# Two-phase Fluid Flow

Twophase flow is the interactive flow of two distinct phases with common interfaces in, say, a conduit. The most common being the simultaneous flow of steam and liquid water (as encountered in steam generators and condensers).

By definition, multiphase flow is the interactive flow of two or more distinct phases with common interfaces in, say, a conduit. Each phase, representing a volume fraction (or mass fraction) of solid, liquid or gaseous matter, has its own properties, velocity, and temperature.

A multiphase flow can be simultaneous flow of:

• Materials with different states or phases (e.g. water-steam mixture).
• Materials with different chemical properties but in the same state or phase (e.g. oil droplets in water).

There are many combinations in industrial processes, but the most common being the simultaneous flow of steam and liquid water (as encountered in steam generators and condensers). In reactor engineering a great deal of study has been performed on the nature of two-phase flow in case of a loss-of-coolant accident (LOCA), which is an accident of importance in reactor safety and in all thermal-hydraulic analyses (DNBR analyses).

## Characteristics of Two-phase Fluid Flow

All two-phase flow problems have features which are characteristically different from those found in single-phase problems.

• In the case of steam and liquid water the density of the two phases differs by a factor of about 1000. Therefore the influence of gravitational body force on multiphase flows is of much greater importance than in the case of single-phase flows.
• The sound speed changes dramatically for materials undergoing phase change, and can be orders of magnitude different. This significantly influences a flow through an orifice.
• The relative concentration of different phases is usually a dependent parameter of great importance in multiphase flows, while it is a parameter of no consequence in single-phase flows.
• The change of phase means flow-induced pressure drops can cause further phase-change (e.g. water can evaporate through an orifice) increasing the relative volume of the gaseous, compressible medium and increasing efflux velocities, unlike single-phase incompressible flow where decreasing of an orifice would decrease efflux velocities.
• The spatial distribution of the various phases in the flow channel strongly affects the flow behavior.
• There are many types of instabilities in multiphase flow.

## Basic Parameters of Two‐phase Fluid Flow

In this section we will consider the simultaneous flow of gas (or vapor) and liquid water (as encountered in steam generators and condensers) in concurrent flow through a duct with cross-sectional area A. The subscripts “v” and “ℓ” indicate the vapor and liquid phase, respectively. Fundamental parameters that characterize this flow are:

Void Fraction
The void fraction, α, is one of the most important parameters used to characterize two-phase fluid flow, especially the gas-liquid flow.
Various geometric definitions are used for specifying this parameter. The void fraction in a two-phase fluid flow may be defined as:

1. The fraction of the channel volume that is occupied by the gas phase. This void fraction is known as the volumetric void fraction.
2. The fraction of the channel cross-sectional area that is occupied by the gas phase. This void fraction is known as the cross-sectional void fraction.
3. The local void fraction refers to that at a one single point or very small volume. Therefore it takes the values of 1 or 0.

For further purposes, we will assume a void fraction to be the fraction of the channel cross-sectional area that is occupied by the gas phase (i.e. cross-sectional void fraction) defined as:

The void fraction is of importance is the two-phase flow, because it influences key physical parameters, such as viscosity, pressure drop and heat transfer.

Static Quality
In general, vapour quality is the mass fraction of vapour in a saturated mixture. Saturated vapour has a quality of 100%, and saturated liquid has a quality of 0%.
The static quality in a two-phase fluid flow is defined as:
Flow Quality
In two-phase fluid flow in is convenient to use the flow quality instead of the static quality. The flow quality in a two-phase fluid flow is defined as:
Mixture Density
In two-phase fluid flow in is convenient to use the mixture density. The mixture density of the two-phase flow which is used to calculate pressure drop. The mixture density in a two-phase fluid flow is defined as:
Slip Ratio - Velocity ratio
In two-phase fluid flow it is convenient to use the slip ratio. The slip ratio (or velocity ratio) in two-phase flow is defined as the ratio of the velocity of the vapor phase to the velocity of the liquid phase. The slip ratio in a two-phase fluid flow is defined as:

In the homogeneous equilibrium model (HEM) of two-phase flow, the slip ratio is by definition assumed to be unity (there is no slip). However, most industrial two-phase flows have different velocity of the gas and liquid phases, these can differ significantly. The models that account for the existence of the slip are called separated flow models.

The relations between x, α, and S can be deducted and the result is:

The reason for defining the void fraction and the slip ratio is that they also make it possible to calculate the pressure drop of the two-phase flow. Several correlations for calculation the slip ratio, S, and the void fraction are presented in literature. The following correlations are given in order of increasing accuracy.

Superficial velocity
Superficial velocity is a hypothetical flow velocity calculated as if the given phase or fluid were the only one flowing or present in a given cross sectional area. The velocity of the given phase is calculated as if the second phase was ignored.

In engineering of multiphase flows and flows in porous media, superficial velocity is commonly used, because it is the value which is unambiguous, while real velocity is often spatially dependent and subject to many assumptions.

Superficial velocity can be expressed as:

For better understanding, let us consider pipe with a 0.1 m2 cross-section of  flow area. Assume that the flow rate is 1 m3/s. For single-phase fluid flow the superficial velocity will be equal to real fluid velocity and that will be 10 m/s.
For two-phase fluid flow (e.g. vapor-liquid flow) the situation will be different. Assuming the slip ratio is unity, both phases taken separately, will have superficial velocities of 5 m/s. The resulting real velocity will be then equal to 10 m/s. If the two phases will have different velocities (with slip), the situation will be more complicated.

## Flow Patterns – Two-phase Flow

One of the most challenging aspects of dealing with two-phase flow or multi-phase flow is the fact that it can take many different forms. Spatial distributions and velocities of the liquid and vapor phases in the flow channel is very important aspect in many engineering branches. Pressure drops and also heat transfer coefficients strongly depends on the local flow structure and thus it is of importance in engineering of nuclear reactors. The observed flow structures are defined as two-phase flow patterns and these have particular identifying characteristics. These different flow patterns have been categorized according to the direction of flow relative to gravitational acceleration.

• Flow patterns in vertical tubes
• Flow patterns in horizontal tubes

The main flow regimes in vertical tubes are shown in the table. It must be noted values of flow quality and flow rate are dependent on the fluid and pressure. In horizontal tubes, there can also be stratified flow (especially at low flow rates), at which the two phases separate under the effect of gravity.

For a constant liquid flow rate, the vapor/gas phase tends to be distributed as small bubbles at low vapor flow rates. Increasing void fraction causes agglomeration of bubbles into larger plugs and slugs. Further agglomeration of slugs, cause by further increasing void fraction causes separation of the phases into annular patterns wherein liquid concentrates at the channel wall and vapor flows in the central core of the vertical channel.

For horizontal channel, gravitational force tends to drain the liquid annulus toward the bottom of the channel, resulting in stratified flow. The gravitational force acting on the liquid phase can be overcome by kinetic forces at high flow rates, causing stratified flows to revert to annular flows. At very high flow rates, the annular film is thinned by the shear of the vapor core and all the liquid is entrained as droplets in the vapor phase. This flow regime is usually known as the mist flow.

See also: Engineering Data Book III, Thome, J.R., Wolverine Tube Inc, 2004.

## Flow Patterns – Vertical Tubes

Bubbly flow
In the bubbly flow, the liquid flow rate is high enough to break up the gas into bubbles, but it is not high enough to cause the bubbles to become mixed well within the liquid phase. The bubbles vary widely in size and shape, but most commonly they are nearly spherical and are much smaller than the diameter of the tube.
Slug flow
Increasing void fraction causes agglomeration of bubbles into larger plugs and slugs. These slugs are similar in dimension to the tube diameter. These slugs travel at a speed that is a substantial fraction of the gas velocity and occur intermittently. Since these large gas slugs are separated from one another by slugs of liquid, they cause large pressure and liquid flow rate fluctuations. In some cases, a downward flow can be observed near the tube wall, even though the net flow of fluid is upward. This is caused by the gravitational force.
Churn flow
Churn flow, also referred to as froth flow is a highly disturbed flow of two-phase fluid flow. Increasing velocity of a slug flow causes that the structure of the flow becomes unstable. The churn flow is characterized by the presence of a very thick and unstable liquid film, with the liquid often oscillating up and down. Due to its nearly chaotic properties, it is one of the least understood of gas-liquid flow regimes.

A typical example of churn flow is boiling flow in nuclear reactors during accidents. Especially for many accident scenarios, boiling may lead to high void fraction including churn-turbulent flow. Its flow structure and induced pressure changes may have a strong impact on the safety.

Annular flow
Annular flow is a flow regime of two-phase gas-liquid flow. It is characterized by the presence of a liquid film flowing on the channel wall (forming an annular ring of liquid) while the gas flows as a continuous phase up in the center of the tube. The flow core can contain entrained liquid droplets. The velocity of the gas core is very large and it is large enough to cause high frequency waves and ripples at the interface. This flow regime is particularly stable and it is desired flow regime for high-velocity, high-quality two-phase fluid flow.

Both adiabatic annular flows (without heat exchange) and diabatic annular flows (with heat exchange) occur in industrial applications:

In the case of BWR, in which a phase transition (evaporation) occurs, detailed knowledge of this flow regime is of high importance. At given combinations of flow rate through a channel, pressure, flow quality, and linear heat rate, the wall liquid film may exhaust and the wall may be dried out. This phenomenon is usually known as “dryout”. Dryout is accompanied with a rapid rise in wall temperature and is of importance in safety of BWRs.

Mist flow
Mist flow is a flow regime of two-phase gas-liquid flow. It occurs at very high flow rates and very high flow quality. This conditions causes, that liquid film flowing on the channel wall is thinned by the shear of the gas core on the interface until it becomes unstable and is destroyed. The flow core in the mist flow entrains all the liquid as droplets in the gas phase. Droplets may wet the tube wall, but this occurs intermittently and only locally. In the heated channel, the presence of mist flow regime is accompanied with significantly higher wall temperatures and high fluctuation of wall temperatures.

## Flow Patterns – Horizontal Tubes

Bubbly flow
In contrast to the bubbly flow in vertical channel, the bubbly flow in horizontal channel is strongly influenced by gravitational force. Due to the buoyancy, bubbles are dispersed in the liquid with a higher concentration in the upper half of the channel. This regime typically occurs at higher flow rates, because at lower flow rates the gravitational force tends to drain the liquid annulus toward the bottom of the channel, resulting in stratified flow.
Stratified flow
In two-phase fluid flow the gravitational force plays very important role, because the fluid with lower density (e.g. gas) is always above the fluid with higher density. Stratified flows are very common in the nature, for example in the ocean and in the atmosphere. In the internal flows the stratified flow occurs at low liquid and gas velocities. As the velocity of the gas increases, the horizontal interface becomes more disturbed and waves may be formed. This flow regime is usual known as the stratified-wavy flow.
Plug flow and Slug flow
Further increasing in the gas velocity causes, that waves reach the top of the tube. Whether the flow will be plug or slug flow, it is dependent especially on the void fraction that causes agglomeration of bubbles into larger plugs and slugs. In the plug flow the diameters of the bubbles are smaller than the tube. Slugs are similar in dimension to the tube diameter. The slugs travel at a speed that is a substantial fraction of the gas velocity and occur intermittently. Since these large gas slugs are separated from one another by slugs of liquid, they cause large pressure and liquid flow rate fluctuations.
Annular flow
Similar to vertical flow, at larger gas flow velocities, the liquid forms a continuous annular film on the channel wall. The horizontal annular flow is characterized by the presence of a thicker liquid film flowing on the bottom of channel wall. The gas flows as a continuous phase in the center of the tube. The flow core can contain entrained liquid droplets. The velocity of the gas core is very large and it is large enough to cause high frequency waves and ripples at the interface. This flow regime is particularly stable and it is desired flow regime for high-velocity, high-quality two-phase fluid flow.
Mist flow
Mist flow is a flow regime of two-phase gas-liquid flow. It occurs at very high flow rates and very high flow quality. This conditions causes, that liquid film flowing on the channel wall is thinned by the shear of the gas core on the interface until it becomes unstable and is destroyed. The flow core in the mist flow entrains all the liquid as droplets in the gas phase. Droplets may wet the tube wall, but this occurs intermittently and only locally. In the heated channel, the presence of mist flow regime is accompanied with significantly higher wall temperatures and high fluctuation of wall temperatures.

## Flow patterns during evaporation

The previous section describes various flow patterns and shortly describes their behavior. These flow patterns were considered to be at constant void fraction and at constant superficial velocities. But there are many industrial applications that have to consider a variable void fraction and variable superficial velocities. In nuclear industry, we have to deal with flow patterns during evaporation (i.e. during changes in the void fraction).

Detailed knowledge of phase changes and the behavior of the flow during the phase change is one of the most important considerations in the design of a nuclear reactor, especially in following applications:

• BWR – Boiling Water Reactors
• A boiling water reactor is cooled and moderated by water like a PWR, but at a lower pressure (7MPa), which allows the water to boil inside the pressure vessel producing the steam that runs the turbines. Evaporation therefore occurs directly in fuel channels. Therefore BWRs are the best example for this area, because evaporation of coolant occurs at normal operation and it is very desired phenomenon.
• In BWRs there is a phenomenon, that is of the highest importance in reactor safety. This phenomenon is known as the “dryout” and it is directly associated with changes in flow pattern during evaporation. At normal the fuel surface is effectively cooled by boiling coolant. However when the heat flux exceeds a critical value (CHF – critical heat flux) the flow pattern may reach the dryout conditions (thin film of liquid disappears). The heat transfer from the fuel surface into the coolant is deteriorated, with the result of a drastically increased fuel surface temperature.
• PWR – Pressurized Water Reactors
• In PWRs at normal operation the flow is considered to be single-phase. But a great deal of study has been performed on the nature of two-phase flow in case of transients and accidents (such as the loss-of-coolant accident – LOCA or trip of RCPs), which are of importance in reactor safety and in must be proved and declared in the Safety Analysis Report (SAR). In case of PWRs, the problematic phenomenon is not the dryout. In case of PWRs, the critical flow is inverted annular flow. This flow occurs when a fuel rod cladding surface is overheated, which causes the formation of a local vapor layer, causing a dramatic reduction in heat transfer capability. This phenomenon is known as departure from nucleate boiling – DNB. The difference in flow regime between post-dryout flow and post-DNB flow is depicted in the figure.
• In PWRs, evaporation occurs also in steam generators. Steam generators are heat exchangers used to convert feedwater into steam from heat produced in a nuclear reactor core. The steam produced drives the turbine.
Difference in the flow regime: Dryout vs. DNB
Convective Evaporation - Vertical Channel
Convective evaporation in a vertical channel is depicted in the figure. This figure shows the typical order of the flow regimes that are encountered from inlet to outlet of a heated channel. At the inlet, the liquid enters subcooled (at the lower temperature than saturation). In this region the flow is single-phase. As the liquid heats up, the wall temperature correspondingly rises. As the wall temperature exceeds the saturation temperature (e.g. 285°C at 6.8 MPa), subcooled boiling begins. Bubbles nucleate in the superheated thermal boundary layer on the heated wall but tend to condense in the subcooled bulk.

Further increase in liquid temperature causes, that the liquid bulk reaches its saturation temperature and the convective boiling process passes through the bubbly flow into the slug flow. Increasing void fraction causes that the structure of the flow becomes unstable. The boiling process passes through the slug and churn flow into the annular flow regime with its characteristic annular film of the liquid. At given combinations of flow rate through a channel, pressure, flow quality, and linear heat rate, the wall liquid film may exhaust and the wall may be dried out. At the dryout point the wall temperature significantly rises in order to dissipate the applied heat flux. The post-dryout flow (mist or drop flow) in the heated channel is undesirable, because the presence of such flow regime is accompanied with significantly higher wall temperatures and high fluctuation of wall temperatures.

Convective Evaporation - Horizontal Channel
Convective evaporation in a horizontal channel is very similar to the evaporation in the vertical channel. But for horizontal channel, gravitational force tends to drain the liquid toward the bottom of the channel and vapor phase concentrates at the top of the channel. Typical flow regimes, including cross-sectional views of the flow structure, are depicted in the figure below.

At the inlet, the liquid enters subcooled (at the lower temperature than saturation). In this region the flow is single-phase. As the liquid heats up, the wall temperature correspondingly rises. As the wall temperature exceeds the saturation temperature (e.g. 285°C at 6.8 MPa), subcooled boiling begins. The convective boiling process passes through bubbly, plug regimes and flow can be either stratified or unstratified (depending on the flow velocity). As can be seen, channel dryout occurs at the top of the tube where the film thickness is thinner due to gravitational force. Dryout then progresses around the perimeter from top to bottom along the channel.

## Two-phase Pressure Drop

In the practical analysis of piping systems the quantity of most importance is the pressure loss due to viscous effects along the length of the system, as well as additional pressure losses arising from other technological equipments like, valves, elbows, piping entrances, fittings and tees.

In contrast to single-phase pressure drops, calculation and prediction of two-phase pressure drops is much more sophisticated problem and leading methods differ significantly. Experimental data indicates that the frictional pressure drop in the two-phase flow (e.g. in a boiling channel) is substantially higher than that for a single-phase flow with the same length and mass flow rate. Explanations for this include an apparent increased surface roughness due to bubble formation on the heated surface and increased flow velocities.

## Pressure Drop – Homogeneous Flow Model

The simplest approach to the prediction of two-phase flows is to treat the entire two-phase flow as if it were all liquid, except flowing at the two-phase mixture velocity. The two-phase pressure drops for flows inside pipes and channels are the sum of three contributions:

The total pressure drop of the two-phase flow is then:

∆ptotal = ∆pstatic + ∆pmom + ∆pfrict

The static and momentum pressure drops can be calculated similarly as in case of single-phase flow and using the homogeneous mixture density:

The most problematic term is the frictional pressure drop ∆pfrict, which is based on the single-phase pressure drop that is multiplied by the two-phase correction factor (homogeneous friction multiplier – Φlo2). By this approach the frictional component of the two-phase pressure drop is:

where (dP/dz)2f is frictional pressure gradient of two-phase flow and (dP/dz)1f is frictional pressure gradient if entire flow (of total mass flow rate G) flows as liquid in the channel (standard single-phase pressure drop). The term Φlois the homogeneous friction multiplier, that can be derived according to various methods. One of possible multipliers is equal to Φlo2 = (1+xglg – 1)) and therefore:

As can be seen this simple model suggests that the two-phase frictional losses are in any event higher than the single-phase frictional losses. The homogeneous friction multiplier increases rapidly with flow quality.

Typical flow qualities in steam generators and BWR cores are on the order of 10 to 20 %. The corresponding two phase frictional loss would then be 2 – 4 times that in an equivalent single-phase system.

Martinelli-Nelson Friction Multiplicator
Separated-phases flow model - Lockhart-Martinelli correlation
An alternate approach to calculate two-phase pressure drop is the separated-phases model.

In this model, the phases are considered to be flowing separately in the channel, each occupying a given fraction of the channel cross section and each with a given velocity. It is obvious the predicting of the void fraction is very important for these methods. Numerous methods are available for predicting the void fraction.

The method of Lockhart and Martinelli is the original method that predicted the two-phase frictional pressure drop based on a friction multiplicator for the liquid-phase, or the vapor-phase:

∆pfrict = Φltt2 . ∆pl (liquid-phase ∆p)

∆pfrict = Φgtt2 . ∆pg (vapor-phase ∆p)

The single-phase friction factors of the liquid fl and the vapor fg are based on the single phase flowing alone in the channel, in either viscous laminar (v) or turbulent (t) regimes.

∆pl can be calculated classically, but with application of (1-x)2 in the expression and ∆pg with application of vapor quality x2 respectively.

The two-phase multipliers Φltt2 and Φgtt2 are equal to:

where Xtt is the Martinelli’s parameter defined as:

and the value of C in these equations depends on the flow regimes of the liquid and vapor. These values are in the following table.

The Lockhart-Martinelli correlation has been found to be adequate for two-phase flows at low and moderate pressures. For applications at higher pressures, the revised models of Martinelli and Nelson (1948) and Thom (1964) are recommended.

Example: Pressure Drop - BWR
Two-phase Minor Loss
In industry any pipe system contains different technological elements as bends, fittings, valves or heated channels. These additional components add to the overall head loss of the system. Such losses are generally termed minor losses, although they often account for a major portion of the head loss. For relatively short pipe systems, with a relatively large number of bends and fittings, minor losses can easily exceed major losses (especially with a partially closed valve that can cause a greater pressure loss than a long pipe, in fact when a valve is closed or nearly closed, the minor loss is infinite).

Single-phase minor losses are commonly measured experimentally. The data, especially for valves, are somewhat dependent upon the particular manufacturer’s design. The two-phase pressure loss due to local flow obstructions is treated in a manner similar to the single-phase frictional lossesvia local loss multiplier.

## Flow Instability

In general, there are a number of instabilities that may occur in two-phase systems. In nuclear engineering, study of the multiphase flow stability is of importance in accident management of pressurized water reactors and of the highest importance in normal/abnormal conditions in boiling water reactors.

In PWRs at normal operation the flow is considered to be single-phase. But a great deal of study has been performed on the flow instability in case of transients and accidents (such as the loss-of-coolant accident – LOCA or trip of RCPs with presence of natural circulation), in which flow oscillations or flow reversals may occur.

Flow oscillations are variations in flow caused especially by void formations and these are undesirable for several reasons.

• Flow oscillations can cause undesirable mechanical stress on fuel components (such as spacing grids). This can lead to failure of those components due to fatigue.
• Flow oscillations affect the local heat transfer characteristics. In case of PWRs, the critical safety issue is named DNB (departure from nucleate boiling), which causes the formation of a local vapor layer, causing a dramatic reduction in heat transfer capability. It has been found through testing that the critical heat flux (CHF) required for departure from nucleate boiling (DNB) can be lowered by as much as 40% when flow is oscillating. This severely reduces the thermal limit and the power density along the length of the reactor core.

Flow oscillations can be a problem during natural circulation operations (e.g. after trip of all RCPs). Natural circulation is an important design feature and ultimate heat removal mechanism. Because of the low flow rates are present, coolant boiling may occur and this may form flow oscillations. During natural circulation, the steam bubbles formed during a flow oscillation may have enough of an effect to actually cause complete flow reversal in the affected channel.

In BWRs evaporation of coolant occurs at normal operation and it is very desired phenomenon. On the other hand convective evaporation in the fuel channel causes that the flow pattern changes along the fuel channel depending on the flow rate and thermal power. It has been found that there are instability regions, in which two-phase flow instabilities may arise. These two-phase flow instabilities are undesirable as they can result in mechanical vibrations and system control problems, affect normal operation, restrict operating parameters and influence reactor safety. It must be noted flow stability in BWRs is not a major issue for many years, because it is well known phenomenon.

In general, there are many classifications of flow instabilities. The following classification is based upon thermal-hydraulic fundamental mechanisms:

The static instabilities are:

• Flow excursion
• Boiling crisis
• Relaxation types, including flow pattern transition

The dynamic instabilities are:

• Density wave oscillations
• Pressure drop oscillations
• Thermal oscillations.

The proper characterisation of the instabilities and the condition for its occurrence can determine optimal and safe operation of the systems. The most accepted explanation for the occurrence of the dynamic type of instabilities called density wave oscillations (DWO).

The density wave causes a delay in the local pressure drop that is caused by a change in inlet flow. Because of this delay, the sum of all local pressure drops may result in a total drop that is out-of-phase with the inlet flow. The basic mechanism causing flow instabilities in BWRs is the density wave. The characteristic periods of these oscillations are associated with the time required for a fluid particle to travel through the entire loop.

Types of instabilities observed in BWRs

• Control System Instabilities. Control system instabilities are related to the action of controllers that, through actuators, attempt to regulate some of the variables of the reactor.
• Channel Flow Instabilities. This type of instability can be described as follows: Let assume a flow perturbation. This perturbation causes a “wave” of voids traveling upward through the channel producing a two-phase pressure drop (pressure drop increases significantly as void fraction increases) that is delayed with respect to the original perturbation. An increase in channel pressure drop (density wave)  may lead to the instability in flow rate.
• Coupled Neutronic-Thermohydraulic Instability. The dominant type of instabilities in commercial BWRs is the coupled neutronic-thermohydraulic instability (also known as the reactivity instability). The power generation in BWRs is directly related to the fuel neutron flux, which is strongly related to the average void fraction in the core channels through. This effect is known as the reactivity feedback. The reactivity feedback caused by changes in void fraction (void coefficient) is delayed as the voids travel upward through the fuel channel. In some cases the delay may be long enough and the void feedback may be strong enough that the reactor configuration becomes unstable. In this case the neutron flux may oscillate.

Special References:

• Francesco D’Auria, The BWR Stability Issue, THICKET 2008 – Session IX – Paper 26
• Dag Strømsvåg, Fundamental mechanisms of density wave oscillations and the effect of subcooling, NTNU, 2011.
• J. March-Leuba, Density -Wave Instabilities in Boiling Water Reactors. NUREG/CR-6003, ORNL, 1992.

References:
Reactor Physics and Thermal Hydraulics:
1. J. R. Lamarsh, Introduction to Nuclear Reactor Theory, 2nd ed., Addison-Wesley, Reading, MA (1983).
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3. W. M. Stacey, Nuclear Reactor Physics, John Wiley & Sons, 2001, ISBN: 0- 471-39127-1.
4. Glasstone, Sesonske. Nuclear Reactor Engineering: Reactor Systems Engineering, Springer; 4th edition, 1994, ISBN: 978-0412985317
5. Todreas Neil E., Kazimi Mujid S. Nuclear Systems Volume I: Thermal Hydraulic Fundamentals, Second Edition. CRC Press; 2 edition, 2012, ISBN: 978-0415802871
6. Zohuri B., McDaniel P. Thermodynamics in Nuclear Power Plant Systems. Springer; 2015, ISBN: 978-3-319-13419-2
7. Moran Michal J., Shapiro Howard N. Fundamentals of Engineering Thermodynamics, Fifth Edition, John Wiley & Sons, 2006, ISBN: 978-0-470-03037-0
8. Kleinstreuer C. Modern Fluid Dynamics. Springer, 2010, ISBN 978-1-4020-8670-0.
9. U.S. Department of Energy, THERMODYNAMICS, HEAT TRANSFER, AND FLUID FLOW. DOE Fundamentals Handbook, Volume 1, 2 and 3. June 1992.
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Fluid Dynamics