P.K. Vijayan
Reactor Engineering Division,
Bhabha Atomic Research Centre, Mumbai, India
IAEA Course on Natural Circulation in Water-Cooled Nuclear Power
Plants, ICTP, Trieste, Italy, 25-29 June, 2007 (Lecture : T-6)

INTRODUCTION
- CLASSIFICATION OF INSTABILITIES
- STABILITY OF SINGLE-PHASE NC
- INSTABILITY DUE TO BOILING INCEPTION
- TWO-PHASE NC INSTABILITY
- CONCLUDING REMARKS

Instabilities are common to both FC and NC systems
NCSs are more unstable. A regenerative feedback is inherent in the mechanism causing NC
5.0
Any change in the driving force will affect the flow which in turn modifies the driving force leading to a transient oscillatory state even in cases which results in an eventual steady state
2.5
0.0
-2.5
-5.0
500 1000
Time - s
1500 2000
Both single-phase and two-phase NCSs exhibit instability whereas only two-phase FCSs are known to exhibit instability
Even two-phase NCSs are more unstable than two-phase FCSs

Following a perturbation if a thermal-hydraulic system returns to its original steady state, then the system is considered to be stable
If the system continues to oscillate with the same amplitude, then it is considered as neutrally stable
If the system oscillates with increasing amplitude or shift to a new steady state, then it is considered as unstable stable
Neutrally stable unstable
Time - s

3
2
1
3
Orientation : HHHC
 P across the bottom horizontal section
2
Orientation : HHHC

P across the bottom horizontal section
180 W
280 W
0.2
nonlinearities leading to limit cycle oscillations
0.1
1
0
-1
80 W
Time series
System nonlinearities
0.0
-0.1
-0.2
-3 -2 -1 0
Limit Cycle,
0
Phase space (plot or trajectory), orbit
-1 60 W
1 2 3
-2
- Void fraction can only vary between zero and unity
-3
0
- Neutron flux cannot be negative
3000 6000 9000 12000 15000 18000
Time - s
(a)
3
2
Orientation : HHHC

P across the bottom horizontal section
-3
0
3
2
180 W 300 W
3000 6000 9000 12000 15000 18000 21000
Time - s
(b)
Orientation :HHHC

P across the bottom horizontal section
1 1
0
-1 350 W
0
-1
60 W 120 W
60 W
240 W 480 W
-2 -2
-3
0 4000 8000 12000 16000 20000 24000 28000
Time - s
(c)
-3
0 5000 10000 15000
Time - s
(d)
20000 25000

Sustained flow oscillations may cause forced mechanical vibration of components
Premature CHF occurrence can be induced by flow oscillations
Induce undesirable secondary effects like power oscillations in a BWR
Instability can disturb control systems and pause operational problems in nuclear reactors.

Several kinds of instabilities are observed in NCSs excited by different mechanisms
A fundamental cause of all instabilities is due to the existence of competing multiple solutions so that the system is unable to settle down to any of them permanently
However, differences exist in the transport mechanism, oscillatory mode, phase shift, the nature of the unstable threshold, and its prediction methods
Loop geometry and induced secondary phenomena also affect the instability

Instabilities are classified according to various bases
- analysis method
- propagation method
- nature of the oscillations
loop geometry,
- disturbances or perturbations

Analysis method (or Governing equations used)
(a) Pure (or fundamental) static instability
(b) Compound static instability
(c) Pure dynamic instability
(d) Compound dynamic instability
Pure static instability
The occurrence of multiple solutions and the instability threshold itself can be predicted from the steady state equations governing the process
Examples are Ledinegg instability and the instability induced by the occurrence of CHF

Compound static instability
In some cases of multiple steady state solutions, the instability threshold cannot be predicted from the steady state laws alone or the predicted threshold is modified by other effects.
In this case, the cause of the instability lies in the steady state laws, but feedback effects are important in predicting the threshold
Typical examples are the instability due to flow pattern transition, flashing and geysering

Pure Dynamic instability
For many NCSs neither the cause nor the threshold of instability can be predicted from the steady state laws as inertia and feedback effects are important
In this case, there are no multiple steady states, but the multiple solutions appear during the transient
The full transient governing equations are required for explaining the cause and predicting the threshold of instability
Typical example is the density wave oscillations commonly observed in NCSs

Compound dynamic instability
In many oscillatory conditions, secondary phenomena gets excited and it modifies significantly the characteristics and the threshold of pure dynamic instability
In such cases even the prediction of the instability threshold requires consideration of the secondary effect
Typical examples are
- neutronics responding to the void fluctuations
- primary fluid dynamics affecting the SG instability

Propagation method
This classification is restricted to only dynamic instabilities.
Dynamic instabilities involve propagation or transport of disturbances. The disturbances can be transported by two kinds of waves
- Pressure or acoustic waves
Density waves (Stenning and Veziroglu)
In two-phase flow, both waves are present, however, their velocities differ by one or two orders of magnitude allowing us to distinguish between the two
DWI is the most commonly observed instability. The frequency of DWI is of the order of 1 Hz in 2-
 flow

Based on the nature of oscillations - Flow excursions,
- Pressure drop oscillations,
- Power oscillations,
- Temperature excursions
Based on the periodicity - Periodic oscillations
- chaotic oscillations
Based on the oscillatory mode
– Fundamental mode
- Higher harmonics
Based on the phase lag - in-phase oscillations,
- out-of-phase oscillations
- dual oscillations
Based on flow direction - Unidirectional oscillations
- Bi-directional oscillations
- Chaotic switching

Based on the loop geometry
- Open U-tube oscillations
- Pressure drop oscillations
- Parallel channel oscillations
C C
F
0
Time
H
Based on the disturbances
Certain two-phase flow phenomena cause a major disturbance and induce or modify the instability in NCSs
- Boiling inception
- Flashing and geysering
- Flow pattern transition
- Occurrence of CHF

Closure
The classification based on the analysis method is the most widely accepted one and covers all observed instabilities
All other classifications addresses only a subset of the instabilities
It does not differentiate between NC and FC systems
Most instabilities observed in FC systems are observed in NCSs
However, certain instabilities associated with NCSs are not observed in FCSs – Single-phase instability and the instability associated with flow direction
Hence a specific discussion of instabilities for single-phase and twophase NCSs is considered useful

Single-phase NC instabilities can be characterised into three different types
- Static instabilities associated with multiple steady states
- Dynamic instabilities
- Compound dynamic instabilities
Multiple steady states in the same flow direction
Pure Static instability
Multiple steady states with differing flow directions
Traditionally pure static instability is associated with multiple steady states in the same flow direction
Theoretically certain single-phase NCSs show multiple steady states in the same flow direction. So far no experimental confirmation exists

Multiple Steady states with differing flow direction
Unlike FC, certain NC systems can exhibit steady flow in both clockwise and anticlockwise directions coolant
Steady
Clockwise flow (Q
2
=0)
Q
2
=0 Q
2
>Q c
Anticlockwise flow with Q
2
>Q c
Q
1
Q
1
- Instability with oscillation growth will set in as Q
2 nears Q c
- The instability will be terminated by flow reversal
Subsequently even if we reduce power below Q c
, instability will not be observed. Also flow will continue in the reverse direction
- The instability cannot be predicted from steady state laws alone

Instability associated with flow reversal in single phase NCS
1415
0.5
2100
26.9
C
800
1180
0.0
-0.5
Orientation : HHVC
1x10
23
L t
=7.19m, L t
/D=267.29, p=0.316, b=0.25
HHVC (ANTICLOCKWISE)
HHVC (CLOCKWISE) stable
Gr m
10
13 u nstable
Clockwise flow
Anticlockwise flow
410
385
H
620
-1.0
9750 10000 10750 11000
10
3
0 5
St m
St m
10
10250 10500
Time - s
620
410
Even though the flow initiates in the anticlockwise direction, steady flow was never obtained
Stability analysis showed that no steady flow exists with upward flow in the cooler for this loop geometry
Upward flow in the cooler or downward flow in the heater can lead to similar behaviour
Similar behaviour is observed in a NCS with throughflow

Figure-of-eight loop with throughflow
Multiple steady states with feed in header 2 and bleed from header 3 compared with test data

Instability near the flow reversal threshold: With small throughflow, the initial NC flow direction is preserved. As the throughflow is large enough, it reverses the NC flow.
Near the flow reversal threshold an instability is observed
The amplitude of the oscillations increase as the flow reversal threshold approaches
F
W+F
W
Typical unstable behaviour near the flow reversal threshold.
F F F
The oscillation amplitude is larger in the low flow branch
Experiments showed that the flow reversal threshold depends on the operating procedure (hysteresis)

Q
Outlet header h l
Q
1
W
1
2
Q
2
W
Q
3
W
3
Q n
W n downcomer
Q d
=0
W d h m h s
Parallel vertical inverted U-tubes relevant to SGs
Inlet header
Vertical parallel channel system relevant to the RPV of BWRs, PWRs, etc. heater h b h m h t cooler
Q t
Q m
Q b
Unequally heated parallel horizontal channels having unequal elevations relevant to PHWRs
Existence of multiple steady states was first explored by Chato (1963)

Unequally heated parallel channel NC systems exhibit an instability associated with flow reversal due to the existence of mutually competing driving forces coolant outlet header
W
There is a driving force between the downcomer and each heated channel promoting upward flow through the heated channels
H
Q
1
W
1
Q
2
W
2
W inlet header
Vertical parallel channel system
There is another driving force between two unequally heated channels (due to the difference in densities) favouring downward flow in the low power channel
The actual flow direction is decided by the greater of the two driving forces

For an unheated channel, upward flow is unstable and stable downward flow can prevail.
Keeping Q
2 constant, if we increase Q
1 channel 1 occurs if Q
1
> Q c then flow reversal in
(Chato (1963)) coolant
H outlet header
W
Q
1
=0
W
1
Q
2
W
2
Keeping Q
2 constant, if we reduce
Q
1
, then the upward flowing channel 1 flow will reverse if
Q
1
<Q c
(Linzer and Walter (2003))
Down comer
The flow reversal threshold is a function of the power of channel 2.
W inlet header
(a) Vertical parallel channel system
Channel flow reversal is undesirable as it can lead to instabilities in two-phase systems

It appears that channel flow reversal can be avoided in a system of vertical parallel channels if all the channels are equally powered
However, with vertical inverted U-tubes (as in SGs) and horizontal channels as in PHWRs, mutually competing driving forces exist even if all the channels are equally powered due to the differences in elevation.
Single-phase NC with reverse flow in some of the longest U-tubes are observed in several integral test loops
Steady state with one of the channels flowing in the reverse direction was observed during thermosyphon tests in NAPS

Metastable
Regime
E
1
0.1
F
A
0.01
B
Rate of change of power: 0.2%/s
Decreasing power
Increasing power
Ch-2 Ch-3
C
D
1E-3
-1.00
-0.75
-0.50
-0.25
0.00
0.25
Mass Flow Rate Ratio (W
1
/W
2
)
0.50
0.75
1.00
Hysteresis curve for single-phase parallel channel NCS computed with RELAP5/MOD3.2

Initial steady state was achieved with equal power to Ch-1,2 & 3 and Ch-4 unheated.

Ch- 1, 2 & 3 are with upflow and Ch-4 with downflow. Power in Ch-1 was decreased keeping other channels’ power constant. The
BLACK curve starts at A. After reaching a power ratio corresponds to B, the flow in Ch-1 reverses from upflow (+ve) to downflow (-ve) and the curve jumps from state B to state C.

For the second case, initial steady state was achieved with equal power to Ch-2 & 3 and Ch-1 & 4 unheated. i.e. Ch-2 & 3 with upflow and Ch-1 & 4 with downflow.

The BLUE curve starts at D. Power in Ch-1 was increased. After reaching a power ratio corresponds to
E, the flow in Ch-1 reverses from downflow (-ve) to upflow (+ve) and the curve jumps from state E to state F.

Essentially the dynamic instability in single-phase systems is also
DWI although it was referred to as DWI only recently (Lahey, Jr)
The frequency of DWI in single phase NC (0.0015 – 0.005 Hz) is significantly lower than that in two-phase systems (1-10Hz) due to the low velocities in single-phase NC
Two types of dynamic instabilities are observed in single-phase NCSs
- Single channel system instabilities
- Parallel channel instabilities

C
H
Flow
Time
Keller (1966)
System Instabilities
1415
Expansion tank cooler
800
305
2200
C
H
3
2
1
0 Flow
-1
-2
-3
25
St m
=7.0, Gr m
=1x10 13 t
75
Time
100
410
Heater
620
385
410 620
All these oscillatory modes can be observed in rectangular loops for different ranges of power

Oscillation growth as a mechanism of single-phase instability proposed by Welander (1967)
Oscillation growth is the usual route to instability from steady state
- Compound static instability also shows oscillation growth, but the instability is terminated by flow reversal
Parallel Channel Instability
Parallel channels also exhibit a dynamic instability mode in single-phase conditions
The instability occurs due to the redistribution of flow
Both in-phase and out-of-phase oscillations are predicted

Pure static instability
Compound static instability
Dynamic instability
Compound Dynamic instability
Pure static instability
Flow excursion or excursive (Ledinegg) instability
Boiling Crisis
Ledinegg instability
Involves sudden change of flow rate to a lower value
The new flow rate may induce CHF
Occurrence of multiple steady states is the fundamental cause of the instability

It occurs during the negative sloping region of the
 p – W characteristic; d(
 p)/dW < 0 is the criterion for the Ledinegg instability
Steam
Two-phase system characteristic a b c
Driving pressure differential
Flow rate riser heater
Feed downcomer
Point ‘b’ satisfies this criterion and is therefore unstable
The instability can be avoided by inlet throttling in FCSs
Inlet throttling may not work as effectively as in FCSs due to the reduction in flow caused by it

Following the occurrence of critical heat flux, a regime of transition boiling may be observed as in pool boiling
Transition boiling
Natural convection
Film boiling
Nucleate boiling
During transition boiling a film of vapour prevents the liquid from coming in direct contact with the heating surface resulting in a steep rise in temperature
T s
- T sat
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 in wall temperature followed by an almost simultaneous occurrence of flow oscillations

Two-phase NCSs also show an instability associated with flow direction as in single-phase NC
All instabilities associated with boiling inception, flashing and geysering, etc. can also be considered as part of two-phase NC instability.
Flow pattern transition instability also belong to this category
Two-phase parallel channel systems exhibit
- Multiple steady states in the same flow direction
- An instability associated with flow reversal as in single-phase flow

Two-phase systems exhibit different flow patterns with differences in pressure drop characteristics which is the fundamental cause of the instability
Bubbly-slug flow has a higher pressure drop compared to annular flow
A system operating near the slug to annular transition boundary is susceptible to this instability
Theoretical analysis of the phenomena is hampered by the unavailability of validated flow pattern transition criteria and flow pattern specific pressure drop correlations
The instability is found to be similar to Ledinegg instability, but occurs at higher power

Fukuda-Kobori
1000
800
Lower threshold
Stable two-phase NC
Type-II instability
600
Single-phase region

P
2035 W
100
80
60
400
Pressure
Upper
200 threshold
1290
W
Type-I instability
0
0 4000 8000 12000
Time - s
Fig. 1a: Typical low power and high power unstable zones for two-phase NC flow
40
20
0
Generally two unstable regions are observed for DWI
The low power unstable region is called the type I instability
The high power instability is called the type II instability
The number of unstable or stable zones depends on the shape of the stability map
In view of the occurrence of islands of instability, the unstable zones can be more than 2.

Stable single-phase flow can become unstable with the inception of boiling
1.0
0.8
0.6
1 ba r
15 b ar
70
ba r
The instability due to boiling inception disappears at high pressure
0.4
0.2
0.0
0.00
0.05
170
ba r
220 b ar
221.2 b ar
0.15
0.20
Large variation of void fraction with small changes in quality at low pressures is the main cause for the instability
0.10
Quality
Flashing and Geysering are also boiling inception instability
This is essentially the Type-I instability discussed by Fukuda and Kobori.

Occurs in NCSs with tall risers. The rising hot liquid experiences static pressure decrease and may reach its saturation value leading to flashing in low pressure systems
The increased buoyancy force increases the flow which in turn reduces exit enthalpy and may even suppress flashing causing reduction in buoyancy force and flow. Reduced flow leads to larger exit enthalpy leading to the repetition of the process.
The necessary condition for flashing is that the fluid temperature at the riser inlet is greater than the saturation temperature at the exit
The instability is observed only in low pressure systems
Thermodynamic equilibrium conditions prevail during flashing

Generally observed in systems with tall risers. Geysering is expected during subcooled boiling taking place at the exit of the heater
As the bubbles move up the riser, bubble growth (due to static pressure decrease) as well as condensation can take place.
Sudden condensation results in depressurisation causing the liquid water to rush to the space vacated by the slug bubble leading to a large increase in the flow rate reducing the driving force
Geysering is a nonequilibrium phenomenon unlike flashing
Both Geysering and flashing instabilities are observed in low pressure systems only

Regenerative feedback and time delay are important for dynamic instability
DWI is the most commonly observed dynamic instability in NCSs and the mechanism causing the instability has already been discussed earlier
A number of geometric and operating parameters affect the instability in addition to certain boundary conditions
Riser height, orificing, length and diameter of source, sink and connecting pipes are the important geometric parameters
Pressure, inlet subcooling, power and its distribution are the important operating parameters
Boundary conditions of interest are wall heat transfer coefficient, heat storage in walls, wall heat losses, constant pressure drop, etc.

If only one instability mechanism is at work, it is said to be fundamental or pure instability. Instability is compound if more than one mechanisms interact in the process and cannot be studied separately
-Thermal oscillations
-Parallel channel instability
-Pressure drop oscillations
-BWR instability
-SG instability

The variable heat transfer coefficient leads to a variable thermal response of the heated wall that gets coupled with the DWO
Thermal oscillations are a regular feature of post dry out heat transfer heat transfer in steam-water mixtures at high pressures
The steep variation in heat transfer coefficient gets coupled with DWO
The dryout or CHF point shift upstream of or downstream during thermal oscillations
The large variation in heat transfer coefficient results in significant variation in heat transfer rate even if the wall heat generation is constant

Interaction of parallel channels with DWO can give rise to interesting behaviours as in single-phase NCS
-In-phase oscillations (system instability)
- Out-of-phase oscillations
-Dual oscillations (overlapping region of IPO and OPO)
In-phase oscillation is a system characteristic and parallel channels may not play a role in this
Out-of-phase oscillation is characteristic of parallel channels. The phase shift depends on the number of channels
- 2 channels : 180 0 ; 3 channels : 120 0 ; n-channels :2

/n
However, in a system of n-channels, all parallel channels need not take part in the instability. Depending on the number of channels taking part the phase shift can be anywhere between 2

/n to

Mechanism of PCI is similar in single-phase and two-phase systems

This is associated with operation in the negative sloping region of
 p-w curve.
Caused by the interaction of a compressible volume at the inlet of the heated section with the pump characteristics
Usually observed in FCSs. DWO oscillation occurs at flow rates lower than the flow rate at which PDO is observed
The frequency of PDO is much smaller than DWO .
Very long test sections may have sufficient internal compressibility to cause PDO
Like Ledinegg instability there is a danger of CHF occurrence during PDO
Inlet throttling can stabilize PDO as in the case of Ledinegg instability

The flow velocity in NBWRs is usually much smaller than
FC BWRs
The presence of tall risers causes the oscillation frequency to be much smaller
The only NCR whose stability has been extensively studied is the
Dodewaard reactor.
The negative void reactivity stabilizes type I instability. But it may stabilize or destabilize type II instability
Pump trip transients in FC BWRs lead to type II instability

Various bases used for classification of instabilities have been reviewed. The most widely accepted classification is based on the method of analysis used in identifying the stability threshold.
While classifying NC instabilities, it was convenient to consider the instabilities associated with single-phase and two-phase NC separately
Natural circulation systems are more unstable due to the regenerative feedback inherent in the mechanism causing the flow.
Besides the instability in single-phase systems, natural circulation loops also exhibit an instability associated with flow reversal in contrast to forced circulation systems.

Density wave instability
DWI is most commonly observed in NCSs. The frequency of DWI is of the order of 1 Hz in 2-
 flow
Due to the importance of void fraction and its effect on flow, the instability is often referred to as ‘flow-void feedback instability’ in two-phase NCSs
Since transportation time delays are crucial to the instability, it is also known as ‘time delay oscillations’
In single-phase, near critical and supercritical fluids, the instability is also known as ‘thermally induced oscillations’

Boiling inception modifies single-phase instability
Boiling inception can induce instabilities in a stable single-phase
NCS
Occurrence of single-phase conditions during part of the oscillation cycle is a characteristic feature of this instability

With power increase boiling inception occurs during the low flow part of the oscillation cycle. Instability continues with flow switching between singlephase and two-phase regimes.
Several flow regimes are observed as shown. The change in power required for attaining the condition with two-phase flow for the entire oscillation cycle can be very significant
0.2
Orientation : HHHC
Plot for 21950 seconds after neglecting initial transients
0.1
0.2
0.1
Orientation : HHHC
Plotted from a time series of 7900 seconds after neglecting initial transients
0.0
0.0
-0.1
-0.1
-0.2
-3 -2 -1 0 1

P - mm of water column
2
(a) Single-phase instability
0.2
Orientation : HHHC
Plotted from a time series of 19500 seconds after neglecting initial transients
0.1
0.0
3
-0.2
-3 -2 -1 0 1

P - mm of water column
2
(b) Instability with sporadic boiling
3
0.3
Orientation : HHHC
Plotted from a time series of 10000 seconds after neglecting initial transients
0.2
0.1
0.0
-0.1
-0.1
-0.2
-0.2
-3 -2 -1 0 1

P - mm of water column
2 3 c) Instability with boiling once in every cycle
-0.3
-3 -2 -1 0 1

P - mm of water column
2
(d) Instability with boiling twice in every cycle
3
Fig. 13: Effect of subcooled boiling on single-phase instability