whole loop without any phase lag among them. Out of phase oscillations were
also observed in the parallel channels of the CIRCUS facility by Marcel et al.
[20].


#### 3.3.2. Theoretical Investigations

Linear
analyses of boiling flow instabilities in natural circulation systems with
homogeneous flow assumptions have been carried out by Furutera [29], S. Y. Lee and D. W. Lee [22], Wang et al. [30], and Nayak et al. [2].
Advantage of homogeneous flow assumption is that it is easier to apply and the
model is also found to predict the stability boundary or the threshold of
instability with reasonable accuracy. Linear stability analyses with
homogeneous flow assumption and empirical model for the slip to calculate void
fraction as a function of mixture quality have been carried out by Fukuda et al. [31].
Linear stability analysis using a four-equation drift flux model has been
carried out by Ishii and Zuber [32], Saha and Zuber [33], Park et al. [34], Rizwan-Uddin and Dorning [35], van Bragt et al. [36],
and Nayak et al. [37]. These models are based on kinematic formulation which
considers the problem of mechanical nonequilibrium between the phases by having
a relationship between the quality and void fraction through superficial
velocities of liquid and vapor phases, vapor drift velocity, and void
distribution parameter. The adoption of drift flux model allows to replace two
separated momentum equations for liquid and vapor as used in the rigorous
two-fluid models, by one momentum equation for the mixture plus a
nondifferential constitutive law for the relative velocity. Besides, it
considers equilibrium phasic temperature as in case of homogeneous model. Saha and Zuber [33]
considered subcooled boiling in the drift flux model and applied the model to
the stability of a natural circulation system. They found that consideration of
thermal nonequilibrium condition results in a more stable system at low
subcooling and a more unstable system at high subcooling as compared to the
thermal equilibrium model. Rizwan-Uddin and Dorning
[35] found that the threshold power for stability in boiling
channel is sensitive to the void distribution parameter considered in the
analysis. They found that with an increase in void distribution parameter, the
stability of boiling channel increases. Similar results were also reported by Park et al. [34]
and van Bragt et al. [36]
for boiling channel systems. Nayak et al. [37] observed that the results are
true not only for forced convection boiling systems, but also for the type I
and type II instabilities observed in boiling natural circulation systems (Figures
10 to 12).Figure 10
Effect of void distribution parameter on threshold of Type I instability observed in HPNCL.Figure 11
Effect of void distribution parameter on threshold of Type II instability observed in the Apparatus-A of Furutera.Figure 12
Effect of void distribution parameter on threshold of Type I and Type II instabilities observed in the Apsara loop.Similar
results were also found for the effect of drift velocity on both type I and type
II instabilities [37]. For any mixture quality, the void
fraction is smaller for larger drift velocity. If the quality is disturbed by a
small amount, the void fraction with smaller drift velocity can have larger
fluctuation than the other due to larger slope of void fraction versus quality.Hence,the flow will be disturbed larger for a smaller fluctuation in quality
in this case. That is the reason for the reduction of type I instability with
increase in drift velocity (Figure 13). An increase in drift velocity is also
found to reduce the unstable region in the type II instability observed in Figures
14 to 15. With increase in drift velocity, the vapor propagation time lag in
two-phase region reduces, which has a stabilizing effect.Figure 13
Effect of drift velocity on threshold of Type I instability observed in High Pressure Natural Circulation Loop (HPNCL).Figure 14
Effect of drift velocity on threshold of Type II instability observed in the Apparatus-A of Furutera.Figure 15
Effect of drift velocity on the Type I and Type II instabilities threshold observed in the Apsara loop.Moreover, from these results it is interpreted thatthe homogeneous model for void fraction,
which considers a zero drift velocity and unity void distribution parameter,
predicts the most unstable region as compared to the slip models. Limited
studies by Nayak et al. [38] and Bagul et al. [39] have shown that the
homogeneous model predicts a more unstable region even as compared to the
two-fluid models such as RAMONA-5 and RELAP/MOD3.2 (Figures 16 and 17). Usually,
the homogeneous model predicts a larger void fraction than the two-fluid model
for the same mixture quality due to the absence of slip between the water and
steam in this model. The larger the void fraction in the system, the greater
the buoyancy force, and consequently a higher flow rate will be encountered. At
high flow rate, the frictional and local pressure drop in the two-phase region
become greater, which has a destabilizing effect.Figure 16
Comparison of stability maps between the homogeneous model and the two-fluid model (RAMONA5).Figure 17
Comparison of stability maps between the homogeneous model and the two-fluid model (RELAP5/MOD3.2).


#### 3.3.3. Effect of Geometric and Operating Parameters on Instability

Almost
all the theoretical and experimental studies agree well that the DWI can be
suppressed in boiling two-phase NC systems by increasing the system pressure.
This is true both for type I and type II instabilities. An increase in power suppresses
the type I instabilities, while enhances the type II instabilities according to
the basic classification of these instabilities. The effects of subcooling on
these instabilities are always debatable. While type I instabilities are always
found to enhance with rise in subcooling, type II instabilities may enhance or
reduce with subcooling depending on its magnitude and the system geometry and
heat input [1, 36, 38]. Invariably, it has been observed that with increase in local losses in
two-phase region, both type I and type II instabilities increase. On the other hand,
with increase in local losses in the single-phase region (such as orificing at
the inlet of channels), the improvement in stability has been found to be
conditional [2, 40] unlike in forced
circulation systems wherein it has been observed that with increase in local
losses in the single-phase region always improves the flow stability. In a
natural circulation system, the flow rate in the channel depends on the heating
power and the channel resistance. With increase in inlet throttling coefficient
for same heating power, the channel flow rate decreases, which in turn causes
an increase in channel exit quality. This may reduce the threshold power for
instability for that channel which may cause the other channel to be unstable.
So increase in orificing at channel inlet does not always increase the
stability of a natural circulation system with multiple parallel channels (Figure 18).Figure 18
Effect of orificing of channels on the stability of boiling natural circulation systems (K1 and K2 refer to the throttling coefficients of channel 1 and 2, resp.).The effect of riser geometry such as riser height and area on
flow stability is important. In a natural circulation system, the low-power type
I instability increases with increase in riser height. But the type II
instability may increase or decrease depending on the flow resistance and
heating power. For smaller riser height, lesser is the channel flow rate and
larger is the channel exit quality for same heating power. This gives larger
two-phase pressure drop due to large channel exit quality. Larger the riser
height, larger is the channel flow rate which may cause larger two-phase
pressure drop due to larger riser length. So a reduction or an increase in
riser height on type II instability of natural circulation system is
competitive [2].The effect of riser area on flow instability has been
discussed in great detail in a companion paper by the authors and hence will
not be repeated here. However, for sake of completeness, only a brief discussion
is presented below. For smaller riser flow area, the flow rate is smaller due
to larger resistance in small riser pipes. As the flow area is increased, the
flow rate increases, which gives rise to small frequency oscillations, typical
of low quality type I density-wave instability (Figure19) due to reduction in
void fraction. Hence, with increase in riser flow area, the type I instability
appears [38]. However,
the type II instability, which occurs at high power or void fraction, disappears
with increase in riser diameter [38] due to reduction in void
fraction or decrease in two-phase pressure drop.With an increase in riser
area,the time period of oscillation reduces due to the increase in flow rate
in the system.Figure 19
Effect of an increase in riser area on stability.


### 3.4. Characteristics of Compound Dynamic Instability

Instability is considered compound
when more than one elementary mechanisms interact in the process and cannot be
studied separately. If only one instability mechanism is at work, it is said to
be fundamental or pure instability. Examples of compound instability are (1)
Thermal oscillations; (2) Parallel channel instability (PCI); (3) Pressure-drop
oscillations; and (4) BWR (boiling water reactor) instability.


#### 3.4.1. Thermal Oscillations

In this case, the variable heat
transfer coefficient leads to a variable thermal response of the heated wall
that gets coupled with the DWO. Thermal oscillations are considered as a
regular feature of dryout of steam-water mixtures at high pressure [4]. The steep variation in heat transfer coefficient typical of
transition boiling conditions in a post CHF scenario can get coupled with the
DWO. During thermal oscillations, dryout or CHF point shift downstream or
upstream depending on the flow oscillations. Hence thermal oscillations are
characterized by large amplitude surface temperature oscillations (due to the
large variation in the heat transfer coefficient). The large variations in the
heat transfer coefficient and the surface temperature causes significant
variation in the heat transfer rate to the fluid even if the wall heat
generation rate is constant. This variable heat transfer rate modifies the pure
DWO.


#### 3.4.2. Parallel Channel Instability (PCI)

Interaction of parallel channels
with DWO can give rise to interesting stability behaviors as in single-phase
NC. Experimentally, both inphase and out of phase oscillations are observed in
parallel channels. However, inphase oscillation is a system characteristic and
parallel channels do not generally play a role in it. With inphase oscillation,
the amplitudes in different channels can be different due to the unequal heat
inputs or flow rates, but there is no phase difference among them. Occurrence
of out of phase oscillations is characteristic of PCI. The phase shift of
out-of-phase oscillations (OPO) is known to depend on the number of parallel
channels. With two channels, a phase shift of 180° is observed. With
three channels, it can be 120° and with five channels it can be 72° [41]. With n-channels, Aritomi et
al. [42] reports that the phase shift can be 2π/n. However,
depending on the number of channels participating, the phase shift can vary anywhere between π and 2π/n. For example, in a 3-channel
system one can get phase shift of 180° or 120° depending
on whether only two or all the three channels are participating.


#### 3.4.3. Pressure-Drop Oscillations (PDO)

Pressure-drop oscillations are
associated with operation in the negative sloping portion of the pressure drop-flow curve of the system. It is caused by the interaction of a compressible
volume (surge tank or pressurizer) at the inlet of the heated section with the
pump characteristics and is usually observed in forced circulation systems. DWO occurs at flow rates
lower than the flow rate at which pressure-drop oscillation is observed. Usually,
the frequency of pressure-drop oscillation is much smaller and hence it is easy
to distinguish it from density-wave oscillations. However, with a relatively
stiff system, the frequency of PDO can be comparable to DWO making it difficult
to distinguish between the two. Very long test sections may have sufficient internal
compressibility to initiate pressure drop oscillations. Like Ledinegg
instability, there is a danger of the occurrence of CHF during pressure drop
oscillations. Also inlet throttling (between the surge tank and the boiling
channel) is found to stabilize PDO just as Ledinegg instability.


#### 3.4.4. Instability in Natural Circulation BWRs

The flow velocity in natural
circulation BWRs is usually smaller than that of forced circulation BWR.
Besides, due to the presence of tall risers in natural circulation BWRs, the
frequency of density-wave oscillation can be much lower due to longer traveling
period of the two-phase mixture in the risers. The effects of negative void
reactivity feedback are found to stabilize the very low frequency type I
instabilities [43, 44]. But
it may stabilize or destabilize type II instabilities depending on its time
period [44].In case of a natural circulation
BWR, the existence of a tall riser or chimney over the core plays a different
role in inducing the instability. Series of experiments carried out by van Bragt et al. [36] in the Dodewaard natural circulation BWR in The Netherlands
showed that instabilities could occur at low as well as at high powers in this
reactor. From measured decay ratio, it was evident that at very low power there
is a trend of increase in decay ratio and similar results are seen at higher
power also. The low-power oscillations are induced by the type I density-wave
instabilities and high power oscillations are induced by the type II
density-wave instabilities. type I and type II instabilities have been predicted
to occur in the Indian AHWR which is a natural circulation pressure tube type
BWR, away from the nominal operating condition [44]. It may be
noted that in case of forced circulation BWRs, instabilities observed under
natural circulation conditions are due to pump trip transients when the core
exit quality is high due to low flow and high power. Hence these are induced by
the type II density-wave instabilities only.


## 3.1. Characteristics of Instabilities Associated with Boiling  Inception

Boiling inception is a large enough disturbance that can
bring about significant change in the density and hence the buoyancy driving
force in an NCS. A stable single-phase NCS can become unstable with the
inception of boiling. Boiling inception is a static phenomenon that can lead to
instability in low-pressure systems. However, feedback effects also are
paramount in the phenomena. Hence the instability belongs to the class of
compound-static instability. In this case, however, the instability continues
with limit cycle oscillations. The oscillatory mode during boiling inception
can also be significantly affected by the presence of parallel channels.


### 3.1.1. Effect of Boiling Inception on Unstable Single-Phase NC

With increase in
power, subcooled boiling begins in an unstable single-phase system leading to
the switching of flow between single-phase and two-phase regimes. Experiments
in a rectangular loop showed that subcooled boiling occurs first during the low
flow part of the oscillation cycle [14]. The bubbles formed
at the top horizontal-heated wall flows along the wall into the vertical limb
leading to an increase in flow rate. The increased flow suppresses boiling
leading to single-phase flow. Several regimes of unstable flow with subcooled
boiling can be observed depending on the test section power such as (a)
instability with sporadic boiling (boiling does not occur in every cycle); (b)
instability with subcooled boiling once in every cycle; (c) instability with
subcooled boiling twice in every cycle; and (d) instability with fully
developed boiling. The change in power required from the first to the last
stage is quite significant and it may not be reached in low-power loops.U-tube manometer-type instabilities have been observed in
boiling NC systems at reduced downcomer level [15] when the
loop is heated from single-phase condition. The flow is found to reverse even
before boiling is initiated (Figure 4). However, with initiation of boiling, no
flow reversal is observed (Figure 5). The
characteristics of oscillation were similar as previous cases (i.e., periodic
large amplitude oscillation with few small amplitude oscillations in between).
However, regular flow stagnation is observed, which is of concern for the
safety of nuclear reactors. Also the amplitude of oscillation was found to be
larger than that under single-phase conditions.Figure 4
Typical flow instability behavior at low power under single-phase condition (power 10 kW).Figure 5
Typical flow instability behavior at low power under two-phase condition (initiation of boiling, power 12 kW).


### 3.1.2. Effect of Boiling Inception on Steady Single-Phase NC

A common characteristic
of the instabilities associated with boiling inception is that single-phase
conditions occur during part of the oscillation cycle. With the bubbles
entering the vertical tubes, the buoyancy force is increased which increases
the flow. As the flow is increased, the exit enthalpy is reduced leading to
suppression of boiling. This reduces the buoyancy force and the flow,
increasing the exit enthalpy resulting in boiling and leading to the repetition
of the process. Krishnan and
Gulshani [16] observed such instability in a figure-of-eight loop. They found
that the single-phase circulation was stable. However, with power increase, the
flow became unstable as soon as boiling was initiated in the heated section. Other
examples of instabilities associated with boiling inception in stable single-phase
NCs are:(a) Flashing Instability
Flashing instability is
expected to occur in NCSs with tall, unheated risers. The fundamental cause of
this instability is that the hot liquid from the heater outlet experiences
static pressure decrease as it flows up and may reach its saturation value in
the riser causing it to vaporize. The increased driving force generated by the
vaporization, increases the flow rate leading to reduced exit temperature and
suppression of flashing. This in turn reduces the driving force and flow
causing the exit temperature to increase once again leading to the repetition
of the process. The necessary condition for flashing is that the fluid
temperature at the inlet of the riser is greater than the saturated one at the
exit [17]. The instability is characterized by oscillatory
behavior and gets suppressed with rise in pressure [18, 19]. Furuya et al. [18] did
systematic analyses to characterize the difference between flashing
instability from other flow instabilities in boiling systems such as geysering,
natural circulation instabilities like a DWI, and flow pattern transition
instability. Their main observations were that the oscillation period in
flashing instability correlates well with the passing time of single-phase
liquid in the chimney section regardless of the system pressure, the heat flux,
the inlet subcooling, and the waveform. Out of phase flashing, oscillations were observed in
the parallel channels of the CIRCUS facility by Marcel et al. [20].(b) Geysering
Geysering was identified by both Boure et al. [4] and
Aritomi et al. [21] as an oscillatory phenomenon which is not necessarily
periodic. The proposed mechanism by both the investigators differ somewhat.
However, a common requirement for geysering is again a tall riser at the exit
of the heated section. When the heat flux is such that boiling is initiated at
the heater exit and as the bubbles begin to move up the riser they experience
sudden enlargement due to the decrease in static pressure and the accompanying
vapor generation, eventually resulting in vapor expulsion from the channel. The
liquid then returns, the subcooled nonboiling condition is restored, and the
cycle starts once again. The main difference with flashing instability is that
the vapor is produced first in the heated section in case of geysering, whereas
in flashing the vapor is formed by the decrease of the hydrostatic head as
water flows up.
The mechanism as proposed by Aritomi et al. [21]
considers condensation effects in the riser. According to him, geysering is
expected during subcooled boiling when the slug bubble detaches from the
surface and enters the riser (where the water is subcooled), where bubble
growth due to static-pressure decrease and condensation can take place. The sudden
condensation results in depressurization causing the liquid water to rush in
and occupy the space vacated by the condensed bubble. The large increase in the
flow rate causes the heated section to be filled with subcooled water
suppressing the subcooled boiling, and reducing the driving force. The reduced
driving force reduces the flow rate. Increasing the exit enthalpy and
eventually leading to subcooled boiling again and repetition of the process.
Geysering involves bubble formation during subcooled conditions, bubble
detachment, bubble growth, and condensation. Geysering is a thermal
nonequilibrium phenomenon. On the other hand, during flashing instability, the
vapor is in thermal equilibrium with the surrounding water and they do not
condense during the process of oscillation. Both these instabilities are
observed during low-pressure conditions only.
Instability due to
boiling inception usually disappears with increase in system pressure due to
the strong influence of pressure on the void fraction and hence the density
(Figure6).Effect of pressure on the void fraction.
(a)
For the entire range of qualities(b)
For low qualities


## 3.1.1. Effect of Boiling Inception on Unstable Single-Phase NC

With increase in
power, subcooled boiling begins in an unstable single-phase system leading to
the switching of flow between single-phase and two-phase regimes. Experiments
in a rectangular loop showed that subcooled boiling occurs first during the low
flow part of the oscillation cycle [14]. The bubbles formed
at the top horizontal-heated wall flows along the wall into the vertical limb
leading to an increase in flow rate. The increased flow suppresses boiling
leading to single-phase flow. Several regimes of unstable flow with subcooled
boiling can be observed depending on the test section power such as (a)
instability with sporadic boiling (boiling does not occur in every cycle); (b)
instability with subcooled boiling once in every cycle; (c) instability with
subcooled boiling twice in every cycle; and (d) instability with fully
developed boiling. The change in power required from the first to the last
stage is quite significant and it may not be reached in low-power loops.U-tube manometer-type instabilities have been observed in
boiling NC systems at reduced downcomer level [15] when the
loop is heated from single-phase condition. The flow is found to reverse even
before boiling is initiated (Figure 4). However, with initiation of boiling, no
flow reversal is observed (Figure 5). The
characteristics of oscillation were similar as previous cases (i.e., periodic
large amplitude oscillation with few small amplitude oscillations in between).
However, regular flow stagnation is observed, which is of concern for the
safety of nuclear reactors. Also the amplitude of oscillation was found to be
larger than that under single-phase conditions.Figure 4
Typical flow instability behavior at low power under single-phase condition (power 10 kW).Figure 5
Typical flow instability behavior at low power under two-phase condition (initiation of boiling, power 12 kW).


## 3.1.2. Effect of Boiling Inception on Steady Single-Phase NC

A common characteristic
of the instabilities associated with boiling inception is that single-phase
conditions occur during part of the oscillation cycle. With the bubbles
entering the vertical tubes, the buoyancy force is increased which increases
the flow. As the flow is increased, the exit enthalpy is reduced leading to
suppression of boiling. This reduces the buoyancy force and the flow,
increasing the exit enthalpy resulting in boiling and leading to the repetition
of the process. Krishnan and
Gulshani [16] observed such instability in a figure-of-eight loop. They found
that the single-phase circulation was stable. However, with power increase, the
flow became unstable as soon as boiling was initiated in the heated section. Other
examples of instabilities associated with boiling inception in stable single-phase
NCs are:(a) Flashing Instability
Flashing instability is
expected to occur in NCSs with tall, unheated risers. The fundamental cause of
this instability is that the hot liquid from the heater outlet experiences
static pressure decrease as it flows up and may reach its saturation value in
the riser causing it to vaporize. The increased driving force generated by the
vaporization, increases the flow rate leading to reduced exit temperature and
suppression of flashing. This in turn reduces the driving force and flow
causing the exit temperature to increase once again leading to the repetition
of the process. The necessary condition for flashing is that the fluid
temperature at the inlet of the riser is greater than the saturated one at the
exit [17]. The instability is characterized by oscillatory
behavior and gets suppressed with rise in pressure [18, 19]. Furuya et al. [18] did
systematic analyses to characterize the difference between flashing
instability from other flow instabilities in boiling systems such as geysering,
natural circulation instabilities like a DWI, and flow pattern transition
instability. Their main observations were that the oscillation period in
flashing instability correlates well with the passing time of single-phase
liquid in the chimney section regardless of the system pressure, the heat flux,
the inlet subcooling, and the waveform. Out of phase flashing, oscillations were observed in
the parallel channels of the CIRCUS facility by Marcel et al. [20].(b) Geysering
Geysering was identified by both Boure et al. [4] and
Aritomi et al. [21] as an oscillatory phenomenon which is not necessarily
periodic. The proposed mechanism by both the investigators differ somewhat.
However, a common requirement for geysering is again a tall riser at the exit
of the heated section. When the heat flux is such that boiling is initiated at
the heater exit and as the bubbles begin to move up the riser they experience
sudden enlargement due to the decrease in static pressure and the accompanying
vapor generation, eventually resulting in vapor expulsion from the channel. The
liquid then returns, the subcooled nonboiling condition is restored, and the
cycle starts once again. The main difference with flashing instability is that
the vapor is produced first in the heated section in case of geysering, whereas
in flashing the vapor is formed by the decrease of the hydrostatic head as
water flows up.
The mechanism as proposed by Aritomi et al. [21]
considers condensation effects in the riser. According to him, geysering is
expected during subcooled boiling when the slug bubble detaches from the
surface and enters the riser (where the water is subcooled), where bubble
growth due to static-pressure decrease and condensation can take place. The sudden
condensation results in depressurization causing the liquid water to rush in
and occupy the space vacated by the condensed bubble. The large increase in the
flow rate causes the heated section to be filled with subcooled water
suppressing the subcooled boiling, and reducing the driving force. The reduced
driving force reduces the flow rate. Increasing the exit enthalpy and
eventually leading to subcooled boiling again and repetition of the process.
Geysering involves bubble formation during subcooled conditions, bubble
detachment, bubble growth, and condensation. Geysering is a thermal
nonequilibrium phenomenon. On the other hand, during flashing instability, the
vapor is in thermal equilibrium with the surrounding water and they do not
condense during the process of oscillation. Both these instabilities are
observed during low-pressure conditions only.
Instability due to
boiling inception usually disappears with increase in system pressure due to
the strong influence of pressure on the void fraction and hence the density
(Figure6).Effect of pressure on the void fraction.
(a)
For the entire range of qualities(b)
For low qualities


## 3.2. Characteristics of Two-Phase Static Instability

Static instability can lead either
to a different steady state or to a periodic behavior. Commonly observed,
static instabilities are flow excursion and boiling crisis.


### 3.2.1. Flow Excursion Orexcursive Instability

The characteristics of the flow excursion instability
or Ledinegg type instability depend very much on the geometry as well as the
system pressure, power, and channel inlet subcooling [22].
Figure 7 shows an example of the stability maps for Ledinegg type instability
at different pressures for a natural circulation boiling water reactor [2]. The Ledinegg-type instability decreases with an increase in
pressure. This may be due to the fact that with an increase in pressure, the
void fraction decreases with quality significantly in the two-phase region,
which can reduce the S-shaped variation of the irreversible losses (i.e., ∂Δpint/∂w) responsible for the occurrence of the
Ledinegg-type instability. Similar to the type I and type II density-wave
oscillations, two types of Ledinegg instabilities are observed at any
subcooling depending on the operating power. With increase in pressure, the
threshold power for the lower instability boundary moves to much higher power
and the upper threshold boundary does not change significantly. The interesting
thing which can be observed from the figure is that this instability almost
vanishes when the operating pressure is more than 0.7 MPa.Figure 7
Characteristics of Ledinegg type instability in NCS.


### 3.2.2. Flow Pattern Transition Instability

While there are
several experimental and analytical studies to understand the characteristics
of Ledinegg-type instability, there are not many studies on flow pattern
transition instability. Nayak et al. [3] were probably the first to clarify
some characteristics of this type instability theoretically. They compared the
stability maps between the Ledinegg and the flow pattern transition instability
(Figure 8). The Ledinegg-type instability is found to occur at a lower
power as compared to the flow-pattern transition instability at any subcooling.
However, both instabilities increase with rise in subcooling. More experimental
and theoretical studies are required to further clarify this instability.Figure 8
Comparison between Ledinegg and flow pattern transition instability maps for NC.The problems
associated with static instability is that the amplitude of oscillations can be
very high and sometimes the static instability can initiate the dynamic
oscillations in the system [17].
There are limited studies on the excursive instability behavior of a parallel
downward flow system (Babelli and Ishii (2001)).
While the mechanism of instability is same for upward- and downward-flow systems,
however, one important finding is that the flow excursion can be the dominant
mode of instability as compared to the density-wave instability in boiling NCs.


### 3.2.3. Boiling Crisis

Following the occurrence of the critical heat flux, a
region of transition boiling, may be observed in many situations as in pool
boiling (see Figure9(a)). During transition boiling a film of vapor can
prevent the liquid from coming in direct contact with the heating surface
resulting in steep temperature rise and even failure. The film itself is not
stable causing repetitive wetting and dewetting of the heating surface
resulting in an oscillatory surface temperature. The instability is
characterized by sudden rise of wall temperature followed by an almost
simultaneous occurrence of flow oscillations. This will not be confused with
the premature occurrence of CHF during an oscillating flow, in which case the
oscillations occur first followed by CHF (see Figure 9(b)).Instability due to boiling crisis.
(a)
Pool boiling heat transfer regimes(b)
CHF occurrence during unstable oscillations


## 3.2.1. Flow Excursion Orexcursive Instability

The characteristics of the flow excursion instability
or Ledinegg type instability depend very much on the geometry as well as the
system pressure, power, and channel inlet subcooling [22].
Figure 7 shows an example of the stability maps for Ledinegg type instability
at different pressures for a natural circulation boiling water reactor [2]. The Ledinegg-type instability decreases with an increase in
pressure. This may be due to the fact that with an increase in pressure, the
void fraction decreases with quality significantly in the two-phase region,
which can reduce the S-shaped variation of the irreversible losses (i.e., ∂Δpint/∂w) responsible for the occurrence of the
Ledinegg-type instability. Similar to the type I and type II density-wave
oscillations, two types of Ledinegg instabilities are observed at any
subcooling depending on the operating power. With increase in pressure, the
threshold power for the lower instability boundary moves to much higher power
and the upper threshold boundary does not change significantly. The interesting
thing which can be observed from the figure is that this instability almost
vanishes when the operating pressure is more than 0.7 MPa.Figure 7
Characteristics of Ledinegg type instability in NCS.


## 3.2.2. Flow Pattern Transition Instability

While there are
several experimental and analytical studies to understand the characteristics
of Ledinegg-type instability, there are not many studies on flow pattern
transition instability. Nayak et al. [3] were probably the first to clarify
some characteristics of this type instability theoretically. They compared the
stability maps between the Ledinegg and the flow pattern transition instability
(Figure 8). The Ledinegg-type instability is found to occur at a lower
power as compared to the flow-pattern transition instability at any subcooling.
However, both instabilities increase with rise in subcooling. More experimental
and theoretical studies are required to further clarify this instability.Figure 8
Comparison between Ledinegg and flow pattern transition instability maps for NC.The problems
associated with static instability is that the amplitude of oscillations can be
very high and sometimes the static instability can initiate the dynamic
oscillations in the system [17].
There are limited studies on the excursive instability behavior of a parallel
downward flow system (Babelli and Ishii (2001)).
While the mechanism of instability is same for upward- and downward-flow systems,
however, one important finding is that the flow excursion can be the dominant
mode of instability as compared to the density-wave instability in boiling NCs.


## 3.2.3. Boiling Crisis

Following the occurrence of the critical heat flux, a
region of transition boiling, may be observed in many situations as in pool
boiling (see Figure9(a)). During transition boiling a film of vapor can
prevent the liquid from coming in direct contact with the heating surface
resulting in steep temperature rise and even failure. The film itself is not
stable causing repetitive wetting and dewetting of the heating surface
resulting in an oscillatory surface temperature. The instability is
characterized by sudden rise of wall temperature followed by an almost
simultaneous occurrence of flow oscillations. This will not be confused with
the premature occurrence of CHF during an oscillating flow, in which case the
oscillations occur first followed by CHF (see Figure 9(b)).Instability due to boiling crisis.
(a)
Pool boiling heat transfer regimes(b)
CHF occurrence during unstable oscillations


## 3.3. Characteristics of Two-Phase Dynamic Instabilities

Unlike the static instabilities,
there are several investigations in the area of dynamic instabilities,
particularly the density-wave oscillations. In fact, numerous experiments and
analytical studies are found in literature to clarify the characteristics of
the density-wave oscillations.


### 3.3.1. Experimental Investigations

Experimental
investigations in two-phase natural circulation loops having single boiling
heated channel have been carried out by [13, 17, 24–27]. They observed density-wave instability in their experiments, which
was found to increase with increase in channel exit restriction and inlet
subcooling. Also low water level in downcomer and low system pressure increases
the density-wave instability in NCs. Such behavior was also observed in
parallel heated channels of a two-phase natural circulation loop by Mathisen
[28]. Lee and Ishii [25] found that the nonequilibrium between the phases
created flow instability in the loop. Kyung and Lee [26] investigated the
flow characteristics in an open two-phase natural circulation loop using
Freon-113 as test fluid.They observed three different modes of oscillation
with increase in heat flux such as (a) periodic oscillation (A) characterized
by flow oscillations with an incubation period. The mean circulation rate and
void fraction at the riser section were found to increase with heat flux; (b)
continuous circulation which is maintained with the churn/wispy-annular flow
pattern. This was found to be a stable operation mode in which the flow was
found to increase with heat flux first and then decrease with increase in heat
flux; and (c) periodic circulation (B) characterized by flow oscillations with
continuous boiling inside the heater section (i.e., there is no incubation
period) and void fraction fluctuates from 0.6 to 1.0 regularly. In this mode,
mean circulation rate was found to decrease with increase in heat flux,
although the mean void fraction kept on increasing. Jiang et al. [17]
observed three different kinds of flow instability such as geysering, flashing,
and density-wave oscillations during startup of the natural circulation loop.
Wu et. al. [27] observed that the flow oscialltory behavior was dependent on
the heating power and inlet subcooling. Depending on the operating conditions,
the oscillations can be periodic or chaotic. Fukuda and Kobori [5] observed
two modes of oscillations in a natural circulation loop with parallel heated
channels. One was the U-tube oscillation characterized by channel flows
oscillating with 1800 phase difference, and the other was the
inphase mode oscillations in which the channel flow oscillated alongwith the
whole loop without any phase lag among them. Out of phase oscillations were
also observed in the parallel channels of the CIRCUS facility by Marcel et al.
[20].


### 3.3.2. Theoretical Investigations

Linear
analyses of boiling flow instabilities in natural circulation systems with
homogeneous flow assumptions have been carried out by Furutera [29], S. Y. Lee and D. W. Lee [22], Wang et al. [30], and Nayak et al. [2].
Advantage of homogeneous flow assumption is that it is easier to apply and the
model is also found to predict the stability boundary or the threshold of
instability with reasonable accuracy. Linear stability analyses with
homogeneous flow assumption and empirical model for the slip to calculate void
fraction as a function of mixture quality have been carried out by Fukuda et al. [31].
Linear stability analysis using a four-equation drift flux model has been
carried out by Ishii and Zuber [32], Saha and Zuber [33], Park et al. [34], Rizwan-Uddin and Dorning [35], van Bragt et al. [36],
and Nayak et al. [37]. These models are based on kinematic formulation which
considers the problem of mechanical nonequilibrium between the phases by having
a relationship between the quality and void fraction through superficial
velocities of liquid and vapor phases, vapor drift velocity, and void
distribution parameter. The adoption of drift flux model allows to replace two
separated momentum equations for liquid and vapor as used in the rigorous
two-fluid models, by one momentum equation for the mixture plus a
nondifferential constitutive law for the relative velocity. Besides, it
considers equilibrium phasic temperature as in case of homogeneous model. Saha and Zuber [33]
considered subcooled boiling in the drift flux model and applied the model to
the stability of a natural circulation system. They found that consideration of
thermal nonequilibrium condition results in a more stable system at low
subcooling and a more unstable system at high subcooling as compared to the
thermal equilibrium model. Rizwan-Uddin and Dorning
[35] found that the threshold power for stability in boiling
channel is sensitive to the void distribution parameter considered in the
analysis. They found that with an increase in void distribution parameter, the
stability of boiling channel increases. Similar results were also reported by Park et al. [34]
and van Bragt et al. [36]
for boiling channel systems. Nayak et al. [37] observed that the results are
true not only for forced convection boiling systems, but also for the type I
and type II instabilities observed in boiling natural circulation systems (Figures
10 to 12).Figure 10
Effect of void distribution parameter on threshold of Type I instability observed in HPNCL.Figure 11
Effect of void distribution parameter on threshold of Type II instability observed in the Apparatus-A of Furutera.Figure 12
Effect of void distribution parameter on threshold of Type I and Type II instabilities observed in the Apsara loop.Similar
results were also found for the effect of drift velocity on both type I and type
II instabilities [37]. For any mixture quality, the void
fraction is smaller for larger drift velocity. If the quality is disturbed by a
small amount, the void fraction with smaller drift velocity can have larger
fluctuation than the other due to larger slope of void fraction versus quality.Hence,the flow will be disturbed larger for a smaller fluctuation in quality
in this case. That is the reason for the reduction of type I instability with
increase in drift velocity (Figure 13). An increase in drift velocity is also
found to reduce the unstable region in the type II instability observed in Figures
14 to 15. With increase in drift velocity, the vapor propagation time lag in
two-phase region reduces, which has a stabilizing effect.Figure 13
Effect of drift velocity on threshold of Type I instability observed in High Pressure Natural Circulation Loop (HPNCL).Figure 14
Effect of drift velocity on threshold of Type II instability observed in the Apparatus-A of Furutera.Figure 15
Effect of drift velocity on the Type I and Type II instabilities threshold observed in the Apsara loop.Moreover, from these results it is interpreted thatthe homogeneous model for void fraction,
which considers a zero drift velocity and unity void distribution parameter,
predicts the most unstable region as compared to the slip models. Limited
studies by Nayak et al. [38] and Bagul et al. [39] have shown that the
homogeneous model predicts a more unstable region even as compared to the
two-fluid models such as RAMONA-5 and RELAP/MOD3.2 (Figures 16 and 17). Usually,
the homogeneous model predicts a larger void fraction than the two-fluid model
for the same mixture quality due to the absence of slip between the water and
steam in this model. The larger the void fraction in the system, the greater
the buoyancy force, and consequently a higher flow rate will be encountered. At
high flow rate, the frictional and local pressure drop in the two-phase region
become greater, which has a destabilizing effect.Figure 16
Comparison of stability maps between the homogeneous model and the two-fluid model (RAMONA5).Figure 17
Comparison of stability maps between the homogeneous model and the two-fluid model (RELAP5/MOD3.2).


### 3.3.3. Effect of Geometric and Operating Parameters on Instability

Almost
all the theoretical and experimental studies agree well that the DWI can be
suppressed in boiling two-phase NC systems by increasing the system pressure.
This is true both for type I and type II instabilities. An increase in power suppresses
the type I instabilities, while enhances the type II instabilities according to
the basic classification of these instabilities. The effects of subcooling on
these instabilities are always debatable. While type I instabilities are always
found to enhance with rise in subcooling, type II instabilities may enhance or
reduce with subcooling depending on its magnitude and the system geometry and
heat input [1, 36, 38]. Invariably, it has been observed that with increase in local losses in
two-phase region, both type I and type II instabilities increase. On the other hand,
with increase in local losses in the single-phase region (such as orificing at
the inlet of channels), the improvement in stability has been found to be
conditional [2, 40] unlike in forced
circulation systems wherein it has been observed that with increase in local
losses in the single-phase region always improves the flow stability. In a
natural circulation system, the flow rate in the channel depends on the heating
power and the channel resistance. With increase in inlet throttling coefficient
for same heating power, the channel flow rate decreases, which in turn causes
an increase in channel exit quality. This may reduce the threshold power for
instability for that channel which may cause the other channel to be unstable.
So increase in orificing at channel inlet does not always increase the
stability of a natural circulation system with multiple parallel channels (Figure 18).Figure 18
Effect of orificing of channels on the stability of boiling natural circulation systems (K1 and K2 refer to the throttling coefficients of channel 1 and 2, resp.).The effect of riser geometry such as riser height and area on
flow stability is important. In a natural circulation system, the low-power type
I instability increases with increase in riser height. But the type II
instability may increase or decrease depending on the flow resistance and
heating power. For smaller riser height, lesser is the channel flow rate and
larger is the channel exit quality for same heating power. This gives larger
two-phase pressure drop due to large channel exit quality. Larger the riser
height, larger is the channel flow rate which may cause larger two-phase
pressure drop due to larger riser length. So a reduction or an increase in
riser height on type II instability of natural circulation system is
competitive [2].The effect of riser area on flow instability has been
discussed in great detail in a companion paper by the authors and hence will
not be repeated here. However, for sake of completeness, only a brief discussion
is presented below. For smaller riser flow area, the flow rate is smaller due
to larger resistance in small riser pipes. As the flow area is increased, the
flow rate increases, which gives rise to small frequency oscillations, typical
of low quality type I density-wave instability (Figure19) due to reduction in
void fraction. Hence, with increase in riser flow area, the type I instability
appears [38]. However,
the type II instability, which occurs at high power or void fraction, disappears
with increase in riser diameter [38] due to reduction in void
fraction or decrease in two-phase pressure drop.With an increase in riser
area,the time period of oscillation reduces due to the increase in flow rate
in the system.Figure 19
Effect of an increase in riser area on stability.


## 3.3.1. Experimental Investigations

Experimental
investigations in two-phase natural circulation loops having single boiling
heated channel have been carried out by [13, 17, 24–27]. They observed density-wave instability in their experiments, which
was found to increase with increase in channel exit restriction and inlet
subcooling. Also low water level in downcomer and low system pressure increases
the density-wave instability in NCs. Such behavior was also observed in
parallel heated channels of a two-phase natural circulation loop by Mathisen
[28]. Lee and Ishii [25] found that the nonequilibrium between the phases
created flow instability in the loop. Kyung and Lee [26] investigated the
flow characteristics in an open two-phase natural circulation loop using
Freon-113 as test fluid.They observed three different modes of oscillation
with increase in heat flux such as (a) periodic oscillation (A) characterized
by flow oscillations with an incubation period. The mean circulation rate and
void fraction at the riser section were found to increase with heat flux; (b)
continuous circulation which is maintained with the churn/wispy-annular flow
pattern. This was found to be a stable operation mode in which the flow was
found to increase with heat flux first and then decrease with increase in heat
flux; and (c) periodic circulation (B) characterized by flow oscillations with
continuous boiling inside the heater section (i.e., there is no incubation
period) and void fraction fluctuates from 0.6 to 1.0 regularly. In this mode,
mean circulation rate was found to decrease with increase in heat flux,
although the mean void fraction kept on increasing. Jiang et al. [17]
observed three different kinds of flow instability such as geysering, flashing,
and density-wave oscillations during startup of the natural circulation loop.
Wu et. al. [27] observed that the flow oscialltory behavior was dependent on
the heating power and inlet subcooling. Depending on the operating conditions,
the oscillations can be periodic or chaotic. Fukuda and Kobori [5] observed
two modes of oscillations in a natural circulation loop with parallel heated
channels. One was the U-tube oscillation characterized by channel flows
oscillating with 1800 phase difference, and the other was the
inphase mode oscillations in which the channel flow oscillated alongwith the
whole loop without any phase lag among them. Out of phase oscillations were
also observed in the parallel channels of the CIRCUS facility by Marcel et al.
[20].


## 3.3.2. Theoretical Investigations

Linear
analyses of boiling flow instabilities in natural circulation systems with
homogeneous flow assumptions have been carried out by Furutera [29], S. Y. Lee and D. W. Lee [22], Wang et al. [30], and Nayak et al. [2].
Advantage of homogeneous flow assumption is that it is easier to apply and the
model is also found to predict the stability boundary or the threshold of
instability with reasonable accuracy. Linear stability analyses with
homogeneous flow assumption and empirical model for the slip to calculate void
fraction as a function of mixture quality have been carried out by Fukuda et al. [31].
Linear stability analysis using a four-equation drift flux model has been
carried out by Ishii and Zuber [32], Saha and Zuber [33], Park et al. [34], Rizwan-Uddin and Dorning [35], van Bragt et al. [36],
and Nayak et al. [37]. These models are based on kinematic formulation which
considers the problem of mechanical nonequilibrium between the phases by having
a relationship between the quality and void fraction through superficial
velocities of liquid and vapor phases, vapor drift velocity, and void
distribution parameter. The adoption of drift flux model allows to replace two
separated momentum equations for liquid and vapor as used in the rigorous
two-fluid models, by one momentum equation for the mixture plus a
nondifferential constitutive law for the relative velocity. Besides, it
considers equilibrium phasic temperature as in case of homogeneous model. Saha and Zuber [33]
considered subcooled boiling in the drift flux model and applied the model to
the stability of a natural circulation system. They found that consideration of
thermal nonequilibrium condition results in a more stable system at low
subcooling and a more unstable system at high subcooling as compared to the
thermal equilibrium model. Rizwan-Uddin and Dorning
[35] found that the threshold power for stability in boiling
channel is sensitive to the void distribution parameter considered in the
analysis. They found that with an increase in void distribution parameter, the
stability of boiling channel increases. Similar results were also reported by Park et al. [34]
and van Bragt et al. [36]
for boiling channel systems. Nayak et al. [37] observed that the results are
true not only for forced convection boiling systems, but also for the type I
and type II instabilities observed in boiling natural circulation systems (Figures
10 to 12).Figure 10
Effect of void distribution parameter on threshold of Type I instability observed in HPNCL.Figure 11
Effect of void distribution parameter on threshold of Type II instability observed in the Apparatus-A of Furutera.Figure 12
Effect of void distribution parameter on threshold of Type I and Type II instabilities observed in the Apsara loop.Similar
results were also found for the effect of drift velocity on both type I and type
II instabilities [37]. For any mixture quality, the void
fraction is smaller for larger drift velocity. If the quality is disturbed by a
small amount, the void fraction with smaller drift velocity can have larger
fluctuation than the other due to larger slope of void fraction versus quality.Hence,the flow will be disturbed larger for a smaller fluctuation in quality
in this case. That is the reason for the reduction of type I instability with
increase in drift velocity (Figure 13). An increase in drift velocity is also
found to reduce the unstable region in the type II instability observed in Figures
14 to 15. With increase in drift velocity, the vapor propagation time lag in
two-phase region reduces, which has a stabilizing effect.Figure 13
Effect of drift velocity on threshold of Type I instability observed in High Pressure Natural Circulation Loop (HPNCL).Figure 14
Effect of drift velocity on threshold of Type II instability observed in the Apparatus-A of Furutera.Figure 15
Effect of drift velocity on the Type I and Type II instabilities threshold observed in the Apsara loop.Moreover, from these results it is interpreted thatthe homogeneous model for void fraction,
which considers a zero drift velocity and unity void distribution parameter,
predicts the most unstable region as compared to the slip models. Limited
studies by Nayak et al. [38] and Bagul et al. [39] have shown that the
homogeneous model predicts a more unstable region even as compared to the
two-fluid models such as RAMONA-5 and RELAP/MOD3.2 (Figures 16 and 17). Usually,
the homogeneous model predicts a larger void fraction than the two-fluid model
for the same mixture quality due to the absence of slip between the water and
steam in this model. The larger the void fraction in the system, the greater
the buoyancy force, and consequently a higher flow rate will be encountered. At
high flow rate, the frictional and local pressure drop in the two-phase region
become greater, which has a destabilizing effect.Figure 16
Comparison of stability maps between the homogeneous model and the two-fluid model (RAMONA5).Figure 17
Comparison of stability maps between the homogeneous model and the two-fluid model (RELAP5/MOD3.2).


## 3.3.3. Effect of Geometric and Operating Parameters on Instability

Almost
all the theoretical and experimental studies agree well that the DWI can be
suppressed in boiling two-phase NC systems by increasing the system pressure.
This is true both for type I and type II instabilities. An increase in power suppresses
the type I instabilities, while enhances the type II instabilities according to
the basic classification of these instabilities. The effects of subcooling on
these instabilities are always debatable. While type I instabilities are always
found to enhance with rise in subcooling, type II instabilities may enhance or
reduce with subcooling depending on its magnitude and the system geometry and
heat input [1, 36, 38]. Invariably, it has been observed that with increase in local losses in
two-phase region, both type I and type II instabilities increase. On the other hand,
with increase in local losses in the single-phase region (such as orificing at
the inlet of channels), the improvement in stability has been found to be
conditional [2, 40] unlike in forced
circulation systems wherein it has been observed that with increase in local
losses in the single-phase region always improves the flow stability. In a
natural circulation system, the flow rate in the channel depends on the heating
power and the channel resistance. With increase in inlet throttling coefficient
for same heating power, the channel flow rate decreases, which in turn causes
an increase in channel exit quality. This may reduce the threshold power for
instability for that channel which may cause the other channel to be unstable.
So increase in orificing at channel inlet does not always increase the
stability of a natural circulation system with multiple parallel channels (Figure 18).Figure 18
Effect of orificing of channels on the stability of boiling natural circulation systems (K1 and K2 refer to the throttling coefficients of channel 1 and 2, resp.).The effect of riser geometry such as riser height and area on
flow stability is important. In a natural circulation system, the low-power type
I instability increases with increase in riser height. But the type II
instability may increase or decrease depending on the flow resistance and
heating power. For smaller riser height, lesser is the channel flow rate and
larger is the channel exit quality for same heating power. This gives larger
two-phase pressure drop due to large channel exit quality. Larger the riser
height, larger is the channel flow rate which may cause larger two-phase
pressure drop due to larger riser length. So a reduction or an increase in
riser height on type II instability of natural circulation system is
competitive [2].The effect of riser area on flow instability has been
discussed in great detail in a companion paper by the authors and hence will
not be repeated here. However, for sake of completeness, only a brief discussion
is presented below. For smaller riser flow area, the flow rate is smaller due
to larger resistance in small riser pipes. As the flow area is increased, the
flow rate increases, which gives rise to small frequency oscillations, typical
of low quality type I density-wave instability (Figure19) due to reduction in
void fraction. Hence, with increase in riser flow area, the type I instability
appears [38]. However,
the type II instability, which occurs at high power or void fraction, disappears
with increase in riser diameter [38] due to reduction in void
fraction or decrease in two-phase pressure drop.With an increase in riser
area,the time period of oscillation reduces due to the increase in flow rate
in the system.Figure 19
Effect of an increase in riser area on stability.


## 3.4. Characteristics of Compound Dynamic Instability

Instability is considered compound
when more than one elementary mechanisms interact in the process and cannot be
studied separately. If only one instability mechanism is at work, it is said to
be fundamental or pure instability. Examples of compound instability are (1)
Thermal oscillations; (2) Parallel channel instability (PCI); (3) Pressure-drop
oscillations; and (4) BWR (boiling water reactor) instability.


### 3.4.1. Thermal Oscillations

In this case, the variable heat
transfer coefficient leads to a variable thermal response of the heated wall
that gets coupled with the DWO. Thermal oscillations are considered as a
regular feature of dryout of steam-water mixtures at high pressure [4]. The steep variation in heat transfer coefficient typical of
transition boiling conditions in a post CHF scenario can get coupled with the
DWO. During thermal oscillations, dryout or CHF point shift downstream or
upstream depending on the flow oscillations. Hence thermal oscillations are
characterized by large amplitude surface temperature oscillations (due to the
large variation in the heat transfer coefficient). The large variations in the
heat transfer coefficient and the surface temperature causes significant
variation in the heat transfer rate to the fluid even if the wall heat
generation rate is constant. This variable heat transfer rate modifies the pure
DWO.


### 3.4.2. Parallel Channel Instability (PCI)

Interaction of parallel channels
with DWO can give rise to interesting stability behaviors as in single-phase
NC. Experimentally, both inphase and out of phase oscillations are observed in
parallel channels. However, inphase oscillation is a system characteristic and
parallel channels do not generally play a role in it. With inphase oscillation,
the amplitudes in different channels can be different due to the unequal heat
inputs or flow rates, but there is no phase difference among them. Occurrence
of out of phase oscillations is characteristic of PCI. The phase shift of
out-of-phase oscillations (OPO) is known to depend on the number of parallel
channels. With two channels, a phase shift of 180° is observed. With
three channels, it can be 120° and with five channels it can be 72° [41]. With n-channels, Aritomi et
al. [42] reports that the phase shift can be 2π/n. However,
depending on the number of channels participating, the phase shift can vary anywhere between π and 2π/n. For example, in a 3-channel
system one can get phase shift of 180° or 120° depending
on whether only two or all the three channels are participating.


### 3.4.3. Pressure-Drop Oscillations (PDO)

Pressure-drop oscillations are
associated with operation in the negative sloping portion of the pressure drop-flow curve of the system. It is caused by the interaction of a compressible
volume (surge tank or pressurizer) at the inlet of the heated section with the
pump characteristics and is usually observed in forced circulation systems. DWO occurs at flow rates
lower than the flow rate at which pressure-drop oscillation is observed. Usually,
the frequency of pressure-drop oscillation is much smaller and hence it is easy
to distinguish it from density-wave oscillations. However, with a relatively
stiff system, the frequency of PDO can be comparable to DWO making it difficult
to distinguish between the two. Very long test sections may have sufficient internal
compressibility to initiate pressure drop oscillations. Like Ledinegg
instability, there is a danger of the occurrence of CHF during pressure drop
oscillations. Also inlet throttling (between the surge tank and the boiling
channel) is found to stabilize PDO just as Ledinegg instability.


### 3.4.4. Instability in Natural Circulation BWRs

The flow velocity in natural
circulation BWRs is usually smaller than that of forced circulation BWR.
Besides, due to the presence of tall risers in natural circulation BWRs, the
frequency of density-wave oscillation can be much lower due to longer traveling
period of the two-phase mixture in the risers. The effects of negative void
reactivity feedback are found to stabilize the very low frequency type I
instabilities [43, 44]. But
it may stabilize or destabilize type II instabilities depending on its time
period [44].In case of a natural circulation
BWR, the existence of a tall riser or chimney over the core plays a different
role in inducing the instability. Series of experiments carried out by van Bragt et al. [36] in the Dodewaard natural circulation BWR in The Netherlands
showed that instabilities could occur at low as well as at high powers in this
reactor. From measured decay ratio, it was evident that at very low power there
is a trend of increase in decay ratio and similar results are seen at higher
power also. The low-power oscillations are induced by the type I density-wave
instabilities and high power oscillations are induced by the type II
density-wave instabilities. type I and type II instabilities have been predicted
to occur in the Indian AHWR which is a natural circulation pressure tube type
BWR, away from the nominal operating condition [44]. It may be
noted that in case of forced circulation BWRs, instabilities observed under
natural circulation conditions are due to pump trip transients when the core
exit quality is high due to low flow and high power. Hence these are induced by
the type II density-wave instabilities only.


## 3.4.1. Thermal Oscillations

In this case, the variable heat
transfer coefficient leads to a variable thermal response of the heated wall
that gets coupled with the DWO. Thermal oscillations are considered as a
regular feature of dryout of steam-water mixtures at high pressure [4]. The steep variation in heat transfer coefficient typical of
transition boiling conditions in a post CHF scenario can get coupled with the
DWO. During thermal oscillations, dryout or CHF point shift downstream or
upstream depending on the flow oscillations. Hence thermal oscillations are
characterized by large amplitude surface temperature oscillations (due to the
large variation in the heat transfer coefficient). The large variations in the
heat transfer coefficient and the surface temperature causes significant
variation in the heat transfer rate to the fluid even if the wall heat
generation rate is constant. This variable heat transfer rate modifies the pure
DWO.


## 3.4.2. Parallel Channel Instability (PCI)

Interaction of parallel channels
with DWO can give rise to interesting stability behaviors as in single-phase
NC. Experimentally, both inphase and out of phase oscillations are observed in
parallel channels. However, inphase oscillation is a system characteristic and
parallel channels do not generally play a role in it. With inphase oscillation,
the amplitudes in different channels can be different due to the unequal heat
inputs or flow rates, but there is no phase difference among them. Occurrence
of out of phase oscillations is characteristic of PCI. The phase shift of
out-of-phase oscillations (OPO) is known to depend on the number of parallel
channels. With two channels, a phase shift of 180° is observed. With
three channels, it can be 120° and with five channels it can be 72° [41]. With n-channels, Aritomi et
al. [42] reports that the phase shift can be 2π/n. However,
depending on the number of channels participating, the phase shift can vary anywhere between π and 2π/n. For example, in a 3-channel
system one can get phase shift of 180° or 120° depending
on whether only two or all the three channels are participating.


## 3.4.3. Pressure-Drop Oscillations (PDO)

Pressure-drop oscillations are
associated with operation in the negative sloping portion of the pressure drop-flow curve of the system. It is caused by the interaction of a compressible
volume (surge tank or pressurizer) at the inlet of the heated section with the
pump characteristics and is usually observed in forced circulation systems. DWO occurs at flow rates
lower than the flow rate at which pressure-drop oscillation is observed. Usually,
the frequency of pressure-drop oscillation is much smaller and hence it is easy
to distinguish it from density-wave oscillations. However, with a relatively
stiff system, the frequency of PDO can be comparable to DWO making it difficult
to distinguish between the two. Very long test sections may have sufficient internal
compressibility to initiate pressure drop oscillations. Like Ledinegg
instability, there is a danger of the occurrence of CHF during pressure drop
oscillations. Also inlet throttling (between the surge tank and the boiling
channel) is found to stabilize PDO just as Ledinegg instability.


## 3.4.4. Instability in Natural Circulation BWRs

The flow velocity in natural
circulation BWRs is usually smaller than that of forced circulation BWR.
Besides, due to the presence of tall risers in natural circulation BWRs, the
frequency of density-wave oscillation can be much lower due to longer traveling
period of the two-phase mixture in the risers. The effects of negative void
reactivity feedback are found to stabilize the very low frequency type I
instabilities [43, 44]. But
it may stabilize or destabilize type II instabilities depending on its time
period [44].In case of a natural circulation
BWR, the existence of a tall riser or chimney over the core plays a different
role in inducing the instability. Series of experiments carried out by van Bragt et al. [36] in the Dodewaard natural circulation BWR in The Netherlands
showed that instabilities could occur at low as well as at high powers in this
reactor. From measured decay ratio, it was evident that at very low power there
is a trend of increase in decay ratio and similar results are seen at higher
power also. The low-power oscillations are induced by the type I density-wave
instabilities and high power oscillations are induced by the type II
density-wave instabilities. type I and type II instabilities have been predicted
to occur in the Indian AHWR which is a natural circulation pressure tube type
BWR, away from the nominal operating condition [44]. It may be
noted that in case of forced circulation BWRs, instabilities observed under
natural circulation conditions are due to pump trip transients when the core
exit quality is high due to low flow and high power. Hence these are induced by
the type II density-wave instabilities only.


## 4. Concluding Remarks

Several decades have been spent on the study of flow
instabilities in boiling two-phase natural circulation systems. A large number
of numerical and experimental investigations in this field have been carried
out in the past. Many numerical codes in time domain as well as in frequency
domain have been developed using various mathematical modelling techniques to
simulate the flow instabilities occurring in the NCSs. It is felt to have a
review and summarize the state-of-the-art research carried out in this area,
which would be quite useful to design and safety of current and future light
water reactors with natural circulation core cooling. With that purpose, a
review of flow instabilities in boiling natural circulation systems has been
carried out. An attempt has been made to classify the instabilities occurring
in natural circulation systems similar to that in forced convection boiling
systems. It was found that the instabilities can be classified based on the mechanism
of their occurrence into broadly two groups such as static and dynamic
instabilities. The analytical tools based on the above mechanisms predicts the
stability threshold and characteristics of instabilities reasonably well. Other
classifications are in fact subcategories of a particular class of the
instabilities covered under this classification. While classifying
instabilities of NCSs, a need was felt to consider the instabilities associated
with single-phase condition, boiling inception, and two-phase condition
separately as a natural circulation system progresses through all these stages
before reaching the fully developed two-phase circulation. Most instabilities
observed in forced circulation systems are observable in natural circulation
systems. However, natural circulation systems are more unstable due to the
regenerative feedback inherent in the mechanism causing the flow. While most of
the work has been devoted to generate data for steady state and threshold of
flow instabilities in NCSs, however, it was felt that more investigations on
characteristics of these flow instabilities must be conducted in future, which
is not understood enough. Moreover, it is found that these instabilities do not
occur in isolated manner in NCSs, however, many times, they occur together
which are known as compound instabilities. Different models of two-phase flow
have been used for modelling these flow instabilities, which range from the
simplest HEM to more rigorous two-fluid model. While the HEM is found to model
the threshold of instability of density-wave type in NCS with reasonably
accuracy, however, there are concerns for using this model since the drift
velocity and void distribution parameters which are indications of slip between
the phases, are found to affect the stability threshold. Computer codes
developed considering more rigorous models such as RELAP5 are yet to be
established for their applicability for simulation of stability in boiling NCS.
In view of this, more research needs to be conducted to explore the capability
of existing mathematical models for prediction of these instabilities in NCSs
in future.



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*Source: 573192-2008-06-16.xml*