COURSE OUTLINE

This course introduces the fundamental concepts, principles and application of multiphase flow.

The course opens with real life examples of such flow and its importance in process industries. In

connection with gas-liquid two phase flow, different flow regimes and flow regime maps are

discussed. Next, various analytical models are introduced to understand the hydrodynamics of

different flow regimes. The phenomenon of choking is explained and relevant formulations are

derived. The concept of bubble formation and bubble dynamics are presented. The important

aspects of hydrodynamics of solid-liquid and gas-solid flows are also discussed. Hydrodynamics

of three phase flows are analyzed and compared with two phase flow situations. Lastly various

measurement techniques used for measuring pressure drop, void fraction and identification of

flow patterns are introduced.

Contents:

Definition of multiphase flow, flow patterns, one dimensional steady homogenous equilibrium

flow, one dimensional steady separated flow model, choking and critical flow rate. General theory

of drift flux model, Bubble formation and bubble dynamics, hydrodynamics of solid-liquid and

gas-solid flow, hydrodynamics of three phase gas-liquid-liquid flows, Measurement techniques in

multiphase flow.

COURSE DETAIL

S.No Topics No. of

Hours

1 Introduction to multiphase flow, types and applications, Common terminologies,

flow patterns and flow pattern maps.

2 One dimensional steady homogenous flow.

3 Concept of choking and critical flow phenomena.

4 One dimensional steady separated flow model.

1. Phases are considered together but their velocities differ.

2. Phases are considered separately, flow with phase change.

5 Flow in which inertia effects dominate, energy equations.

6 The separated flow model for stratified and annular flow.

7 General theory of drift flux model.

8 Application of drift flux model to bubbly and slug flow.

9 Hydrodynamics of solid-liquid and gas-solid flow.

1. Principles of hydraulic and pneumatic transportation.

10 An introduction to three phase flow.

11 Measurement techniques for multiphase flow.

1. Flow regime identification, pressure drop, void fraction and flow rate

measurement.

Total

PREREQUISITES

Course on basic fluid mechanics at the undergraduate level.

REFERENCES

1. One dimensional Two Phase Flow by G. B. Wallis.

2. Measurement of Two Phase Flow Parameters by G.F.Hewitt.

3. Flow of Complex Mixtures by Govier and Aziz.

4. Two Phase Flow by Butterworth and Hewitt.

5. Handbook of Multiphase systems by Hetsroni.

ADDITIONAL READINGS

Current issues of International Journal of Multiphase Flow.

Introduction

The simultaneous flow of two or more phases through a conduit where the phases interact at the

interface is termed multiphase flow. Although simultaneous flow of as many as four phases

namely, water, crude oil, gas and sand is not uncommon during oil exploration, flow of two phase

mixtures is the most common occurrence in industry. It covers a diverse range of flow phnomena

involving various combinations of phases like solid, liquid and gas. The presence of an interface

varying over space and time renders the hydrodynamics of two phase flow substantially different

from single phase. For example two-phase flow in a fluidized bed can be differentiated from

single phase flow of a fluid through a packed bed of particles by considering the fact that in the

former case, geometrical arrangement of phase boundary (i.e particle spacing) is function of fluid

flow while in the second case, the geometry is fixed.

The different variations of two phase flow are

a) Gas–liquid flow – involves boiling, condensation as well as adiabatic flow. They are common

in power and process industries, refrigeration, air-conditioning and cryogenic applications.

b) Gas–solid flow – pneumatic conveying, combustion of pulverized fuel, flow in a cyclone

separators are examples of this category of two phase flow.

c) Liquid–solid flow – this type of flow is encountered in slurry transportation, food processing

as well as in various processes in biotechnology.

d) Liquid–liquid flow – This type of flow is also characterized by the presence of a deformable

interface (similar to gas–liquid flow) and processes several features similar to other two phase

flow phenomena. Liquid–liquid flow is common in petroleum industries and chemical reactors.

Method of analysis of single and two-phase flow: A comparison

It is interesting to note that two-phase flow occurs when an additional fluid is introduced in the

flow passage, but a straightforward extension of single-phase momentum equation does not give

us information about two-phase hydrodynamics. For example single-phase pressure drop for flow

of an incompressible fluid through an inclined pipe can be obtained from the following equation:

Where, , A, S, G, ρ and v are the wall shear stress, cross sectional area, interfacial area, mass

flux, density and specific volume of the fluid respectively.

However when we apply eqn (1) to two-phase flow, the corresponding equation is:

.... (2)

where ρ has been replaced by ρM and ν by νM. It may be noted that ρM ≠ 1/νM since ρM is an

additive function of volumetric composition while νM is additive in terms of mixture quality.

Therefore, during two phase flow ρM and νM can be expressed in terms of individual phase

properties as,

Further, there is no obvious relationship between the wall shear stress in single and twophase flow

and we need information about the interfacial shear stress . In addition, S includes S1 and S2 while

A includes A1and A2 where 1 and 2 are the two-phases. Single phase flow can be categorised as

laminar, turbulent or a transition between the two. On the other hand, in two phase flow the

phases can distribute themselves in a wide variety of ways which is not under the control of an

experimenter or designer and the phase distribution can vary with:-

Flow geometry (size and shape) and orientation (vertical, horizontal and inclined)

Flow direction in vertical or inclined flows (up or down)

Phase flow rates and properties (density, viscosity, interfacial tension, wettability)

n addition during two phase flow, the lighter fluid tends to flow past the heavier one. As a

result, the in-situ volume fraction is different from the inlet volume fraction of the two fluids. So

any analysis of two-phase requires an accurate knowledge of:

a) The distribution of the two phase

b) The in-situ composition, which has no direct relationship with the inlet composition and varies

with phase physical properties, their flow rates and interfacial distribution.

Thus it can be concluded that the hydrodynamics becomes more complex by the mere

introduction of a second phase in the flow passage and this can be attributed to the following

factors:-

1. Existence of multiple, deformable and moving interfaces

2. Multi scale physics of the flow phenomena

3. Significant discontinuities of fluid properties and complicated flow field

near interface

4. Compressibility of the gas phase (for gas-liquid and vapor liquid flows)

5. Different wall interactions for different fluids

Prior to an analysis of two phase flows it is important to understand the distribution of the two

phases in the test passage. The next chapter presents a comprehensive discussion on the flow

patterns which occur in circular conduits for different fluid pairs, conduit orientation and so on. In

Chapter 3 the different methods of analysis and the conventional notations used in studies of

multiphase flow have been elaborated in order to ensure that consistent notations are used in

subsequent analysis of multiphase flow in the following chapters. In chapter 3, 4 and 5 simple

analytical models namely the homogeneous flow model, the drift flux model and the separated

flow model have been elaborated and specific application to different relevant flow patterns have

been discussed. Henceforth, chapter 6 discusses the measurement schemes of different

hydrodynamic parameters during two phase flow in order to provide a flavor of the additional

difficulties encountered during experimentation with two phase/multiphase flow situations. In

order to maintain conciseness, three parameters have been selected for the discussion. They are (i)

two phase pressure drop in order to highlight the additional complexities involved in measuring

two phase as compared to single phase pressure drop and two parameters characterizing two phase

flow namely (ii) in-situ composition and (ii) estimation of flow pattern

During the simultaneous flow of two phases through any conduit, the two fluids can distribute

themselves in a wide variety of ways, which is not under the control of the experimenter or the

designer. There could be a large number of possible distributions, depending on the geometry and

orientation of the tube as well as physical properties and velocity of the two phases.Nevertheless,

a few factors restrict the variety of interfacial distribution. These include (a) the surface tension

effects which tend to create curved interfaces and keeps the channel wall always wet with liquid

during gas-liquid flows (unless the wall temperature is above the saturation temperature) and (b)

gravity which tends to pull the heavier phase at the bottom in a non-vertical channel. A close

observation of the different interfacial distributions reveals that they can be broadly delineated

into different flow regimes or flow patterns which are characterised by typical topographical

distribution of the two phases.

An accurate estimation of the different patterns is essential for the understanding and analysis of

two phase flow since all the transport processes like momentum, heat as well as mass transfer are

strongly influenced by the phase distribution. Therefore, a large number of studies, both

experimental and theoretical, have been reported on the characterization of flow patterns for

different combinations of the two phases.

From a survey of the past studies, it is observed that many of the two phase systems have a

common geometrical structure. Accordingly, two phase flow can be classified into several major

groups such as separated flow, transitional or mixed flow and dispersed flow. The different flow

patterns which confirm to the aforementioned descriptions for different fluid types are listed in

Table2.1.

Table 1: Generalised flow patterns for different fluid types

Flow

pattern

Classification of

flow

pattern

Schematic Description Application

Separated

flow

• Film flow

• Gas‐liquid

stratified flow

• Liquid‐liquid

stratified flow

• Gas‐liquidliquid

three layer flow

Liquid film in

gas/Gas film in

liquid.

Lighter fluid

flowing over the

heavier one

• Film condensation

• Film boiling

• Annular flow

• Core annular

flow (for

liquid‐liquid

cases)

Gas core Liquid

film

Viscous liquid

core and water

film

Annular Flow,

Rewetting, Film

boiling,

Transportation of

crude oil

• Jet flow

Liquid jet in gas/

Gas jet in liquid

Jet condenser

Dispersed flow Bubbly

Gas

bubbles in

liquid

Chemical

reactors

Droplet

flow namely

Oil droplet in

water

Water droplet in

oil

Liquid

droplets in

an

immiscible

liquid/ gas

Spray cooling

Particulate

flow

Solid

particles in

gas/liquid

Transportation

of powder

Mixed/transitional

flow

Cap, slug,

churn

Sodium boiling

in forced

convection

Bubbly

annular flow

Gas

bubbles in

liquid film

Gas core

Evaporators

with wall

nucleation

Droplet

annular flow/

Wispy annular

flow

Gas core

with

droplets

and

annular

liquid film

Irregular

liquid

chunks in

continuous

gas core

which is

separated

from pipe

wall by an

annular

liquid film

Steam

generator

Bubbly

droplet

Annular

flow

Gas core

with

droplets

Liquid

film with

gas

bubbles

Boiling nuclear

reactor channel

Three

layer flow

Oil at top

Water at

bottom

Oil-water

droplets at

middle.

Oil

transportation

Depending on the type of interface, the class of separated flow can be divided into plane flow

which includes film and stratified flow and quasi-axisymmetric flow consisting of the annular and

jet flow regimes. The class of dispersed flow is usually subdivided by considering the phase of

dispersion. Accordingly, three regimes are distinguished: bubbly, droplet or mist and particulate

flow. In each regime, the geometry of dispersion can be spherical, spheroidal, distorted, etc. Since

the change of interfacial structures occur gradually, we have a third class which is characterised

by the presence of both separated and dispersed flow. In this case too, it is more convenient to

subdivide the class of mixed flow according to the phase of dispersion. The flow patterns thus

obtained are depicted in Table 2.1.

In the following section, the typical flow patterns for different fluid combinations (gas-liquid,

liquid-liquid, gas-solid and gas-liquid-liquid), pipe orientations (vertical/horizontal) and flow

conditions (heated or unheated) have been discussed in order to understand the influence of

operating variables on phase distribution. A short discussion on the influence of pipe fittings has

been provided to compliment the chapter. In conclusion, a discussion on the different ways of

representing the range of existence of various flow patterns viz the flow pattern maps have been

presented.

1.Vertical co-current gas-liquid upflow:

A schematic of the different air-water flow regimes observed in a vertical tube are shown in

Fig.2.1 and described below:

a. Bubbly flow- Liquid flows as a continuous phase in which gas bubbles of approximately

uniform size are observed. The bubble diameter is not comparable to the diameter of the tube.

b. Slug flow- As the gas flow rate is increased, number of bubbles increase and they coalesce

to form elongated bubbles having spherical nose and cylindrical tail. These bullet shaped

axisymmetric bubbles are termed as Taylor bubbles in two phase terminology. Such bubbles are

also observed during the draining of water from bottles with a narrow neck and when a volume of

air rises through a stationary column of liquid. In slug flow, the Taylor bubbles are separated by

liquid slugs which may or may not be aerated. The periodic passage of Taylor bubbles and liquid

slugs across any cross-section (Fig.2.1) characterises slug flow. In the Taylor bubble regions, the

liquid flows downward as a thin annular film from the preceding to the succeeding liquid slug.

This forms a wake region when it meets the liquid slug. The vorticity induced in the wake shears

bubbles from the tail of the Taylor bubble and aerates the liquid slugs.

c. Churn flow-With a further increase in airflow, the Taylor bubbles become longer till they

break and cause a random and chaotic mixture propagating through the tube. This pattern is

known as churn flow. It is highly unstable and oscillatory in nature and can be differentiated from

slug flow by the absence of the periodic character.

d. Annular Flow- With further increase in gas flow, the gas bubbles coalesce to form a

continuous gas core and the liquid is forced to flow as an annular film between the gas core and

the pipe wall. Some liquid gets sheared from the film and forms a bridge in the gas phase. Several

researchers have identified this as a different flow pattern and named it as wispy annular flow.

Fig.2.1. Gas-liquid Flow patterns in vertical upflow

e. 2. Horizontal co-current gas-liquid flow

In a horizontal pipe, the effect of gravity causes stratification of the two phases and accounts for

the differences in flow regimes. The different patterns are presented in Fig.2.2 and described as

follows:

(a) Bubbly flow - In case of horizontal flow, the bubbles accumulate on the top for moderate

liquid velocity.

(b) Plug/Slug flow - As the air flow rate increases, the bubbles coalesce and form long plugs

which are also confined to the upper region of the tube. The intermittent liquid slugs may or may

not be aerated.

(c) Stratified flow - With further increase of air flow rate, plugs coalesce to form stratified flow.

At relatively lower flow rates the interface is smooth while at higher flow rates, the interface

becomes wavy and the wave amplitude increases with phase velocities.

(d) Annular Flow - This has the same appearance as mentioned in vertical flow and is

characterised by a continuous gas core and an annular liquid film between the gas core and the

pipe wall. However, the film thickness is not uniform and the liquid film is substantially thicker at

the bottom of the pipe.

Fig.2.2 Flow pattern in horizontal flow

. Flow Patterns in vertical heated tubes:

The flow patterns observed in a vertical heated tube are different from those observed in an

unheated tube under the same flow conditions due to the presence of heat flux at the channel wall.

As a result of heat transfer through the wall, thermodynamic non- equilibrium exists at a particular

cross section. This is evident from the simultaneous presence of sub-cooled liquid and

superheated vapour. Further, as the quality changes along the direction of flow, different flow

regimes appear along the flow direction. For a long tube there could be transition from sub-

cooled liquid regime to super heated vapour regime through a number of flow patterns. A

schematic representation of vertical tubular channel heated by a uniform low heat flux and fed at

its base with liquid below its saturation temperature is shown in Fig.2.3. It shows the absence of

the chaotic churn flow pattern and the appearance of mist/ droplet flow at high vapour velocities.

Such a distribution is not formed in an unheated tube.

Fig. 2.3 Flow regimes in vertical evaporator tubes

4. The corresponding situation in horizontal heated tubes –

The influence of gravity makes the situation more complex in a horizontal heated channel. There

is departure from hydrodynamic and thermal equilibrium as in vertical flows through heated

channels as well as asymmetric phase distribution and stratification due to horizontal orientation.

Therefore several important features can be observed namely:

1. Possibility of intermittent drying and rewetting of upper surface of tube in wavy flow.

2. Progressive drying out over long tube length of upper circumference of tube wall in

annular flow.

3. Less obvious effect of gravity at higher inlet liquid velocities give more symmetrical flow

patterns and closer similarities to vertical flows.

Unique flow patterns can also be observed during condensationas shown in Fig. 2.4.

Fig. 2.4

5. Flow patterns for liquid-liquid systems:

Certain interesting features are noted when the gas phase is replaced by a second immiscible

liquid (say oil). For horizontal pipes, the stratified flow pattern gives way to three layer flow with

increase in phase flow rate. This pattern is characterised by an oil layer at the top and a water

layer at the bottom with a dense dispersion of droplets separating the two as shown in Fig. 2.5 (a).

Such a distribution has not been observed for gas-liquid cases under any flow conditions.

Moreover, liquid-liquid dispersed flow can comprise of either oil in water dispersion or water in

oil dispersion depending on the flow conditions unlike the presence of only gas-in liquid

dispersions for the previous case. This is evident from Fig. 2.5 (c) which presents flow patterns

for a vertical pipe of the same dimension. The transition between the two types of dispersed flow

is termed as phase inversion and is unique to liquid-liquid flows. It has received much academic

interest and industrial concern due to its uniqueness and complexity. For vertical tubes, the flow is

either dispersed or core-annular with the tendency of formation of the core-annular pattern

increasing with the viscosity of the oil. This is an extremely fortunate situation since it results in a

drastic reduction of the power required to pump the liquid. A comparison of Figs 2.5 (a) and

(b) highlights the tendency of slugging at reduced tube dimensions.

Fig. 2.5 Typical flow patterns during oil-water flow through (a) horizontal pipe of

25.4 mm (b) horizontal pipe of 12.7 mm (c) vertical pipe of 25.4mm

6. Flow patterns for gas-liquid-liquid three phase flows: Simultaneous flow of two immiscible

liquids and a gas is not uncommon in industry. A large variety of flow patterns can be observed

during such three phase flow. A brief description of the typical flow pattern in three phase flow is

provided here. In horizontal pipes, a three layer flow pattern is observed at low flow rates

(Fig.2.6). At higher phase velocities, the air usually exists as plugs which alternate with liquid

slugs. The distribution of the two liquids in the

slug can be either stratified or dispersed depending on the flow rates. The slug flow pattern is also

the most predominant flow pattern for vertical pipes where they are characterised by axisymmetric

bullet shaped air Taylor bubbles intercepted by liquid slugs. The distribution in the liquid slug can

be either oil in water dispersed flow, water in oil dispersed flow or an emulsified flow at high

phase velocities as shown inFig. 2.7.

Fig.2.6 Flow patterns in horizontal co-current upward air-water-kerosene flow

7. The commonly encountered patterns in gas-solid flows (pneumatic conveying and

fluidisation):

Traditionally, flow regimes have been divided into two main groups: dilute and dense. The

transition between these two regimes for vertical conveying systems is defined by the choking

velocity. The dense flow regime is usually divided into specific flow regimes such as slugging,

bubbling, fluidizing and plugging (Fig.2.8). The accumulated and classical choking presents two

possible transitions from dilute flow regime. When the gas velocity is reduced at a fixed solid

flow rate, the dilute flow turns into slugging flow or a non-slugging dense phase flow. The

condition when the dilute flow becomes non-slugging is called accumulated choking and is related

to the accumulation of solid at the bottom of the pipe line. The condition when the dilute flow

becomes a slugging flow is called classical choking and is related to the formation of slugs.

Although pneumatic conveying and fluidized bed systems are designated for different tasks, they

nonetheless have many similarities. For example, for both systems dilute flow, fast fluidization,

turbulent fluidization, slugging fluidization, bubbling flow and fluidized flow regimes occur. The

dilute flow regime is characterized by suspension flow at high gas velocity and low solid mass

flow rate. For pneumatic systems, the dilute flow regime is most commonly used, while for

fluidized bed systems this regime might occur as a bypass process for emptying the column or

when the inserted sample has a wide size distribution. For a wide particle size distribution, the

large particles are fluidized at the lower part of the column while the fine powders might be

carried over by a dilute flow regimes. By reducing the gas velocity, suspension flow is halted and

particle clusters might appear. The flow regime occurring after the appearances of particle clusters

is termed as fluidization. The turbulent fluidization regime is characterized by extreme particle

turbulence without large discrete bubbles or voids. The slugging flow regime is characterized by a

particle dense phase transport that is facilitated by bubbles whose size is comparable to the pipe

diameter. The bubbling flow regime, on the other hand, is characterized by smaller bubbles.

Two more possible flow regimes occur in a pneumatic conveying system, but are not common in

fluidized system. The first is the plug flow regime which is characterized by particles that are

transported as plugs separated by air gaps. Sometimes these particles fall from the bottom of one

plug and collect at the front of consequent plug. This phenomenon is known as “particle rain” and

occurs when the cohesion force between the particles is smaller than the particle weight. The

worst case scenario for designers of pneumatic conveying systems is blockage. The flow

conditions which causes blockage can be defined as a kind of flow regime.

Fig. 2.8 Typical flow patterns during gas-solid flow

8. Gas-liquid flow patterns in other applications:

a. Vertical downward flow: Downward flow of a gas-liquid mixture is unstable as the gas

phase tends to move up. However, in certain range of the operating condition such flow can be

established. Annular flow regime occurs at low liquid flow rates while a falling liquid film occurs

with no gas flow. Slug and bubbly flow occur only at liquid velocities greater than bubble rise

velocity.

b. Inclined Channels: Usually stratification occurs only for very low superficial velocities

and inclinations close to the horizontal. Smooth stratified flow disappears on slight deviation from

the horizontal orientation and stratification disappears completely for inclinations beyond 300. In

addition, the shape of the Taylor bubbles changes as the inclination is increased from horizontal to

vertical. The nose becomes more pointed and the bubble more asymmetric as the inclination

increases from 0 to 450 (approximately) from the horizontal as shown in the fig.2.9.

Subsequently, the nose of the bubble assumes the nice rounded shape observed for Taylor bubbles

in vertical tubes. This results in higher rise velocity of the bubble with increase in inclination from

0 to 450 and a subsequent decrease with a further increase in inclination.

Fig.2.9 Taylor bubble in (a) Vertical tube (b) Inclined tube

c. Rectangular Channels: In these channels flow is similar to circular channels when the

aspect ratio is not very different from unity. Nevertheless, unique flow regimes may be observed

for extreme values of the aspect ratio. Presence of corners in the flow geometry influences the

flow regime as the corner regions tend to retain the liquid film. However, such effects are

pronounced in flow channels of smaller cross-sections.

d. Annular channels: A very interesting phenomena occurs when a rod is inserted in the

flow passage of circular tubes. The rod induces gross asymmetry in the slug flow pattern. This

asymmetry arises due to the asymmetric shape of the Taylor bubbles. They partially enclose the

inner tube and form open annular rings as shown in Fig 2.10.

e. f. Fig 2.10 Taylor bubble in (a) circular tube (b) concentric annulus

Bends and Coils: A bend or a coil acts to separate the phases due to the presence of

centrifugal force. For example, a bend will induce coalescence of bubbles to form slug flow and

will separate entrained droplets in annular flow. At low superficial velocity, the action of

gravitational forces and the fact that vapour phase tends to flow faster than the liquid phase

greatly complicates the picture. In a vertical pipe joined to a horizontal pipe via a 90 degree bend

the momentum of the upflowing liquid tries to carry it to the outside of the bend and gravitational

forces tend to make it fall to the inside of the bend. Fig 2.11 presents a few photographs to

highlight the effect of pipe fittings on oil-water flow. The phenomena of film inversion as oil-

water stratified flow turns round a return bend is evident from Fig. 2.11 (a). The change of

interfacial distribution as liquid-liquid flow encounters an abrupt contraction or expansion is

evident from the following two figures. They emphasis upon the onset of dispersion as the flow

encounters an expansion and the reverse phenomena as the cross-section reduces abruptly.

Effect of Pipe fittings

Fig.2.11 Effect of pipe fittings on oil-water two phase flow

(a) Film Inversion at a hairpin bend (b) Onset of dispersion at an expansion

(c) Coalescing effect at a contraction

9. Flow Pattern Maps:

It is very important to predict the pattern which is likely to occur for a given set of flow

parameters. One method of representing the various transitions is in the form of a flow pattern

map which is a two dimensional graph segregated into areas representing the range of existence of

the different patterns. Different dimensional as well as dimensionless parameters have been

frequently used as the co-ordinate axes of the maps. Some of the frequently used non-dimensional

parameters include two phase Froude number, Eotvos number and Weber number, Reynolds

number of the individual phases/mixture for gas-liquid and liquid-liquid systems and Reynolds

number and Archimedes number for fluid-particle systems. However the most commonly used

axes for gas-liquid and liquid-liquid flow maps are the actual or superficial phase velocities of the

two phases defined as the volumetric flow rate per unit cross-sectional area of the pipe. The flow

pattern maps commonly used for horizontal and vertical gas-liquid flow are shown in Figs 2.12

(a) and (b). It may be noted that although the use of superficial velocity restricts the application

of the maps to fluids with a limited variation of properties, it is preferred due to its simplicity. In

addition, one dimensionless parameter which may be adequate to represent one particular

transition may not be suitable for a different transition governed by a different balance of forces.

Fig 2.12 (a) Flow Regime map for horizontal gas-liquid flow

Fig 2.12 (b) Flow Regime map for vertical gas-liquid flow

An alternative and more flexible method to overcome this difficulty is to examine each transition

individually and derive a criterion for that particular transition based on the principle underlying

the mechanism. For example bubble to slug flow transition is generally modelled on the basis of

bubble coalescence which depends on a critical void fraction whereas flooding and flow reversal

are responsible for slug-churn and churn-annular transition.

Recently, soft computing that includes Artificial Neural Network, Fuzzy logic or Genetic

Algorithm is being increasingly used to produce generalised flow pattern maps from known input

parameters..

Two phase flow obeys all basic laws of fluid mechanics. However, the hydrodynamics is

substantially complex as there are interaction between the phases and between any individual

phase and the conduit wall. As a result, the conservation equations are more in number unless one

goes for a very gross averaging. In addition, the conservation equations should contain extra terms

to take care of the interphasic interactions. With the current state of understanding and the

available computation power, direct numerical simulation (DNS) is not possible in most of the

cases. So along with the conservation equations, constitutive relations or closure relationships are

also to be used. Moreover, as there are more than one phase present, fluid properties influencing

the hydrodynamics becomes more than twice of that in case of single phase flow. Nevertheless,

over the years different methodologies have evolved to analyse two phase flow. The commonly

adopted methods can be classified as follows:

1. Empirical Correlations:

Due to the complexity of two phase flow, experiments often become the only method for

investigating such phenomena. As a result, many analyses are solely based on experimental data.

In spite of the advancement of computational techniques, use of empirical correlations based on

experimental observations is very important for specific problems. Correlations are obtained

either by dimensional analysis or by grouping of several variables together on a logical basis. The

main advantages of this technique are (a) Easy to use and (b) Satisfactory within statistical limits

as long as applied to situations similar to those used to obtain the original data. The limitations of

this technique are: (i) can be misleading if used indiscriminately in a variety of applications and

(ii) since little insight is achieved into the basic phenomena, there is no indication of ways to

improve its performance or accuracy of prediction. The Lockhart -Martinelli correlation which

shall be discussed later is an example of a widely used correlation in gas-liquid flows.

2.Simple Analytical Models:

These do not take into account the details of flow due to which foam and stratified flow of gas

and liquid are treated exactly alike. They can be successful for organizing experimental results or

predicting design parameters with minimum computational effort. A few such models are:

(i) Homogenous model: where the components are assumed to be intimately mixed with one

another such that the two phase mixture can be treated as a single pseudo fluid with suitable

average properties without bothering about the detailed description of flow pattern. This model

gives accurate results for suspension of droplet in gas, well dispersed gas-liquid bubbly flow, gas-

solid or liquid-solid particulate flow and liquid-liquid dispersed flows.

(ii) Drift flux model: which modifies the homogeneous model by incorporating the relative

motion between the phases. This is achieved by introducing the concept of drift flux which shall

be discussed in the next chapter. It has been observed to give good results for the mixed flow

patterns discussed in Chapter 2.

(iii) Separated flow model: where the phases are assumed to flow side by side. Accordingly,

separate equations are formulated for each phase and the interaction between phases is considered

separately. This gives accurate results for annular and stratified flow.3. Integral Analysis:

One dimensional integral analysis can be performed for two phase flow situations by assuming

forms of certain functions which describe, for example, velocity or concentration distribution in a

duct. These functions are made to satisfy appropriate boundary conditions and basic fluid

mechanics equation in integral forms. The techniques are similar to that used for analyzing single

phase boundary flow problems.

In this case, velocity and concentration fields are deduced from suitable differential equations

which follow one dimensional flow idealization. The equations are then written for time averaged

quantities. More sophisticated theories may even consider temporal variations. They are similar to

the single phase theories of turbulence.

5. In this case, constitutive equations are formulated for the two phases based on their topological

distribution inside the flow conduit and the interfacial interaction is considered accordingly.

6They are a class of very powerful techniques based on universal phenomena that are independent

of flow regime, analytical model or particular system.

Typical methods are various theories of wave motion, extremism techniques for obtaining locus of

limiting behavior of a system.

Usually simpler theories are applied for multiphase flow analysis and more complex theories are

used for inclusion of suitable addition effects and prediction of numerical values of correction

factors which can be applied to the simpler theories to increase their accuracy. The complex

theories may also lead to analytical rather than empirical relationship between important variables.

Thus the sequential levels of analysis forms a pyramid in which the broader and more general

theories serve to support the more approximate and simpler techniques.

The subsequent chapters describe the simple analytical models and their applications to different

flow patterns. Considering the large number of parameters one has to deal in multiphase flows, the

relevant nomenclatures which shall be adopted for different two phase flow situations throughout

the web course has first been discussed in the next chapter.

Prior to an analysis of two phase flow, it is necessary to define some of the relevant terminologies

which shall be used frequently in subsequent chapters. In this regard, the present chapter initially

deals with the nomenclatures which have been adopted from single phase flow and subsequently,

defines the terminologies unique for two phase flow. It may be noted that even for the common

notations from single phase flow, the number of parameter are more than twice since each

property has to be specified for either of the phase as well as for the mixture and at times for the

interface.

In general the two phases are denoted by subscripts 1 and 2 or l and g for liquid-gas, l and s for

liquid – solid and g - s for gas-solid flows. Usually, for adiabatic cases, the continuous phase is

denoted by 1 and the discontinuous/dispersed phase by 2. A continuous phase is defined as one in

which any two points can be joined by a line (straight or curved) which passes through the same

phase without crossing any interface. When both phase are continuous (eg. in stratified flow),

phase 2 is the lighter phase.

The following table lists the common notations for single and two phase flow for one dimensional

steady state flow conditions through a pipe of diameter D and length L In the table, the properties

of the either of the phase is subscripted as 1and 2 while the mixture properties are denoted by

subscript TP and any interfacial characteristic with subscript i.

Table 4.1: Common notations for single and two phase flow

Parameter Notation used in

Single Phase Flow

Two Phase Flow

Mass Flow Rate W W₁, W₂

WTP = W1 +W2

Density ρ ρ1 ρ2

Specific volume v v1, v2, vTP

Viscosity μ μ1, μ2 μTP

Volume Flow Rate Q = W/ρ Q₁ = W1/ ρ1, Q₂ = W2/ρ2

QTP = Q1+Q2

Cross-Sectional Area of the

pipe

A =πD2/4 A , A₁ , A₂

Wetted Area S S₁ , S₂, Si

Mass Flux G = W/A G₁ = W₁/A, G₂ = W₂/A

GTP = G1 + G2

Volume flux j = Q/A j1 = Q1/A, j2 = Q2/A

j = j1+j2

Velocity u = Q/A u₁ = Q₁/A₁, u₂ = Q₂/A₂

u₁S = Q₁/A, u₂S = Q₂/A

Pressure gradient (-dp/dz) = (-dp/dz)g +(-dp/dz)f

(for adiabatic equilibrium flow)

(-dp/dz) = (-dp/dz)g +(-dp/dz)f

+(-dp/dz)a (for heated tubes)

(-dp/dz) = (-dp/dz)g +

(-dp/dz)f+(-dp/dz)acc

(for adiabatic equilibrium

flow as well as heated

tubes)

Pressure drop

In the table, u1 and u2 are termed as the in-situ velocities of phase 1 and 2 while u1s and u2s are the

respective superficial velocities where the superficial velocity is defined as the velocity which the

fluid would have had it flowed along in the pipe . Usually pressure drop in single phase adiabatic

equilibrium case arises due to hydrostatic (∆pg) and frictional loss (∆pf). In heated tubes, the

acceleration pressure drop (∆pa) is also important. For two phase flow all the three components

are significant as we shall see in the subsequent chapters.

In addition to the terms defined above, the following parameters are also necessary for a

comprehensive understanding of two phase flow. This arises because in the study of the

hydrodynamics of single phase flow, the most important parameter is the mass flow rate or the

volumetric flow rate of the fluid. The same is also true for two phase flow phenomena. However,

the variables important from the engineering point of view of any two phase flow situation include

not only the flow rate but also the volumetric proportion of the two phases in the flowing mixture

because all the average properties of the mixture are functions of its composition.

For gas-liquid two phase flow, the composition is expressed either in terms of the gas

voidage α which is a measure of the fractional volume of the flow channel occupied by the gas

phase or as liquid holdup HLwhich is a measure of the volumetric content of the liquid in the

mixture. For two-phase gas-liquid flow, the average value of liquid holdup can be expressed

mathematically as

(4.1)

and it is related to the average gas voidage by the equation

(4.2)

When the flow is not uniform, often it is not possible to measure <α> over a long length of pipe.

In this case a large number of instantaneous readings over a length dL gives the time average α at

a given location. The average value of α both in space and time is then

(4.3)

Usually the symbol α is used loosely to represent an average volumetric concentration without

bothering exactly how the average is to be taken. Therefore one has to take extra care when

periodic phenomena and non uniform concentrations are important.

The average gas voidage <α> is usually not equal to the inlet or outlet volume composition, <β>

even under steady state conditions where <β> is obtained in terms of phase superficial velocities

as

(4.4)

This is because the phases travel at different velocities due to a difference in their density,

interfacial distribution and other properties. This causes the fluid of lower density to slip past that

of higher density and reduces the gas-liquid ratio in the flow channel over that of the entering or

leaving mixture. Since the existence of holdup arises due to the distribution and properties of the

two phases and does not depend upon the entry conditions only, it cannot be manipulated

according to the convenience of the experimenter or the designer.

Though the term liquid holdup refers to the volume averaged property by definition, it is not

always possible to measure holdup over a volume element. The measurements then obtain

averaged values with respect to different space dimensions and with time which are subsequently

converted to the volumetric averaged parameter. This gives rise to the following definitions of

liquid holdup for gas-liquid flow through a conduit.

Volume average holdup: It is obtained as the fraction of the conduit volume occupied by the

liquid phase at any instant of time. In mathematical terms the definition can be expressed as

Volume of liquid in total volume of mixture (4.5)

This is the most useful definition of holdup in industrial designs and gives the overall composition

of the flowing mixture.

Area average holdup: It is the fraction of the conduit cross sectional area occupied by the liquid

phase at any instant of time. This average property is usually determined by impedance method

and optical techniques. This is the volume average value for infinitesimal length of the test section

and is equal to the volumetric average holdup when the holdup does not vary with the length of

the conduit geometry. Mathematically this can be expressed as:

(4.6)

Chordal average holdup: It is defined as the composition of the mixture along a particular chord

of known length. It is obtained when it is difficult to measure the area or volume average values.

It is used particularly in connection with the radiation attenuation and scattering techniques. It is

converted to the area average values either by mathematical manipulation or by the use of

multiple beams and is mathematically expressed as:

(4.7)

Time average holdup: This is obtained by measuring the holdup of the mixture at a particular

point in the two phase flow field as a function of time. This is usually required to obtain the void

fraction profile in any system since knowledge of it adequately describes the structure of the flow

field. Any point in the field can only be occupied by one phase at a time. Therefore, by definition,

the point average holdup of a phase varies from zero to one and vice versa instantaneously and

assumes a square wave form with respect to time. The point average holdup or local holdup with

respect to time is, therefore, meaningless. The holdup under these situations is defined as the

average fraction of a particular interval of time during which the point is occupied by the liquid

phase. It does not carry the sense of volume or area and can be mathematically expressed as

(4.8)

Where i = 1,2,…..n and n is the number of periods during which liquid phase exist at a particular

point.

In our discussion, the holdup HL and the gas voidage α refers to the volume average value unless

otherwise mentioned.

In boiling /condensation applications, it is often necessary to have a measure of the fraction of

total mass across a given area which is composed of each component. This is given by the quality

of the mixture which is defined as

(4.9)

For unsteady or non uniform flows, x is subject to averaging and the average is taken over a

specified surface for a period of time or

(4.10)

This is constant for unheated tubes and a function of heat flux Φ for heated tube where it can be

expressed as

(4.11)

(4.12)

if both liquid and vapor phase are in thermodynamics equilibrium i.e. they exist at Tsat

Due to a difference in individual phase velocities, two terms namely slip and relative velocity

arise. They are expressed as

Slip ratio (4.13)

Relative velocity (4.14)

And (4.15)

Where α ,k = function (W1,W2, fluid properties, geometry)

Interestingly, the subscript convention is the reverse for thermodynamic properties where

(4.16)

(4.17)

The aforementioned notations are tabulated in Table 4.2 for ease of reference.

Table 4.2: Additional terms in two phase flow

In situ void fraction α= A2/A

Water Holdup H1=1-α

Inlet volume fraction β=Q2/(Q1+Q2)

Quality x=W2/(W1+W2)

Relative velocity u21=u2-u1

Slip ratio k=u2/u1

The homogeneous flow model considers the two fluids to be mixed intimately as shown in Fig-

5.1 such that they can be approximated as a pseudo fluid with suitable average properties. Thus

the single phase equations for continuity, momentum and energy can be applied to the two phase

mixture by merely replacing the fluid property with mixture property.

The assumptions of the model are as follows:-

The slip velocity between the two phases is negligible or α = β.

Two fluids are uniformly mixed and moving as a pseudo fluid at the mixture velocity or

u1 =u2=j.

There is thermodynamic equilibrium between the phases.

(Fig-5.1)

Accordingly, the mass, momentum and energy equations for two phase homogeneous flow

inclined at an angle θ from the horizontal can be written as

Continuity: →5.1)

Momentum: → 5.2)

Energy: → 5.3)

The pressure gradient can thus be obtained from eqn (5.2) as:

→ 5.4)

Where,

In the above equations

P = perimeter of the pipe = πD for a circular pipe

A = the cross sectional area of the pipe = for a circular pipe

D = pipe diameter

θ is the pipe inclination from the horizontal orientation

W TP= total mass flow rate = W1+W2

uTP = j

From single phase flow theory:

→ 5.5)

When both density and area changes with length:-

→5.7)

→ 5.8)

Thus from eqns (5.4) to (5.8) we get

Or

Or,

→5.9)

In order to estimate pressure gradient from the aforementioned equation (5.9), the only unknown

is fTP , the equivalent friction factor during two phase flow under homogeneous equilibrium

condition. The other input parameters include the physical properties and flow rates of the two

phases as well as conduit dimension, inclination and taper. It may be noted that dx/dz can be

obtained from the heat balance equation as discussed in chapter3. Therefore, it is also an input

parameter.

Nevertheless, in presence of significant flashing, x cannot be obtained from enthalpy balance

alone since it changes both with enthalpy and pressure or

Substituting this in equation (5.9), we get:-

→ 5.11)

An Estimation of fTP : There are different approaches to estimate the two phase friction factor fTP

1) fTP is assumed to be equal to that which would occur if total flow is assumed to be all

liquid. This is applicable for low quality flows of vapour-liquid or gas-liquid mixtures. Thus:

→ 5.12)

Where, → 5.16)

In eqns (5.12–5.16) subscript L refers to the liquid phase in gas-liquid /vapour-liquid flow and LO

refers to the condition where the entire two phase mixture flows as liquid through the conduit.

Thus

frictional pressure gradient calculated from Fanning’s equation for total mixture

(liquid + vapour) assumed to flow as liquid.

A similar approach is adopted for high quality flows. In this case

→ 5.17)

Where → 5.18)

Subscript GO refers to the condition where the entire two phase mixture flows as gas/vapour

through the conduit.

Nevertheless, in both the cases, the definition of fLO or fGO in the evaluation of does not

allow extrapolation to the correct value when x=1 (single phase vapour flowing through conduit)

or x =0 (single phase liquid flow through conduit).

2) Estimation of fTP : Considering this problem, a second approach attempts to estimate fTP for

the two phase mixture on the basis of a suitable definition of two- phase Reynolds number viz

→ 5.19)

where → 5.20)

μTP is the suitable two phase viscosity of the mixture. To find μTP for a suspension of fluid spheres

at low concentrations,

→ 5.21)

If suspensions consist of solid sphere, μ2 is very large then

→ 5.22)

This is called Einstein’s equation.

If emulsion consists of bubbles containing gas of a low viscosity

→ 5.23)

These equations are valid at concentrations below about 5% for which change in viscosity due to

second phase is small. Numerous rheological models to account for larger values of α and

particles of various sizes and shapes are available. Nevertheless, since the details of many two

phase mixtures (which are non-Newtonian) are not available and information about two phase

flow patterns are not known, idealized rheological models cannot be defined.

Accordingly, expressions for viscosity are chosen to fit limiting cases when either phase is present

in the form of relationship between μTP and x such that for x=0 and, for

x=1.

The possible forms of relation are:-

(McAdams Relation) → 5.24)

(Cicchitti Relation) → 5.25)

(Dukler Relation ) → 5.26)

For laminar flow:- → 5.27)

Using the Mc Adam’s equation for μTP, we get:

→ 5.28)

and → 5.29)

or, → 5.30)

For turbulent flow is assumed as a rule of thumb. This expression is not very

accurate for single phase flows in commercial situation where pipes are subjected to distortion and

scaling. But in absence of a better correlation this value of fTP is chosen as a first choice for

turbulent flow.

From the above equation the two phase frictional pressure drop has been expressed in terms of

related single phase pressure drop that is the pressure drop encountered when the entire mixture

flows as liquid at the total mass flow rate. Such ratios are termed as two phase multipliers in two

phase terminology. There are four types of two phase multipliers. Two of them ΦLO2 and φ L

2 are

expressed in terms of liquid flow only through the pipe and the other two ΦgO2 and φ g

2 are in

terms of gas flow only through the same conduit under the same conditions of temperature and

pressure.

The definitions are as follows:

→ 5.31)

where is the frictional pressure drop when the entire mixture flows as liquid in the

pipe. Mathematically,

→ 5.32)

and is the frictional pressure drop when the liquid portion

of the two phase mixture flows alone in the pipe. Mathematically,

→ 5.33)

It may be noted that the friction factors used in eqns (4.32) and (4.33) are not equal since

whereas

→ 5.34)

Considering these, it can be noted that, for laminar flow and

→ 5.35)

Hence and → 5.36)

And for turbulent flow if Blausius equation is assumed:

And → 5.37)

Both Φlo2 and ΦL

2 can be used to find the two phase frictional pressure gradient but generally

ΦLO2 and ΦGO

2 are used in problems of boiling (and condensation) when saturated liquid (or

vapour) enters from one end of the pipe and changes phase as it flows. ΦL2 and Φg

2 are generally

used in separated flows(Lockhart-Martinelli correlation) which shall be discussed later.

Significance of the denominator term:

It may be recalled that a similar expression is obtained for predicting pressure drop during single

phase compressible flows through closed conduits. The expression as given below comprises of a

frictional term, gravitational term and an acceleration term arising due to area change of the

conduit in its numerator and (1-Ma2) as the denominator where Ma is the Mach

number (u/a) with u being the velocity of flow and athe velocity of sound in the same medium at

the same conditions of temperature and pressure.

Proceeding in a similar manner, it can be postulated that the denominator in eqns (5.9) and (4.11)

should also correspond to (1-MaTP 2) where MaTP refers to Mach number of the two phase

mixture under homogeneous flow.

In the denominator of eqn (5.9)

→ 5.38)

→ 5.39)

→ 5.40)

→ 5.41)

→ 5.42)

→ 5.43)

Or, → 5.44)

→ 5.45)

→ 5.46)

→ 5.47)

→ 5.48)

For air-water mixture: → 5.49)

→ 5.50)

For → 5.51)

For this value of α, the double differential of αTP is positive. Hence, αTP is minimum for α=1/2

Limitations of the Homogenous Flow Model:-

• Inapplicable for flow through rapid change in area where no slip condition fails.

• Not applicable for counter-current flows, which are driven by gravity acting on the different

densities of phases because a suitable average velocity cannot be determined in this case.

• Applicable for well dispersed flow with a limited void fraction of the

dispersed phase.

• Properties of the phases do not vary widely.

• Body force filled does not segregate the phase.

• Re-circulatory flow should be absent.

Model valid for bubbly and wispy annular flows especially at high phase flow rates and pressure.

The One Dimensional Drift Flux Model

The drift flux model modifies the homogeneous flow model discussed in Chapter-5 by

incorporating the relative motion of one phase with respect to the other. It is essentially a

separated flow model in which attention is focussed on relative motion rather than on motion of

individual phases. The model thus considers the mixture as a whole and expresses the motion of

the mixture by the mixture momentum equation. The relative motion and energy difference

between the phases is expressed by additional constitutive equations or dynamic interaction

relations replaced by constitutive laws. Accordingly, the formulation of the drift flux model needs

only four conservation equations namely mixture continuity, momentum balance, energy equation

and gas continuity instead of the six equations describing the mass, momentum and energy

balance of either of the phases in two-fluid model. The formulation of the model is therefore much

simpler as compared to two–fluid model. However, it requires some drastic constitutive

assumptions. This often causes some of the important two phase characteristics for example the

dynamic interactions between the two phases to be lost.

The use of the drift flux model is appropriate when the motion of the two phases is closely

coupled. From the same argument it can also be used for macroscopic two phase flows even when

the two phases are weakly coupled locally because the relatively large axial dimension of the

system usually gives sufficient interaction time. Despite being an approximate formulation in

comparison to the more rigorous two-fluid formulation, it is of considerable interest due to its

Simplicity

• Applicability to a wide range of two phase flow problems of practical interest namely bubbly,

slug and drop regimes of gas-liquid flow as well as fluidised bed of fluid particle system.

• The model is a key to rapid solution of unsteady flow problems of sedimentation and foam

drainage and is useful for the study of system dynamics and instabilities caused by low velocity

wave propagation namely void propagation.

• The model serves as a starting point for extension of theory to complicated problems of fluid

flow and heat transfer where two and three dimensional effects such as density and velocity

variations across a channel are significant.

• An important aspect of the drift flux model is concerned with the scaling of systems. This has

direct application in the planning and design of two phase flow experimental and engineering

systems. The similarities of two different systems can be studied effectively by using the drift flux

model formulation and mixture properties.

• It may be noted that the analysis of the dynamics of two phase flow systems in engineering

problems usually requires information on the response of the total mixture and not of each

constituent phase and the detailed analysis of the local behaviour of each phase can be carried out

more easily if the mixture responses are known.

However, the model is not suitable for acoustic wave propagation, choking phenomena and high

frequency instabilities.

Prior to development of the theory of drift flux model, it is necessary to introduce the concepts of

volumetric flux, drift velocity and drift flux.

The volumetric flux (j) is defined as the volume flow rate per unit area or

(6.1)

For two phase flows, the individual fluxes are given as:

(6.2a)

(6.2b)

Where the brackets denote the cross-sectional average values. These are usually omitted unless

variation across the flow is being considered and henceforth, the notations used without square

brackets shall denote area average values unless otherwise mentioned.

The volumetric flux is a vector quantity but for the present situation it will be used exclusively to

represent the scalar component in the direction of motion along the pipe. Under such conditions, it

is numerically equal to the superficial velocity of the individual phases as defined in Chapter-3.

However, the basic definitions of the two terms are different but they can be used interchangeably

when temporal and cross-sectional variations across the flow are ignored. For one dimensional

two phase flow, the volumetric fluxes of the individual components are related to the local

component concentration and velocity as:

(6.3a)

(6.3b)

And the total local flux is:

(6.4)

This gives rise to the following relations:

(6.5a)

(6.5b)

The mass and volumetric fluxes are then related as flows when change of phase occurs in the flow

passage:

(6.6a)

(6.6b)

Thus, (6.7)

From eqns (6.6) and (6.7) the void fraction can be related to quality as:

(6.8)

The drift velocity is defined as the difference between component velocity and average as follows:

(6.9a)

(6.9b)

According to Wallis (1969), drift flux j21 is the volumetric flux of either component relative to a

surface moving at the volumetric average velocity j.

(6.10a)

(6.10b)

From eqns (6.4) and (6.10a)

(6.11)

From eqns (6.3) and (6.10a)

(6.12)

(6.13)

(6.14)

Where u21 is the relative velocity of phase 2 with respect to phase 1. From eqn. (6.14), the drift

flux of component 2 with respect to 1 is proportional to their relative velocity. The drift flux can

be expressed in terms of component fluxes as:

(6.15a)

or

(6.15b)

Similarly,

(6.16a)

Or

(6.16b)

From eqns (6.15b) and (6.16b),

(6.17)

The symmetry evident from eqn (6.17) is an important and useful property of drift flux. From

eqn(6.15a)

(6.18a)

(6.18b)

rom equations (6.18) volumetric flux of component 1 is the sum of volumetric concentrations

times the average volumetric flux and a flux

due to relative motion.

Eqn (6.18) expresses the void fraction α as :

(6.19a)

or

(6.19b)

It may be noted that the first term on the right hand side of eqn (6.19b) depicts the void fraction

under no-slip condition which is the α predicted from the homogeneous flow theory and the

second term is a correction factor in terms of drift flux . This shows that the drift flux

theory has modified the expression of void fraction as obtained from the homogeneous flow

model by incorporating the ratio of drift flux and volumetric flux of component 2. In this way all

properties of flow such as mixture density, momentum flux, kinetic energy, etc. can be expressed

as the homogeneous flow value corrected by a multiplicative or additive factor which is a function

of the ratio of j21 to the component fluxes (j1or j2 ).

For example, the liquid holdup (HL = 1-α) for a gas-liquid mixture can be expressed as:

(6.20a)

(6.20b)

Where the correction term is

Similarly, the mixture density ρM can be expressed as :

(6.21)

In this case the correction term is an additive factor

The velocities of phase 1 and 2 can be expressed in terms of the velocity obtained from the

homogeneous flow theory (j) as:

(6.22)

(6.23)

The above equations show that the drift flux model follows the standard approach used to analyse

the dynamics of a mixture of gases or of immiscible liquids and provides a convenient way of

modifying homogenous theory to account for relative motion.

Kinematic Constitutive Relations:

The equations discussed above for evaluation of different mixture properties using the drift flux

model contains the drift flux term(j21) as a ratio of the individual volumetric flux. Therefore, the

equations cannot be solved without an additional equation to express j21 in terms of known input

parameters. In fact, the drift flux model is particularly useful when the relative motion can be

determined by a few key parameters and is independent of the flow rate of each phase.

There are two distinct approaches to obtain an expression for the relative motion. In the first

approach, the analysis is started from mixture field equations and then various constitutive axioms

are directly applied to the mixture independent of the two-fluid model. In the second method the

necessary constitutive equations are obtained by the reduction of the two-fluid model formulation.

Apparently the first approach seems to be more logical because it is a self sufficient and

independent formulation of the model for mixtures. However, in reality it has several limitations.

The main drawback arises from the fact that the two phases are generally not in thermal

equilibrium. Therefore it is not possible to specify a mixture temperature and suggest a simple

equation of state in terms of macroscopic mixture properties. In addition, the kinematic and

mechanical state between two phases is greatly influenced by the interfacial structures and

properties. In order to incorporate these effects into the drift flux model formulation, it is simpler

and more realistic to use the reductions from the two-fluid model rather than the former approach.

Accordingly, in the present chapter the momentum equation for each phase has been solved to

obtain the relative velocity law.

In the two-fluid model, the momentum balance equations for unit volume of the individual phases

in three dimensional vector form is:

(6.24a)

(6.24b)

For one dimensional flow, eqns (6.24a) and (b) can be resolved in the direction of motion to give:

(6.25a)

(6.25b)

In equation (6.25) b1 and b2 are the body forces per unit volume of components 1 and 2 which act

on the respective component, is the average pressure gradient and f1 and f2 are the ‘left over’

forces per unit volume of the corresponding phase which are incorporated to complete the

momentum balance equation. For example, if the aforementioned equations refer to elements of

incompressible Newtonian fluid containing only one component which is not undergoing phase

change,

And the usual results of viscous flow are obtained. They represent the average total force per unit

volume that is not contained in the pressure gradient eg components due to hydrodynamic drag,

apparent mass effects during relative acceleration, particle-particle forces, forces due to

momentum changes during evaporation/condensation, etc. To obtain a quasicontinuum model of

two phase flow, the f’s are calculated using an element of flow larger than particles, drops or

bubbles that occupy the flow field.

The way in which they are evaluated depends on the particular flow regime and conditions of the

problem.

Under steady state inertia dominant conditions the aforementioned equations become:

(6.26a)

(6.26b)

Where F1 and F21 are the equivalent f’s per unit volume of the whole flow field thus

(6.27a)

(6.27b)

FW1 and FW2 are drag forces from the duct wall on component 1 and 2 per unit volume of flow.

F12 arises due to mutual hydrodynamic drag. It acts on component 1 in direction of flow and in

opposite direction on component2. In the absence of wall effect and under steady state conditions

without phase change, the F’s arise due to mutual hydrodynamic drag only. Since action and

reaction are equal

(6.28)

In this case, the averaged void and velocity profiles become flat and the multiparticle dispersed

system in an infinite medium essentially reduces to a gravity and drag dominated one dimensional

flow.

Therefore the equations become:

(6.29a)

(6.29b)

On subtracting eqn (6.29b) from eqn(6.29a) we get

(6.30)

or

(6.31)

or

(6.32)

Equation (6.32 ) represents the balance between buoyancy and fluid dynamic drag. It shows that

in the absence of wall effect, the interfacial drag per unit volume F12 is a function of component

properties, interfacial geometry, void fraction and relative motion. Therefore for a given system,

(6.33)

Accordingly

(6.34)

And for a given system

(6.35)

From eqn (6.35), the relative velocity u21 as well as drift flux j21 between the two phases depends

upon the drag force acting at the interface as well as the interfacial geometry. The relative velocity

is thus expected to vary whenever the interfacial structure of the mixture changes. This suggest

that the need for separate equations to express relative velocity and j21 for different flow pattern.

From a survey of the past literature [Zuber,1964b,Zuber et al,1964,Zuber and Staub,1966], it is

evident that the drift velocity is a function of the terminal velocity u∞ of a single discontinuous

phase in an infinite medium and the void fraction of the continuous phase. The drift effect due to

the continuous phase has been accounted for by a linear constitutive law in the following form:

(6.36)

his gives:

(6.37)

Where u∞ and n have different values for different flow regimes and different two phase system.

In the absence of infinite relative velocity, the following limiting conditions must hold for the

variation of j21 with α as expressed by eqn (6.37):

(6.38)

(6.39)

In a dispersed two phase flow system, the drag correlation should be expressed in terms of drift

velocity and Reynolds number based on that flow. This suggests that the kinematic constitutive

equation on the relative motion between the phases can be best studied in terms of the drift

velocity of the dispersed phase u2j. It is noted that the drift flux model is useful when the drift

velocity is comparable with the total volumetric flux .

Thus it is particularly useful for the bubbly, slug and churn flow patterns. The values of u2j for a

few representative cases are as follows:

For the viscous regime,

(6.40)

Where

(6.41)

(6.42)

Where rd is the radius of the dispersed phase

For Newton’s regime

(6.43)

For distorted fluid particle regime where Nμ the

viscosity number is given as:

(6.44)

(6.45)

For churn turbulent flow regime

(6.46)

It may be noted that in the aforementioned expression for u2j, the proportionality constant is

applicable for bubbly flows and 1.57 for droplet flows.

For slug flow regime (6.48)

Graphical technique for solution of drift flux model:

From the above discussion, it is evident that a simultaneous solution of eqns (6.18 ) and ( 6.37)

shall yield the value of α and j21 which can then be used to estimate other mixture properties. In

order to simplify calculations, the present section proposes a graphical technique for the

simultaneous solution:

Equation (6.37) with the boundary conditions ( 6.47) and (6.48 ) when plotted as j21 vs α yield a

curve as shown in fig 6.1.

Fig 6.1 : j21vs α from equation 6.37 for vertical flow under gravity dominated situations

Eqn. (6.18) when plotted with the same axis is a straight line with intercepts j21=j2 for α=0 and

j21=-j1for α=1. Fig.6.2 depicts the line for different flow conditions namely upflow of two phases,

downflow of two phases and counter‐current flow of two phases with phase 2 moving up and

phase 1 down and vice versa (phase 2 moving down and phase 1 up).

Fig 6.2 : j21vs α from equation (6.18) for different flow directions of the two phases

It is expected that the point of intersection of the straight line of Fig. 6.2 with the curve of Fig.

6.1 should yield the value of j21 and α for the given set of phase flow rates Q1 and

Q2 where and . The superimposed curves are presented in Fig. 6.3.

Fig 6.3a

Fig 6.3 c

Fig 6.3d

Fig 6.3 : Solution for void fraction in vertical flow using the one dimensional drift flux

method for

a) Vertical cocurrent up flow

b) Vertical cocurrent downflow

c) Vertical countercurrent flow with liquid down and gas up

d) Vertical countercurrent flow with liquid up and gas down

Fig 6.3b

A close observation of the figure reveals that it not only enables us to locate the point of

intersection for different directions of two phase flow but also indicates the influence of changing

one flow rate keeping the other constant. For example, in co current upflow (Fig. 6.3 a), by

keeping j2 constant and increasing j1, we get the stable solution at a lower α and vice versa.

Similarly for counter current flow with gas flowing down and liquid flowing up (Fig. 6.3 d), the

curve and the line does not intersect at any point thus indicating that no solution is possible as

expected. Again for gas flowing up and liquid down (Fig. 6.3 c),there are either two solutions at

two different values of α or none depending on the flow rates. The two different flow situations

are:

1) At low α where bubbly flow is observed.

2) At high α where droplet or annular flow is observed.

If at constant j2 , j1 is increased steadily, the line depicting eqn (6.18) becomes tangential to the

curve inFig. 6.3 c. Henceforth no intersection between the line and the curve occurs indicating

that no stable solution is possible beyond the critical value of α in figure 6.3 c. Thus the condition

of tangency gives the limit of counter current operation with gas flowing up and liquid flowing

down and this point (in Fig. 6.3c) denotes the flooding point beyond which either the flow pattern

changes or the excess material is ejected.

It may be noted that curve (2) obtained from eqn (6.37) when generated for gas‐solid and liquid‐

solid systems cannot be extended beyond a critical α. This is termed as solid loading. Beyond

αcritical a sudden discontinuity is noted in the curve as shown in Fig. 6.4.

Fig. 6.4 : Dimensionless drift flux vs. particle volumetric concentration ε for

fluidisation(solid‐liquid) conditions

In the figure, α is the volumetric concentration of dispersed (solid) phase. If particles are

completely inflexible and incompressible, a packed bed with particle concentration αCritical must be

supported from above/below depending on the value of . Beyond this, solid packing becomes

important and solidsolid interaction cannot be neglected in eqn (6.26) which merely shows a

balance between fluid dynamic drag and buoyancy. So equation (6.37) needs to be modified to

account for F22, the interparticle interactions. It is seen that the value of αcritical beyond which the

curve of vs α cannot be extended depends upon the system. Usually 0.58 ≤a critical ≤ 0.62, but

for a foaming system,αcritical≤0.1. Tapping and shaking has been observed to give a higher α.

For very flexible particles, particle‐particle interaction can be neglected till α≈1 An analytical

solution of the flooding point can be obtained from the fact that eqn (6.18) is the equation of the

tangent to the curve obtained from eqn(6.37). Mathematically,

Or,

Thus

And

or

Eqns (6.51) and (6.53) express the corresponding flow rates at flooding point. Both the equations

show that the flooding point flow rates are functions of α for a particular flow distribution. They

can be evaluated for different values of α and used to generate the curve of Fig. (6.1).

A second way of representing different modes of operation is by expressing eqn (6.15) as:

Graphically the above equation represents a straight line with a slope of and

intercepts at j1=0 and at j2=0 for a given system as shown in Fig.6.5.

Fig 6.5 : Various regimes of operation in one dimensional vertical flow in terms of component

fluxes for a flow regime in which the drift flux is a function of void fraction but independent of

j1 & j2 and ρ1>ρ2

Sign Convention:

Since the solution of drift flux model formulation depends on the direction of velocity and flux of

the two phases it is often necessary to follow a sign convention in one dimensional flow. Although

there is no hard and fast rule for the sign convention, a consistent direction should be taken as

positive throughout a particular analysis. This should be mentioned at the beginning of the

formulation and followed throughout.

Usually one is more interested in describing the motion and concentration of component 2 rather

than component 1.Its volumetric concentration is denoted as α and while that of component 1 is 1-

α where 2 is assumed to be the discontinuous or dispersed phase and 1 the continuous phase.

Further j21 rather than j12 is selected for analytical purposes although the defining equations are

symmetrical. In addition, u∞ is chosen as the terminal velocity of a single dispersed phase particle

in an infinite medium rather than the velocity of the continuous phase which will bring the particle

to rest. From the same arguments, a positive sign is assigned to the direction of drift of component

2 or j21 is usually taken as positive.

When force balance is used to deduce j21, the direction of gravitational field is usually taken as

positive.

It may be noted that during instances when say a fluid is flowing around stationary particles in

fluidised bed, attention is focussed on the motion of the continuous component. Under such

condition the reverse sign convention is adopted and analysis is performed using HL and j12 rather

than j21. However these do not change the analytical techniques and the choice of direction is

governed according to convenience and ease of solution of a problem.

Corrections to the one dimensional model:

It may be recalled that eqn (6.18) is true for one dimensional flow where all the quantities are

assumed to be averaged across the cross section normal to the direction of flow or the quantities

are same for all local points in the flow. A rational approach to obtain a one dimensional drift flux

model is to integrate the three dimensional drift flux model over a cross-sectional area and then to

introduce proper mean values. A simple area average over the cross-sectional area for any

property p is defined as:

(6.55)

Where denotes the average over the cross section defined for the property p and the void

fraction weighted mean value is

(6.56)

Where k refers to the kth phase

From the aforementioned discussion, the assumption of one dimensional flow can be relaxed by

rewritingeqn (6.18) by averaging all the terms across the duct , viz

(6.57)

The second term in the right hand side of eqn (6.55) is a covariance between the concentration

profile and volumetric flux profile and can be expressed mathematically as:

(6.58)

It may be noted that

(6.59)

Since

(6.60)

Thus for estimating we need to find the local void fraction α and local mixture flux

at each point and multiply them and then integrate it over the flow field.

To keep matters simple, the past researchers have incorporated suitable correction factors by

taking the queue from single phase flow. In single phase flow, momentum flux in a pipe with

velocity profile but uniform density is:

(6.61)

This is not equal to

(6.62)

Usually for simplicity, a correction factor which is the ratio of and is

introduced such that it is equal to unity for truly one dimensional flow and not far from unity for

the general case.

Accordingly, for two phase flow a distribution parameter has been defined by Zuber and

Findlay(1967) as: (6.63)

where is the ratio of average of product of flux and concentration to product of averages or

(6.64)

Physically, this effect arises from the fact that the dispersed phase is locally translated with the

drift velocity with respect to the local mixture volumetric flux and not the average

volumetric flux . For example, if the dispersed phase is more concentrated in the high flux

region, the mean translation of the dispersed phase is promoted by a higher value of in-situ .

Accordingly equation (6.57) becomes (6.65)

is defined as weighted average velocity to distinguish it from where

(6.66)

It is more convenient to use rather than because can be directly related to input

parameters like overall flow rate and volumetric mean concentration which can be

obtained experimentally using quick closing valve technique or gamma ray scanning technique.

The equation relating to the measurable parameters is:

(6.67)

Thus equation (6.64) becomes

(6.68)

Or

(6.69)

Or

(6.70)

In order to evaluate one has to know:

1) Dependence of on α

2) Variation of α across cross‐section.

If is small compared to

(6.71)

where the second term of equation (6.71) is the in‐situ composition obtained from homogenous

flow theory.

This shows that in order to account for concentration variation but not relative velocity, one has to

multiply mean concentration calculated from homogenous theory by

Accordingly, the different mixture properties can be expressed as:

(6.72)

Assuming ρ1 and ρ2 are uniform within any cross‐sectional area. For most practical two phase

flow problems this assumption is valid since the transverse pressure gradient within a channel is

relatively small. The axial component of weighted mean velocity of phase k

is (6.73)

Where the scalar expression of velocity corresponds to the axial component of the vector.

(6.74)

And the volumetric flux is given by:

(6.75)

The appropriate mean drift velocity is

(6.76)

The experimental determination of drift velocity is possible if volume flow rate of each phase and

mean void fraction are measured. Thus

(6.77)

Where

Estimation of C0:

The value of C0 can be determined either from experimental data using eqn (6.64) or from

assumed profiles of void fraction α and total volumetric flux <j> using eqn (6.63). Equation (6.64)

suggests a plot of weighted mean velocity versus the average volumetric flux <j>. Generally

for two phase flow patterns with fully developed void and velocity profiles, the data points yield a

straight line particularly when the local drift velocity is constant or negligibly small. In that case,

the slope and the intercept of the best fit line provides the respective value of distribution

parameter C0 and weighted mean local drift velocity <u2j>. If concentration profile is uniform

across the channel, Co=1. In addition, if is negligibly small, the flow becomes essentially

homogeneous. In this case, the relation between mean velocity and flux reduces to a straight line

through the origin at a angle of The deviation of experimental data from homogeneous flow

line shows the magnitude of drift of dispersed phase with respect to volume centre of the mixture.

An important characteristic of such plot in that for two phase flow with fully developed void and

velocity profile the data points cluster around a straight line. This trend is particularly useful when

the drift velocity is constant or negligibly small. Hence for a given flow regime, the value of

C0 may be obtained from the slope of these lines whereas the intercept with the mean velocity axis

can be interpreted as the weighted mean local drift velocity . In order the determine C0 from

equation (6.63), power law profiles are generally assumed for void fraction α and total volumetric

flux j.

(6.78)

Where the subscripts o and w denote the corresponding values at the centre and the wall

respectively and r denotes the radical distance from the centre. Substituting these values in

equation (6.63), Co has been obtained as:

(6.79)

Zuber et al (1967) have shown that Co depends on pressure, channel geometry and flow rate. It is

also influenced by subcooled boiling and developing void profile. Ishii() has shown that for fully

developed bubbly flow

(6.80)

where is the Reynolds Number based on liquid properties. The values of Co as available in

literature for different conduit geometries and flow conditions are as follows:

For fully developed flow in a round tube

(6.81)

For fully developed flow in a rectangular channel

(6.82)

For developing void profile

(6.83) round tube

(6.84) rectangular channel.

For boiling bubbly flow in an internally heated annulus

(6.85)

In downward two‐phase flow for all flow regimes

(6.86)

(6.87)

Where, (6.88)

It needs mentioning that heat addition in two phase systems causes the void profile to change from

concave to convex due to wall nucleation and delayed transverse migration of bubbles towards

centre of the channel. Under these conditions, most of the bubbles are initially located near the

nucleating wall. Such a situation does occur even in adiabatic flow when small bubbles tend to

accumulate near the wall region at low void fraction. The concave profile is particularly more

pronounced in the subcooled boiling regime since in this case only the wall boundary layer is

heated above the saturation temperature and the core liquid is subcooled. In this region where

voids are still concentrated close to the wall, the mean vapour velocity can be less than the mean

liquid velocity because the bulk of liquid moves with high concentration velocity. However as

more and more vapour is generated along the channel, the void fraction profile changes from

concave to convex and becomes fully developed. A similar concave profile can also be obtained

by injecting gas into flowing liquid through a porous tube wall. For flow with void generation at

the wall either due to nucleation or gas injection, C0 assumes a near zero value at the beginning of

two phase flow. This is also evident from the definition of C0 in equation (6.63). With increase in

cross‐sectional mean void fraction, the peak of the local void fraction moves from near wall

region to central region. This results in an increased value of C0 with development of void profile.

Further for droplet or particulate flow in turbulent region, the volumetric flux profile is more or

less flat due to turbulent mixing and particle slip near the wall which increases the volumetric

flux. The concentration of dispersed phase also has a tendency to be uniform or a weak peaking

near the core.

This causes C0 to lie close to unity and the local slip becomes important.

The Separated Flow Model

The Separated Flow Model considers each phase individually and formulates separate mass,

momentum and energy balance equations for either of them. The balance equations contain

interaction terms to incorporate the transfer of mass, momentum and energy from the interface to

the ith phase. Thus for two phase systems, a complete formulation of the model comprises of six

differential equations and the interfacial transfer conditions. This makes the two-fluid model more

complicated than the drift flux model not only in terms of the number of field equations involved

but also in terms of the necessary constitutive relations. In addition, the accuracy of the

constitutive equations governs the usefulness of the model. This is particularly applicable to the

interaction terms since they determine the degree of coupling and consequently the transfer

processes in each phase. Without these interfacial exchanges in the balance equations, the two

phases are essentially independent and can be analysed by mere single phase flow equations.

The analysis is, in general, more useful when the two phases are weakly coupled and the inertia of

each phase changes rapidly. Due to separate conservation equations, the two-fluid model can

predict more detailed changes and phase interactions as compared to the drift flux model. It can

also account for the dynamic and non-equilibrium interaction between the phases. This is

particularly useful for the analysis of transient phenomena, local wave propagation and related

stability problems as well as flow regime transition. For general three-dimensional flow, the

twofluid model is better than the mixture model since it is extremely difficult to develop the

relative velocity correlation in a general three dimensional form.

Nevertheless, if one is concerned with the total response of the two phase mixture rather than the

local behaviour of the individual phases, the drift flux model is usually more effective for solving

problems. Further for practical applications, if the two phases are strongly coupled, the two-fluid

model introduces unnecessary complications into the system.

The present chapter deals with a general formulation of the two-fluid model as well as various

constitutive equations as closure to the set of equations. However, it may be noted that equating

the number of equations and number of unknowns does not imply the existence of a solution or

guarantee its uniqueness. It is merely a necessary condition for a properly set mathematical model

that represents the physical system to be analysed.

A useful starting point for two phase flow analysis is to formulate conservation equations for

mass, momentum and energy for each phase. Each pair of balance equations can then be added to

give the overall balance equation for the mixture. In order to develop the conservation equations, a

schematic of the generalized separated flow situation is depicted in Fig 7.1. Although it is

primarily applicable to the stratified and annular flow patterns, the basic equations are not

dependent on the particular flow configuration.

Fig 7.1 A schematic representing the flow of two phases under separated flow conditions

Continuity Equation: For two phase flow the model is characterised by two independent velocity

fields to specify the motion of each phase. The mass balance equations in differential form are:

(7.1a)

(7.1b)

where S1 and S2 are external sources of matter and are almost invariably zero. S12 is a source term

which represents the mass rate of phase change per unit volume.

For steady state flow, the equations become:

(7.2a)

(7.2b)

If each phase is incompressible, the mean density of each phase is constant and the equations

become

(7.3a)

(7.3b)

And for no phase change and incompressible fluids

(7.4a)

(7.4b)

Equations (7.4a) and (7.4b) can be used for low speed two phase flow without phase change.

Under these conditions, the kinematics is completely governed by phase redistribution namely

convection and diffusion.

For no phase change under steady state conditions but with change in area:

(7.5)

And (7.6)

This gives:

(7.7)

Since (7.8a)

(7.8b)

In one dimensional form, Equations (7.1a) and (b) after integration across the duct assumes the

following form

( 7.9a)

(7.9b)

dding equations 7.9 (a) and (b) and using the definition of mass flux as:

(7.10)

gives the separated flow equation of the mixture for no phase change as:

(7.11)

It may be noted that the form of the equations (7.9 and 7.11) resembles single phase compressible

flow.

Momentum Balance Equation: From principle of conservation of momentum,

(7.12)

Applying eqn (7.12) to phase 1 in section of channel as shown in fig. 7.1, we obtain for one

dimensional flow:

Rate of momentum out flow – rate of momentum in flow + rate of momentum storage

........ (7.13)

Since ,

Rate of creation of momentum = ........

(7.14)

And sum of force acting on control volume of phase 1

.... (7.15)

where, is the wall shear stress for fluid 1, is the rate of momentum transfer from gas to

liquid per unit interfacial area , or in other words the interfacial shear stress and Si is the interfacial

periphery. The first two terms in eqn (7.15) are the pressure force on the ends of the element and

the third term is the pressure force on the curved surface which occurs when there are changes in

the cross-sectional area of the liquid.

On equating the expressions of eqn (7.14) and (7.15), the momentum balance equation for phase 1

can be expressed as follows after division by and considering limit as and A=

Constant.

........

(7.16)

A similar equation for the gas phase is

............(7.17)

For one dimensional flow under steady state conditions, eqns (7.16 and 7.17) become:

(7.18a)

(7.18b)

The aforementioned derivation is based on the assumption that no mass transfer occurs between

the phases. In presence of mass transfer from one phase to the other, the interfacial shear can

be significantly different from in absence of interphase mass transfer where the two friction

factors can be defined in terms of friction factors and as:

.............(7.19)

........ (7.20)

can be evaluated from a knowledge of using the "equivalent-laminar-film model"

described by Bird et al (1960). The model postulates a laminar boundary layer in the gas phase

adjacent to the interface over which the velocity changes from the mean value for the gas phase

u2 to the liquid interfacial velocity (assumed to be u1 as a first approximation). The model gives

the following results relating the two shear stresses on the basis of the supposition that the

thickness of the boundary layer does not change appreciably due to mass transfer.

(7.21)

(7.22)

Where is the interfacial perimeter and is the rate of conversion of liquid to gas per unit

length.

For vapour-liquid two phase flow, for the evaporation of liquid or condensation of vapour

under conditions of thermodynamic equilibrium can be obtained as

........ (7.23)

is the heat flux from the surface of area S and h12 is the latent heat of vaporization.

For (low condensation rate), eqn (7.21) reduces to ........ (7.24)

Substituting eqn (7.24) in eqn (7.21) gives ........(7.25)

Eqn (7.25) implies that for evaporation, the interfacial shear is reduced by an amount

corresponding to the product of the evaporation rate per unit interfacial area and half the velocity

difference. For condensation is negative thus implying that the interfacial shear is enhanced

similarly.

Mixture Momentum Balance

The two equations (7.18 a and b) for the separate phases can be combined several ways. The

following two results are particularly convenient under steady flow conditions. If eqn (7.18 b)

divided by α is subtracted from eqn (7.18a) divided by (1-α), the equation of motion for the

mixture is obtained as

(7.26)

Where

And

Equation (7.26) does not include the pressure gradient term and can be considered to be the

relative motion equation, since it describes the difference between the rates at which the two

phase gain kinetic energy.

On the other hand, adding eqns (7.18a) and (b) gives:

........ (7.27)

Which is the momentum balance equation for the mixture with the three terms on the right hand

side corresponding in order to the gravitational, frictional and acceleration pressure drop as

mentioned.

Mixture Energy Balance:

The energy balance equation for the mixture can similarly be expressed in terms of quality as:

(7.28)

Where q and w are the rates of heat added and work done by the system.

Pressure Drop Calculation from known Input Parameters:

In order to calculate pressure drop from input variables, the acceleration pressure drop in eqn

(7.27), viz

(7.29)

can be expanded further by using the following relations derived in Chapter-4.

And

This gives the acceleration pressure drop as:

(7.30)

Where (7.31a)

(7.31b)

Accordingly, the acceleration pressure gradient becomes:

(7.32)

Now and or

(7.33)

Substituting in equation (7.27) becomes

or

(7.34)

(7.34)

If the conduit cross-sectional area changes with z, then the term

needs to be included in eqn (7.34).

This gives:

(7.35)

This is the separated flow model equivalent of eqn (5.9) derived from homogeneous conditions

and is widely used as a basis for pressure drop calculation in two phase flow. It has two unknowns

viz frictional pressure gradient and void fraction. Therefore, its solution requires two additional

equations besides the relationship between thermodynamic properties.

Condition of choking for separated flow:

The condition for choked flow is obtained from the momentum equation of compressible flow as

follows:

(7.36)

From analogy with the definition suggested by eqn (7.36), it is evident that the Mach number for

two phase separated flow (MaTP) can be expressed from the denominator of eqn (7.35) as:

This gives:

(7.37a)

Accordingly, the denominator of eqn (7.35) is of the form (1-MaTP2) and the choked flow

condition can be obtained by equating it to zero. Mathematically,

(7.37b)

Considering the equations:

And substituting them in eqn (7.37b) we obtain

(7.38)

This gives the condition of choking in absence of flashing. However, the equation is misleading

since the term is usually derived from a correlation obtained at moderate values of

pressure gradient when frictional forces dominate the inertia terms. For better results, it is

necessary to consider the two phases separately and deduce choking by combining the individual

momentum balance equations.

In the subsequent section, we shall consider the general case including phase transfer between the

two fluids and deduce the condition of choking from the two-fluid model.

The situation for phase change:

The general momentum equation for the individual phases in one dimensional separated flow in a

duct can be expressed as:

( 6.25a)

(6.25b)

If gravity is the only body force,

In the presence of phase change, f's would include (a) drag forces from duct wall on components 1

(Fw1 ) and 2 (Fw2) per unit volume of flow (b) drag forces between components F12 acting on

component 1 in the direction of motion and in the opposite direction on component 2. Further,

since the two components have different velocity, any phase change will also result in change of

momentum.

In order to evaluate the aforementioned term, we consider the mass rate of phase change per unit

length and the velocity change This gives the force necessary to account

for momentum increase due to phase change per unit volume as:

(7.39)

At this juncture it is difficult to decide what proportion of the force expressed by eqn (7.39) is to

be shared by the two phases. This assignment depends on the process involved

(boiling/condensation), interaction during the phase change and the hydrodynamic mechanism

which gives rise to F12 for example, drag forces on an evaporating droplet will depend on its rate

of evaporation. In general, a fraction η of the force represented by eqn (7.39) is assigned to stream

2 and (1- η) to stream 1. The choice of η is different for different systems. For an isentropic

process, η = ½ from condition of reversibility because unless the force is shared equally, the

equations will differ for evaporation and condensation and fluids cannot be made to return to their

initial states by reversing process in a symmetrical way. Accordingly, the two f's are:

(7.40)

This gives the two momentum equations for steady flow as:

(7.41a)

(7.41b)

Similar to eqns 7.18(a) and (b), eqns 7.41(a) and (b) can also be combined in several ways and the

following two results are particularly convenient.

Subtracting 7.41 (b) from (a) we get:

(7.42)

Eqn (7.42) contains no pressure gradient term and is the relative motion equation describing the

rates at which the two phases gain kinetic energy.

Again multiplying (7.41a) by (1-α) and (7.41b) by α and adding them, we get

(7.43)

This is the equation of motion for the mixture and is identical to eqn (7.27) which is obtained by

considering the momentum balance for both components taken together. Combining the last term

on the right hand side of eqn (7.43) with the left hand side, we get

(7.44)

Or

Or

(7.45)

Condition of choking for phase change considering two phases separately:

Combining the momentum equations with continuity equations, results which parallel

corresponding developments in gas dynamics can be obtained.

Considering component 2

(7.46)

Substituting eqn (7.46) in eqn (7.41b) we get

(7.47a)

Similarly for phase 1

(7.47b)

From equation of continuity:

(7.48)

By logarithmic differentiation,

(7.49)

Substituting eqn (7.49) in eqn (7.47a), we get

or,

Similarly for phase 1

(7.50b)

Both equations resemble one dimensional steady flow equations of single component flow, the

only points of difference being effects of phase change and additional degree of freedom

introduced by α.

In individual components, choking occurs when u1=c1 or u2=c2

Nevertheless, this does not correspond to compound choking of the combined flow since α can

adjust to local conditions.

To investigate compound choking has to be eliminated in equations (7.50a) and (b).

For this eqn 7.50 (a) is multiplied with α and eqn 7.50 (b) with (1- α). This gives:

….. (7.51a)

(7.51b)

Adding equations (7.51a) and (b), we get,

(7.52)

Considering F12 and to be independent of pressure gradient, the choking condition is:

(7.53)

All factors except those in parenthesis are positive. Therefore for choking to occur, either of the

ratios or have to be less than unity and the other greater than unity or one

phase has to be subsonic and the other supersonic.

For flashing, choking condition is modified since it introduces a dependence of

This gives:

(7.54)

And can be solved by evaluating if the thermodynamic path is known. The result depends on

η except when

Due to the practical importance, flow of boiling water has been studied more than any other two

phase system and the correlation based on Martinelli method is highly developed with tables and

curves generalised for convenient use. It expresses the frictional pressure gradient in terms of two

phase multipliers denoted as Φ2. The correlation is usually established for adiabatic flow with low

pressure gradients and serious errors can occur if it is extended for cases involving rapid phase

change and acceleration. Nevertheless, despite the limitations, most of the technical problems are

still solved using this method.

Assuming constant heat flux, negligible work, kinetic or potential energy changes or property

variation, the generalised expression to estimate pressure drop for two phase flow under separated

flow conditions is:

(7.55)

For x increasing linearly with z

(7.56)

If x=0 at z= 0 and = constant

(7.57)

Eqn (7.57) is considerably simplified for straight pipes with no area change (constant G) and can

be solved using graphical relations presented in Figs 7.2 to 7.4. Fig 7.2 presents vs X as

function of pressure where is:

(7.58)

The second term on the right hand side of eqn (7.57) is evaluated from Fig 7.3 which plots

parameter r2as a function of exit quality and pressure with r2 expressed as :

(7.59)

And the last term on the right hand side is estimated from Fig. 7.4 which expresses void fraction

as a function of quality and pressure.

It may be noted that more accurate results for high pressure water can be obtained from tables

developed by Thom to estimate as a function of steam quality and pressure.

Fig 7.2 : vs X as function of pressure Fig 7.3 Parameter r2 as a function of exit quality and

pressure

Fig 7.4 Void fraction as a function of quality and pressure

When inertia effects dominate:

In certain cases when a two component separated flow is accelerated rapidly through a nozzle,

inertia and pressure drops terms dominate (f's are zero) and in steady one-dimensional flow

without phase change, we get from eqns (7.41a) and (b)

(7.60)

In this case, velocity changes of two components can be related. If both components start with a

low velocity and density changes are small, or,

Similarly for phase 2,

Their final velocities after expansion are related by equation or, .

The expression has no universal validity but is true only under conditions of rapid expansion at

low Mach number.

Estimation of frictional pressure gradient and void fraction:

For the solution of eqn. (7.34), the two additional equations for frictional pressure

drop and void fraction can be derived by analyzing each component separately.

However, a common technique is to use empirical correlations to predict frictional pressure

gradient and void fraction. When one component is in contact with the wall, the appropriate wall

shear stress can be obtained from a correlation. The drag force (F12) between the components is a

function of the relative velocity and can be estimated as discussed in chap 6. However, a detailed

solution of the resulting equation is quite often a formidable task and a simplified approach is

assumed for an easy yet accurate solution.

The Empirical approach:

The simplified approach involves empirical correlations to express the frictional pressure gradient

and void fraction in terms of phase flow rates, fluid properties and conduit geometry. The method

of solution is mostly determined by form of the correlation. In order to use the correlations, the

frictional pressure gradient for two phase separated flow is expressed as

where is the equivalent wall shear for the two phase flow in question. Accordingly, the

basic differential momentum equation for simplified one dimensional approach becomes:

(7.39)

A comparison of eqn (7.39) with eqn (5.4) describing the momentum equation for homogeneous

flow shows that the aforementioned equation can be obtained by allowing different velocities for

the two phases i.e. by relaxing one assumption of homogeneous equilibrium flow.

The basic assumptions for the simplified one dimensional approach can thus be summarised as:

1. Constant but not necessarily equal velocities for the gas and liquid phase. (For equal velocities

the model reduces to the homogeneous flow model)

2. Attainment of thermodynamic equilibrium between the phases.

3. Use of empirical correlations /simplified concepts to relate two phase multiplier (Φ2) and void

fraction (α) to independent variables of flow

Where And

The Separate Cylinder Model as proposed by Lockhart and Martinelli for air-water

systems:

Historically, the most widely used correlation is due to Lockhart and Martinelli (1949). The

correlation expresses the two phase pressure gradient for air-water systems under adiabatic

conditions in terms of observed pressure gradient when either of the components flows

alone in the pipe under the following conditions:

(a) No phase change

(b) No acceleration pressure drop and

© Negligible body force effects

It may be noted that both and defined in terms of the total mixture flowing as liquid and

the liquid portion of the mixture flowing alone in the pipe can be used to find for two

phase separated flow. is advantageous in problems of boiling and condensation when

saturated liquid enters the bottom of the pipe and changes phase as it flows. However, Lockhart

and Martinelli have used because and bear a relationship to each other and it is

easier to evaluate them. Further, they confirm to certain limiting conditions, viz:

For no gas flow

and (7.44)

For no liquid flow

(7.45)

At the critical point and and can be related as follows:

(7.46)

Where, . Since at

critical point, the following equation can be expressed as follows for laminar flow

(7.47)

For turbulent flow using Blausius equation

or

(7.48)

Thus in a nutshell, the limiting conditions relating and under critical flow are:

for laminar flow and for turbulent flow .

The correlation is based on the following assumptions:

(1) The two phase do not interact with each another (This is the most severe assumption and is the

primary cause for the mismatch between the experimental and predicted values.

(2) They flow in two separate cylinders such that the cross-sectional area of the two cylinders is

equal to the pipe cross section or

(3) The pressure drop in each imagined cylinder is same as in actual flow

or (7.49)

(4) The pressure drop is mainly due to the frictional component or

(7.50)

From assumptions (3) and (4) can be calculated from single phase theory

since (7.51)

Lockhart and Martinelli have defined four flow patterns on the basis of flow behaviour

(laminar/turbulent) when the respective phases flow alone in the pipe. Accordingly, the flow can

be classified as (a) both phases in laminar flow (b) both phases in turbulent flow (c) liquid in

laminar and gas in turbulent flow (d) gas in laminar and liquid in turbulent flow. It may be noted

that the last combination seldom occurs and so it has not been studied extensively.

The correlation is expressed graphically as vs. corresponding to laminar/turbulent flow

of either of the phases in Fig.7.5.

Fig 7.5 vs.

The graphical representation can be expressed in terms of an analytical expression which can be

derived as follows:

We know (7.52)

Expressing the frictional pressures drop for the liquid occupied portion of the pipe as:

(7.53)

where, Dl is the hydraulic diameter of the liquid occupied portion of the pipe cross-section and

ρ1 and u1are the liquid density and in-situ liquid velocity respectively.

The friction factor for the liquid occupied pipe fraction fliq portion can be expressed in terms of

liquid Reynolds number as:

or (7.54)

Since

Dl is the hydraulic diameter of the liquid flow and can be related to the cross-sectional area (AL)

through which the phase is flowing any instant as:

(7.55)

The friction factor fl may be expressed by a Blausius type equation

Where (7.56)

Similarly, (7.57)

And

(7.58)

In eqns (7.55) and (7.58) γ and δ are the shape factors to account for the equivalent areas occupied

by the liquid and the gas phase.

Accordingly:

(7.59)

Again the pressure gradient for the liquid of the two phase mixture flowing alone in the pipe can

be expressed as:

(7.60)

Where (7.61)

(7.62)

(7.63)

(7.64)

Eqn (7.64) shows that the ratio of two phase frictional pressure gradient to that which would exist

if liquid phase flows alone in the pipe is a function only of fraction and shape of flow area

occupied by liquid phase. Similarly for gas phase, (7.65)

Where and are the equivalent diameters of the liquid and gas phase

respectively. If n=1 and γ=λ, then for laminar

flow

Or for laminar flow (7.66a)

And for turbulent flow considering Blausius equation

(7.66b)

Thus it has been proved that the separate cylinder model predicts the following correlation

between the two phase multipliers, viz

(7.67)

With m=2 for laminar flow And m=19/8= 2.375 for turbulent flow using Blausius equation.

The above correlation and fig. 7.5 shows that both the correlating parameters have two phase

pressure drop which is an unknown. This calls for a trial and error solution. In order to alleviate

this problem, Martinelli and co-workers argued that the two phase friction multipliers Φl2 and

Φg2 can be correlated uniquely as a function of parameter X2 defined as:

(7.68)

Where X2 is a measure of the degree to which the two phase mixture behaves as liquid rather than

gas. The resulting graphical correlation between is shown in Fig.(7.6). The figure

shows different curves depending on whether the phase flowing alone is in laminar or turbulent

flow and the multipliers are subscribed accordingly. For example, the multiplier applies to

the case in which the liquid phase flowing alone in the channel is in laminar (viscous) flow and

the gas phase flowing alone is turbulent. This can be used to calculate the pressure drop for

laminar /turbulent flow of both liquid and gas.

From Fig. (7.6) it is evident that a relationship should exist between α and (X2). Due to the greater

specific volume of the gas, the variation of void fraction is not symmetrical about X = 1 and tends

to be lopsided at low pressure. The correlation is presented in the same figure Fig. (7.6) and

shows that the relationship of α vs. X is independent of flow regime and at low pressure it can

mathematically be expressed as:

(7.69)

Fig 7.6 Lockhart ‐Martinelli's correlation

The correlation in essence balances frictional shear stress and pressure drop. Although the

correlation was specifically derived for horizontal flow without phase change or significant

acceleration, it is used to calculate both void fraction and frictional pressure drop even when these

effects are not negligible. However, one encounters progressively increasing errors as the

frictional component decreases in proportion to other terms. The past researchers have noted two

important observations about application of the separate cylinder model.

1) The curves of Φl and Φg vs X are not smooth. They show discontinuity of slope which may be

associated with change of flow pattern

2) There is a definite influence of mass velocity (G) on X-(Fig - 7.7) where the original Martinelli

– Nelson correlation line corresponded to

Fig 7.7 : Influence of Mass Velocity on Lockhart Martinelli Parameter

Other correlations:

Subsequently other correlations observed to be more accurate than the Lockhart and Martinelli

correlation have been proposed. Chisholm's Correlation: A simple and accurate analytical

representation of the Lockhart and Martinelli graphic relationships for the multipliers has been

proposed by Chisholm (1967).

(7.70)

(7.71)

Where c is a dimensionless parameter whose value is independent of quality but depends on the

nature (i.e. laminar or turbulent) of the phase-alone flows. The values for C as suggested by

Chisholm is given inTable 7.1.

Table 7.1 Values of c to fit the empirical curves of Lockhart and Martinelli

It may be noted that the correlation was developed for horizontal two phase two component

systems at low pressure (close to atmospheric) and its application to systems outside this range of

condition is not recommended.

Martinelli and Nelson Correlation:

Martinelli and Nelson (1948) noted that the Lockhart - Martinelli correlation is applicable for

horizontal two phase flow of two component systems at low pressure. It was represented in a

generalised manner to enable application of the model to single component systems and to steam-

water mixtures in particular. For prediction of pressure drops during forced circulation boiling

Martnelli and Nelson assumed the flow regime to be always turbulent-turbulent. The correlation

of frictional pressure gradient is expressed in terms of Φ2LO which is more convenient for boiling

and condensation than ΦL2. Assuming thermodynamic equation at all points and applying

ΦLtt correlation arbitrarily to atmospheric pressure stream – water flow, a relationship between

ΦL and Xn was established for initial pressure level by nothing that as pressure is increased

towards critical point, densities and viscosities of two phases became similar. The relationship

may be expressed as:

with C=1.36 (7.72)

From this, curves of Φ2LO vs x (mass quality) was plotted and the curves did not become

asymptotic to the correct value as the critical pressure was approached. Accordingly, they

proposed a revised multiplier correlation to fit data for steam-water mixtures over a range of

pressures. However, the relationship was found to be inaccurate for other fluids at the same ratio

of pressure p to the critical pressure pc.They attributed this to the fact that the Lockhart -

Martinelli correlation did not contain surface tension as a parameter and a part of the pressure

effect observed by the researchers was due to the variation of surface tension of water with

pressure.

Subsequently, Hetsroni () also showed that the systematic effect of mass flux where the Martinelli

and Nelson curve is approached at low mass flux (G<1360 kg/ m2 s) and the homogeneous model

fits more closely at high mass flux (G>2000-2500 kg/m2 s) has also not been accounted for in the

Martinelli correlation. Thus, the traditional Martinelli‐type correlations are inadequate in

representing a wide range of two‐phase flow pressure gradient data, and a large mean and

standard deviation are observed when the models are compared with large banks of experimental

data.

Baroczy correlation: This is the most widely used advanced correlation and extends the

Martinelli correlation to systems other than air‐water flows. It employs two separate sets of curves

plotting Φ2LO as a function of physical property index with mass quality x as

parameter for a mass flux of 1356 kg/m2s (Fig7.8) and a plot of correlation factor ωexpressed as a

function of the same physical property index for mass velocities of 339,678,2712 and 4068 kg/m2s

with x (mass quality) as parameter (Fig. 7.9a and b). Fig. (7.9) serves to correct Φ2LO obtained

from Fig. 7.9a to the appropriate value of mass velocity. This gives the expression of frictional

pressure gradient as:

(7.73)

The Baroczy correlation has the disadvantage of being graphic in nature.

Fig 7.8 Φ2LO as a function of physical property index with mass quality x as parameter for

a

mass flux of 1356 kg/m2s

Fig 7.9 (a)

Fig 7.9 (b) Correlation Factor ω ωexpressed as a function of the same

physical property index for mass velocities of 339,678,2712

and 4068 kg/m2s with x (mass quality) as parameter

Chisholm's Method ‐

A correlation that fits Baroczy curves quite well and extends over the range of data covered is that

of Chisholm (1973), viz

(7.74)

Where n is the power in the friction factor‐Reynolds number relationship (0.25 for the Blausius

equation) and the parameter B is given by (7.75)

(7.76)

(7.77)

where (7.78)

and and are the pressure gradients for the total mass in the channel

flowing with gas phase and liquid phase properties, respectively.

Friedel's correlation: In recent studies, Friedel (1979) has compared a data bank of 25,000 data

points with existing correlations and with a new correlation that he developed, as follows:

(7.79)

Where (7.80)

(7.81)

(7.82)

(7.83)

(7.84)

and and are the friction factors for the total mass flux flowing with gas and liquid

properties, respectively. For this particular correlation, is given by

(7.85)

Recent evaluations (based on Heat Transfer and Fluid Flow Service proprietary data bank) have

led to the following tentative recommendations with respect to the published correlations

(Whalley 1980):

1. For the Friedel (1979) correlation should be used.

2. For and the Chishlom (1973) correlation should be used.

3. For and for the Martinelli correlation (Lockhart and Martinelli 1949;

Martinelli and Nelson 1948) should be used.

It should be emphasized that these recommendations are tentative and may change as further data

appear and new correlations are developed. However, the fundamental fact remains that, unless a

better physical basis is developed, an irreducible error is involved in the prediction of two‐phase

pressure drop. It may also be noted that the correlations discussed above were developed primarily

for round tubes. They can be applied to other channel shapes by introducing the appropriate

equivalent diameter instead of the tube diameter (the equivalent diameter is given by four times

the crosssectional area divided by the wetted perimeter). An important case is that of cross flow

over tube banks particularly that occurring in the context of shell‐and‐tube heat exchangers.

Measurements and correlations of pressure drop in shell‐and‐tube heat exchangers are discussed

by Grant (1975). For the cross‐flow zone, Grant suggests the use of the Chisholm(1973)

correlation with values B and n as given in Table 7.2. For the window flow zone, Grant suggests

the following expression for the pressure drop multiplier,

(7.86)

where B=0.25 for vertical up‐and‐down flow and for horizontal side‐to‐side flow.

Table 7.2: Values of B and n as suggested by Grant (1975)

Flow type B n

Vertical up and down spray and bubble 1 0.37

Horizontal side to side spray and bubble 0.75 0.46

Horizontal side to side stratified and stratified

spray 0.25 0.46

Over the past 30 years the deficiencies of earlier correlations have become apparent due to

availability of more data. As a result further correlations have been proposed. The process is still

continuing and it reflects the fact that, for a situation as complex as two‐phase flow, it is very

difficult to formulate relationships that have a general physical basis. The main difficulty is that

the empirical correlations are based on assumption that the frictional pressure gradient is a

function only of channel cross‐sectional geometry, mass flux, and physical properties. However,

in two phase flow, the effects of flow development are considerable, and any wide‐ranging data

bank on two‐phase flow contains data with a variety of inlet configurations and channel lengths,

which will give a range of pressure gradients for the same nominal conditions. The lengths

required to reach equilibrium in two‐phase flow correspond typically to several hundred

diameters, and thus most experiments never reach equilibrium conditions. Furthermore, in

practical situations, equilibrium conditions themselves are not necessarily relevant, particularly

when there is a phase conversion (i.e., evaporation or condensation) along the channel. Here it

may often be preferable to use a more basic physical model (although that too presents

difficulties).

Flow Regime Based Models

Subsequent to the discussion on simple analytical models, we now attempt to apply the concepts

developed for individual flow patterns. We confine ourselves to the bubbly, slug and annular flow

patterns during gas-liquid flow through pipes. Accordingly, subscripts L and G instead of 1 and 2

denote the two phases in this chapter. However a student is free to adopt symbols of his choice but

he should specify them at the beginning of the analysis and use consistent nomenclature

throughout. A similar analysis will be applicable for the corresponding flow distributions during

liquid- liquid flow. In that case subscripts 1,2 or L1and L2 can be used as per convenience.

8.1 The Bubbly Flow pattern:

This distribution is characterised by a suspension of discrete bubbles in continuous liquid as

shown in fig. 8.1

Engineering applications include bubble columns for promoting mass transfer, high pressure

evaporators, flash distillation columns etc.

It is noted that the bubbles exhibit a drift relative to the continuous phase at low velocities. The

relative motion reduces as we increase the mixture velocity and at high phase velocities, the two

phases present a uniform homogenous appearance with a dense dispersion of fine bubbles which

travel at the same velocity as the continuous liquid medium. The bubbly mixture under these

conditions can be analysed by the homogenous flow model while at lower phase velocities, when

there is substantial relative

motion between the phases, the drift flux model is a better approximation. We further note that the

wall shear and momentum fluxes can be neglected at low phase velocities when drift flux model is

suitable but have a significant contribution at high fluid flow when the homogenous flow model is

accurate. Thus even the entire range of the relatively simple bubbly flow can not be analysed by a

unified model and this has often prompted researchers to distinguish the two types as "dispersed

bubbly" flow at high mixture velocities under no‐slip condition and "bubbly" flow at low phase

flow rates.

8.1.2 The analysis for dispersed bubbly flow:

As mentioned earlier both friction and momentum effects are important under these conditions,

and the homogenous flow analysis gives:

Where

for (5.23)

Eqn (5.23) is applicable at low Reynolds number when the bubbly mixture in laminar flow is

Newtonian. At higher bubble concentration, the mixture rapidly becomes non‐Newtonian and

exhibits a yield stress, a decreasing apparent viscosity with increasing shear rate, etc. Foams

exhibit considerable rigidly at high void fraction and bubbles behave like atoms in a crystal. The

influence of α on μTP is also enhanced in contaminated liquids due to tendency of bubbles to

behave as solid spheres. For turbulent flow, we can use liquid viscosity in Reynolds number and

employ single phase flow correlations. fTP can be taken as 0.005 till Reynolds number of 105 . The

remaining analysis can be performed based on the discussions of chapter 5.

8.1.3 The analysis for bubbly flow at low mixture velocities:

If component fluxes are not very large compared to drift flux, the in‐situ velocities can be

expressed as:

Where And

This gives

And the momentum flux:

Thus, in order to predict void fraction, momentum flux and pressure drop, one need to have an

estimation of j21 As mentioned in chapter 6, j21 can be expressed as

(6.37)

In equation (6.37) u∞ is the velocity of a single bubble in an infinite medium and n is a constant

based on a suitably defined Reynolds number. From the aforementioned discussion, it is evident

that bubbly flow analysis needs a prior estimate of u∞ and n. Since the past literature reveals

bubbles with a wide variation of size and shape ( as depicted in fig 8.2a and b) to exhibit different

rise velocities in infinite medium, a unique value of u∞ and n cannot be used for the entire range of

bubbly flow. For example the

smallest bubbles which are approximately perfect spheres due to dominant effect of surface

tension on their shape, have u∞ from Stokes law as

(8.1)

This is also valid for solid spheres and is applicable for bubbles when it is assumed that liquid

velocity goes to zero at bubble surface. For fluid spheres containing liquid with viscosity μg and

having a completely non rigid surface

(8.2)

If (8.3)

in the complete absence of impurities which tend to collect at bubble surface and give a certain

resistance to shear stress. Generally in most practical cases some contamination is present and

u∞ lies between the values given by eqns (8.1) and (8.3).

Again when bubbles are very large, the effect of surface tension and viscosity is negligible and

u∞can be expressed as: (8.4)

Where Rc is the radius of curvature in the region of bubble nose. In terms of volume of gas in

bubble, eqn (8.4) becomes

(8.5)

when the shape of the bubble is approximately a spherical cap with an included angle and

a relative flat tail.

An alternate definition of equivalent radius is Rb which is the radius the bubble would have if it

were spherical. Accordingly, (8.6)

Which gives:

(8.7)

For bubbles of intermediate size, the effects of a) surface tension b) liquid inertia c) viscosity d)

cleanliness e) and whether bubbles rise in straight lines or describe a spiral path or oscillate are

important and needs to be accounted for.

For this, several correlations are available. The range of applicability of each equation is

determined in terms of dimensionless groups (proposed by Peebles and Garber)

and expresses rise velocity and n for the different ranges as presented in Table

8.1.

In region 4, a better value of the constant for gas‐liquid systems is 1.53. In this region, the

terminal velocity is independent of bubble size. Region 5 is the transition region between bubbly

and slug flow. In this region the three dimensional effects become important and there is

significant entrainment of bubble in each other's wake. Due to streamlining/channelling, the

relative velocity increases with increase in number of bubbles and the value of n reduces from

unity. The region is commonly termed as"churn‐turbulent "bubbly flow. From the table, an

estimate of u∞ and n needs knowledge of equivalent bubble radius Rb which is the radius of a

sphere with the same volume as the bubble.

It is noted that Rb is a function of the way the bubbles are formed. Accordingly, different

expressions of Rb depending on the source of bubble formation is mentioned below.

Bubble formation at a circular orifice facing upward in a stationary fluid:

• Assuming an approximately spherical bubble of radius b R attached to an orifice of radius

Ro by

a cylindrical neck, the dimension of the largest bubble which can be in static equilibrium

is

expressed as: (8.8)

Radius of bubble formed by blowing through‐small orifice at low flow rates is

approximated as:

A more accurate version derived from experimental data gives:

(8.10)

For bubbles formed at a finite rate, many other factors namely all liquid and gas properties, details

of orifice design, gas supply, etc. become important. When gas flow rate through orifice is

increased, bubble size at first increases since the bubble takes a finite time to break from the

orifice after reaching size given by equation (8.10). For a system in which flow rate through

orifice is carefully maintained, the constant bubble size at departure is predicted by knowing the

time for which the bubble remains attached to the orifice. This time calculated from equation of

motion of rising bubble gives the volume of a bubble at detachment in an inviscid liquid as:

(8.11)

Where Qg is the gas volumetric flow rate through orifice. In viscous

liquids (8.12)

For very large gas velocity, bubbles no longer form individually but gas leaves orifice as a jet

which eventually breaks into individual bubble. The condition for formation of gas jet is then

given by:

(8.13)

ug = gas velocity through orifice

Generally bubbles formed in this way have Rb=2Ro

In commercial applications bubbles are not formed at a single orifice but a group of orifices or a

porous plate. Then single orifice theory is useful only as a first approximation.

Formation of bubble by Taylor instability:

For bubbles formed by detachment from blanket of gas or vapour over a porous or heated

surface, (8.14)

Although the formation is not identical with "Taylor instability" of a fluid below a denser

fluid, the physics is similar and the bubble is scaled by the same dimensionless parameter

This is particularly important to describe film boiling.

Bubbles are formed by evaporation of surrounding liquid or release of gases dissolved in

liquid.

Bubbles form around nucleation centres which are impurities in fluids and pits, scratches

and cavities on wall.

The equivalent diameter of the bubble Db just large enough to break away from a

horizontal surface is given by:

(8.15)

Where β is the contact angle in degrees

This is valid only for the quasi‐static case and not for bubbles formed during boiling.

Influence of shear stress:

In forced convection or mechanically agitated systems, the bubble size is determined by

shear stress which determines the size of bubbles which form away from the point of formation as

well as the maximum bubble size which is stable in flow.

A balance between surface tension forces and fluid stresses i.e. a suitably defined Weber

number determines bubble size. The maximum bubble size stable in flow is then given by:

(8.16)

Where is the mechanical power dissipated per unit mass.

Influence of containing walls:

When bubbles rise in a finite vessel, its velocity is generally lower than predicted from Table 8.1

or .

In a tube of diameter D,

The functional form varies with bubble characteristics. In region 5 where the large bubbles in

inviscid liquid behave like slug flow bubbles, the functional forms are as follows (Wallis, 1969):

(8.17a)

(8.17b)

(8.17c)

In viscous fluids for bubbles behaving as solid spheres:

(8.18)

And for fluid spheres when

(8.19)

If the bubbles behave as slug flow bubbles and obey the equation

(8.20)

A reasonable fit to eqn (8.19) which is tangential to eqn (8.20) at is

(8.21)

This may be used to estimate for .

Influence of Vibrations:

If bubble is placed in a vertical vibrating column, it experiences a downward force opposing

gravity. If circumstances are suitable, then the bubble can be oscillated steadily about a mean

stationary position or even forced to move downward.

8.2 The Slug Flow Pattern:

Slug flow is one of the basic flow patterns that characterize gas/vapour –liquid flow in closed

conduits. It occurs over a fairly wide range of gas and liquid flow rates in small and medium size

tubes. The most important characteristic of slug flow is its intermittent nature, which is due to a

unique phase distribution. The gas flows as a series of bullet‐shaped Taylor bubbles that are

separated by liquid slugs. These slugs span the entire tube cross‐section and contain dispersion of

small gas bubbles.

The

dispersion is denser in the wake region just below the elongated bubble as compared to the

remaining portion of liquid slug. The wake is formed by the mixing of the liquid film in the

Taylor bubble region with the liquid slug behind it. As a result, both the void fraction, and hence

the two‐phase mixture density, and the pressure at any tube cross section vary periodically at an

average frequency that is governed by the Taylor bubble and liquid slug velocities. Since the

hydrodynamics of flow are different in a vertical and a horizontal pipe, they shall be treated

separately.

8.2.1 Analysis of slug Flow in a vertical pipe:

A schematic of the slug flow pattern in a vertical pipe is shown in Fig.8.2.

Fig. 8.2 Slug Flow Pattern in a vertical tube

Considering the intermittent character, the analysis is best done by considering the flow passage to

be divided into several “unit” cells where each cell consists of one bubble and part of the liquid

slug on each side of it as shown in the figure.

For a simplified analysis, we begin with the simplest representation of slug flow comprising of

Taylor bubbles and pure liquid slugs stacked one above the other in the flow passage. The liquid

flows downward as a thin annular film in the Taylor bubble region and subsequently translates

upward as liquid slugs.

For a specified total volumetric flux the mean velocity of liquid in the slug is simply

The bubble dynamics is then a function of j, the corresponding velocity profile, bubble length,

pipe geometry and fluid properties. Apart from the effect of the wake from the preceding bubble,

the velocity profile in the liquid slug is a function of pipe roughness and Reynolds number

expressed as:

This shows that the bubble dynamics is dependent on j but not on the individual fluxes jL and jG of

the liquid and gas. Again if each unit cell is independent, bubble dynamics is not a function of

void fraction because dynamics of nose and tail govern bubble motion entirely and bubble length

is not important. Therefore the bubble velocity ub should be independent of α and a function of j

only.

Now considering unit cell, the bubble rises with a velocity ub where the liquid ahead of it

translates upward at a velocity j. The drift velocity with respect to the preceding slug, ugj of the

bubble can be expressed as:

(8.22)

Since the entire gas translates as Taylor bubble, ub=ug and ug is the in-situ velocity of the gas.

From chapter 6 on drift flux model, it was deduced that for one dimensional flow with no wall

shear effects, the drift flux and drift velocity is a function of void fraction α only and does not

depend on j for a particular system and tube characteristics.

Both the aforementioned conditions can be satisfied only when ugj varies neither with j nor with α.

In other words ugj is constant.

This implies that the value of ugj calculated for the special case of a single bubble rising in

stagnant liquid (u∞) can be used for all values of liquid slug velocity (j).

Mathematically, this gives

ugj=u∞ ( for no net liquid flow) (8.23)

Therefore for all values of α , the definition of drift flux gives

(8.24)

With liquid flow ahead of bubble, we obtain (8.25)

From the above discussion, the void fraction for each unit cell can be expressed as:

(8.26)

This value of α can then is used to predict the pressure drop and other hydrodynamic parameters

of slug flow in a vertical pipe. Equation (8.26) shows that the only parameter necessary to

estimate α for the slug flow pattern is uα,the velocity of single Taylor bubble rising through

stationary liquids.

8.Bubble rises through a denser liquid due to buoyancy. Therefore, velocity u∞ with which a

single bubble rises through stagnant liquid is governed by interaction between buoyancy and other

forces acting on bubble due to its shape and motion. If viscosity of gas/vapour in bubble is

negligible, there are only three forces besides buoyancy. These are liquid inertia, liquid viscosity

and surface tension. A balance between buoyancy and these three forces can be expressed in terms

of three dimensionless groups, namely

D is the characteristic dimension of the duct cross-section.

A general solution is the function of the three parameters which may be combined to generate new

dimensionless groups as long as bubble length is greater than tube diameter and the length does

not affect rise velocity. Alternatively bubble equivalent diameter must exceed 60% of the tube

diameter. Nevertheless, the simplest solutions are obtained when one of the dimensionless groups

govern the motion.

The limiting Cases are –

(a) Inertia dominant, when viscosity and surface tension can be neglected. Under this condition,

Froude number the first dimensionless group is important. Mathematically,

(8.27)

or (8.28)

It may be noted that u∞ can be obtained from the above equation for bubbles of different lengths

if the top end of the tube is closed. There is an apparent dependence of u∞ on bubble length in

tubes open at the top since gas expands as it rises in hydrostatic pressure gradient. This expansion

leads to a non- zero value of j ahead of bubble and rise velocity is augmented by the velocity of

the liquid. For tubes closed at top, there is no liquid motion ahead of bubble and a more or less

constant value of k1 is obtained.

Different researchers have obtained different values of k1 For vertical round tubes, the most

widely accepted value of k1=0.345. Not much work has been done to estimate k1 for non-circular

channels.

Griffth has shown for rectangular channels k1 is a function of

and (8.29)s

where Db is the larger dimension of the rectangular channel.

For bubble rising in an annulus with D0 as the outer diameter, Griffith has proposed k1 a function

of . The past researchers have also reported that a bubble rises faster for smaller annulus

spacing. For tube bundles, the characteristic dimension is the overall housing diameter rather than

the diameter of individual tubes.

(b) Viscosity dominant flow where u∞ is obtained from the second dimensionless group. This

gives (8.30)

or (8.31)

Where k = 0.01 (Wallis, 1969) and 0.0096(White and Beardmore, 1962) for vertical round

tubes.

© Surface tension dominant: A bubble does not move at all when surface tension dominates. The

static interface adopts a particular shape such that the hydrostatic forces are completely balanced

by the surface forces. For round vertical tubes this happens when

(8.32)

An alternative group often used is Bond number defined as

The general case is governed by three parameters and can be presented as a two dimensional plot

of sany two chosen dimensionless groups with a third independent dimensionless group as

parameter. The actual manner of choosing groups is according to convenience. For example, the

dimensionless bubble velocity k1 may be plotted as a function of inverse viscosity number Nf,

which is obtained by eliminating u∞ from the first two groups and can be expressed as:

(8.33)

A convenient third group is obtained by eliminating both U∞ and D to get only fluid properties

and g. This group termed as Archimedes number and expressed as:

(8.34)

is a constant for a given fluid at a particular temperature.

The aforementioned plot is shown in figure 8.3. The three asymptotic solutions viz

1) Inertia dominant: and

2) Viscosity dominant : and

3) Surface tension dominant : are clearly satisfied in the figure.

Fig. 8.3 General dimensionless representation of bubble rise velocity in slug flow

Alternative methods of plotting have also been used by combining the dimensionless quantities in

various ways. For example, a property group used by White and Beardmore (1962) is

When gas density is low compared to liquid density, this gives: (8.35)

A plot of k1 versus Eo as function of Y is shown in figure 8.4.

Fig 8.4 An alternative representation of Fig 8.3 .

A general mathematical equation in the intermediate range of Nf when surface tension effects are

negligible is

(8.36)

Surface tension effect can be incorporated in eqn (8.36) by a further modification as:

(8.37)

where m = m (Nf) and for Nf>250 m=10

For 18< Nf< 250 m=69 Nf−0.35

For Nf < 18 m = 25 (8.38)

Equation (8.37) is a correlation for bubble rise velocity in terms of all relevant variables. In the

inviscid region for large Nf,

It may be noted that different bubble shapes are reported for the various regimes. In a highly

viscous fluid (Nf < 2), the bubble nose as well as the tail are rounded and the bubble wake is

laminar. For a fluid of low viscosity (Nf > 300), the bubble tail is flat and the wake is turbulent.

8.2.3 Corrections to the expression of uα

The bubble drift velocity ugj is not strictly constant as assumed above. It is influenced by

(a) The velocity profile in liquid slug which is a function of j or more strictly

(b) Wake of preceding bubble.

To account for these effects, a modified expression of rise velocity is

(8.40)

Were C1 is a measure of fact that bubble does not simply move relative to average velocity j but a

weighted average velocity. It has the same significance as C0 in drift flux model but the physical

reasoning behind the deviation of these coefficients is not identical. C2 is a measure of change in

relative velocity due to approaching velocity profile

For fully developed flow in circular pipe (Rej > 8000), C1=1.2 C2=1 (8.41)

For laminar flow, accepted correlations for C1 and C2 are not available. A commonly accepted

form is:

(8.42)

Where Ls= liquid slug length ( bubble separation length)

For high velocity the limiting value of

In channels where boiling occurs

With C1 and C2 , the modified equation of void fraction is

(8.43)

8.2.4 The pressure gradient for ideal slug flow in vertical pipes:

In view of the assumption of negligible wall shear stress the pressure gradient in

non accelerating flow in vertical pipes can be expressed as:

(8.44)

where (8.45)

In the above equation: ub= u∞ +j where u∞ is the velocity of bubble in stationary liquid. This is

applicable for Nf >300 and when frictional pressure drop estimated from homogeneous theory is

small compared to the gravitational pressure drop.

The correction necessary for significant wall shear is difficult to estimate. The average shear stress

is either positive or negative since some liquid is actually running down the wall around the

bubble. A possible procedure is to calculate shear stress in liquid slug from single phase friction

factor based on j and neglecting the wall shear stresses around the Taylor bubble. This is justified

considering the fact that the weight of the liquid in the film is supported by the wall shear in this

region. As a result, the wall shear does not contribute to the total pressure gradient. In addition,

the bubble can be considered to be a region of constant pressure since the density and viscosity of

the gas phase is negligible as compared to the liquid phase and the bubble can be approximated as

a cylinder of constant curvature which gives negligible interfacial shear between the bubble and

the film.

The wall shear in the liquid slug can be expressed as: (8.46)

Where (8.47)

Assuming that approximately a fraction (1-α ) of the pipe length is occupied by liquid slug , eqn

(8.44) with the addition of drag on liquid slugs become,

(8.48)

If

This gives: (8.50)

The above equation is the equation for the homogenous flow frictional pressure drop in which

mean density is calculated from < α > expressed as:

(8.51)

Note: The acceleration pressure drop can be treated as in bubbly flow but since slug flow is not as

homogenous as bubbly flow, choking conditions obtained for bubbly flow will give erroneous

results for slug flow. Moreover, since some of the liquid is moving with velocity j in the slugs and

the rest is moving downward in the falling film, the assumption of uniform liquid velocity is

incorrect. Further studies are required in this area.

8.2.5 Corrections for long bubbles:

For long bubbles, a substantial amount liquid exists as falling film in the Taylor bubble region as

shown inFig. 8.3. Falling film around long bubbles in a vertical tube . This needs to be included in

the expression of void fraction which is otherwise expressed as

Fig.8.3: Falling film around long bubbles in a vertical tube

This needs to be included in the expression of void fraction which is otherwise expressed as

(8.52)

In this case area of bubble is not equal to area of tube in Taylor bubble region. If δ is liquid film

thickness, then area occupied by Taylor bubble is

(8.53a)

Which gives the volume occupied by Taylor bubble in unit cell as:

(8.53b)

Thus the actual void fraction for slug flow with long Taylor bubbles is:

(8.54)

The liquid film reaches a terminal velocity within a short distance from the nose tip. The film then

attains a steady speed and a uniform thickness which can be calculated from falling film theory.

The weight of the film is balanced by wall shear stress and hence does not contribute to the

pressure drop as mentioned above. Nevertheless, this reduces the net amount of liquid contained

in the liquid slug. Both these effects can be accounted for by treating the liquid in the film as if it

were gas. This gives the effective void fraction α' as:

(8.55)

or,

(8.56)

And

(8.57)

as obtained from equation (8.57) should be substituted in pressure drop equations where the

film thickness δ is calculated as follows:as obtained from equation (8.57) should be substituted in

pressure drop equations where the film thickness δ is calculated as follows:

From equation of continuity, total volumetric flow across any pipe cross‐section is

constant. Therefore, at section AA' of Fig. 8.3

(8.58)

Where Qg and Qf are the volumetric flow rates of the gas and the liquid film in the Taylor bubble

region

or (8.59)

or, (8.60)

where jf is the volumetric flux of the liquid film.

or (8.61)

which relates or

Q'f is also related to γ by falling film theory valid for

when (8.62)

The falling film theory also proposes several equations relating to for different ranges

of

8.2.6 Horizontal Slug Flow:

There is no drift flux during slug flow through horizontal pipes (Fig. 8.4) due to buoyancy effects.

Therefore u∞ loses its significance although bubbles do not move with the same average velocity

as liquid or bubble velocity ub ≠ j. Since there is no pressure drop along bubble length, liquid film

on the wall is substantially stationary with mean thickness δ .

Fig. 8.4: Horizontal slug flow

This gives the cross-sectional area occupied by the Taylor bubbles as: (8.63)

For continuity of volumetric flux at any cross‐section, (8.64)

Or (8.65)

For (8.66)

From eqn (8.66) and (8.67)

In the absence of effects due to gas viscosity and inertia and for bubbles which move

independently, factors influencing bubble velocity can be combined into the following

dimensionless groups.

=liquid velocity in slug/Bubble velocity

= Reynolds number of liquid slug

=Viscous Force/Surface Tension Force

= Buoyancy Force /Surface Tension.

These are analogous to groups which describe balance between inertia, viscosity, surface tension

and buoyancy in vertical flow.

At very low velocities,

and (8.68)

At high velocities and high

(8.69)

Or

Or, (8.70)

The pressure drop in liquid slug can be calculated by single component flow techniques. Pressure

drop along cylindrical part of bubble is zero. Therefore, the only additional pressure drop per

bubble is due to effects at nose and tail. Since different number of bubbles can make up the

overall flow rates in the same pipe, the pressure cannot be determined unless bubble length is

specified separately.

Assuming pressure drop per bubble equals pressure drop in a length of about four pipe diameters,

the pressure drop for a typical unit cell comprising of one bubble and one slug for all Reynolds

number is

(8.71)

Ls is the liquid slug length. This gives the mean pressure gradient as

(8.72)

The evaluation of the last term depends on the condition of the problem. If volume of each bubble

Vb is known, length of unit cell can be obtained from knowledge of void fraction as

(8.73)

or (8.74)

For long bubble

Or (8.75)

or (8.76)

Thus (8.77)

(8.78)

The aforementioned analysis is also applicable in vertical pipes for low values of

or when viscous effects are important or buoyancy effects are small compared to

viscous surface tension effects.

8.3 Annular flow:

A schematic of the annular flow pattern as shown in Chapter-2 reveals a continuous liquid film

along the pipe wall while gas flows as a central core which may or may not contain significant

number of droplets. It is the predominant flow pattern in evaporators, natural gas pipelines and

steam heating systems. In the following section, the pattern is analysed based on two‐ fluid

formulation.

Considering the one dimensional momentum equations for the gas and liquid phases

(8.79a)

and (8.79b)

where for vertical upflow (8.80a)

(8.80b)

and for horizontal flow

(8.81)

If Dg is the diameter of the gas core flowing through a pipe of diameter D, then the gas void

fraction α is given as:

or (8.82)

This gives the equivalent f's per unit volume of the whole flow field as:

(8.83a)

And

(8.83b)

Accordingly (8.84a)

and (8.84b)

Substituting eqns (8.80) and (8.84) in eqns (8.79) we get the equation of motion in one

dimensional flow for two fluids in annular flow as:

(8.85a)

(8.85b)

for vertical upflow and

(8.86a)

(8.86b)

for horizontal flow

In steady horizontal flow with no acceleration, force balances for the gas core and the combined

flow relates the interfacial and wall shear stresses to the pressure gradient as:

(8.87)

It is expected that τi will depend on the difference between gas velocity and some characteristic

interface velocity. If then

(8.88)

For the same gas flowing alone in the pipe, the wall shear stress in terms of friction factor is:

(8.89)

This gives pressure drop as:

(8.90)

Combining equations (8.87), (8.88) and (8.90) with the definition of proposed in chapter

(5) (8.91)

At very high flow rates of both the phases, almost the entire liquid is entrained as droplets. This

gives the pressure drop and void fraction from homogeneous flow theory as:

(8.92)

and (8.93)

Since the volumetric flow rate of gas is usually much greater than the liquid,

(8.94)

And (8.95)

(8.96)

Since fTP in homogeneous flow is relatively unchanged from fWG , we get

(8.97)

(8.98)

Where the wall friction is factor for the film and is the equivalent friction factor for

the same liquid flowing alone in the pipe.If since wall roughness is same in both

cases, . Further is almost the same fraction of liquid Reynolds number as it is

for single phase flow apart from the transition region. For liquid Reynolds number defined as

a

(8.99)

Where for laminar flow and =0.005 for turbulent flow (8.100).

Two Phase Flow with Phase Change

Till now, the discussion contains two phase flow without any change of phase. The flow

phenomena considered are adiabatic and the phases have different chemical compositions.

Sometime this typical class of flow is also termed as two‐component two phase flow. On the other

hand, there is flow phenomenon where different phases appear due to change of phase (generally

caused due to heat transfer). These can be distinguished as single component two phase flows, for

example the flow of steam and water through a condenser used in a power plant.

Phase change phenomena can be broadly classified into two categories namely solid‐liquid and

gas (vapour) ‐liquid phase change phenomena. The first category includes the process of melting

and solidification.In both these processes one of the phases is a solid which has little movement.

The other phase is liquid and can have a velocity field mainly due to natural convection. In many

cases the problem of melting and solidification is analysed as a static problems of heat transfer.

Two different types of processes‐ boiling and condensation involve vapour‐liquid phase change.

In these two phenomena, as both the phases are fluid as there could be a large density difference

between the phases and there movement cannot be ignored. In fact, the complex interplay between

the phase change phenomenon and the fluid movement during boiling and condensation constitute

many interesting and challenging problems of two phase flow. There are innumerable examples of

boiling and condensation in engineering. Some of the industries/applications where these

processes play an important role are as follows:

• Steam power cycle in both conventional and nuclear power plant

• Refrigeration and air conditioning

• Cryogenics

• Oil and Chemical industries

• Material Processing

• Electronic Cooling

• Biochemical and food Engineering

The above list is only indicative but not exhaustive.

Though both boiling and condensation involve vapour‐liquid phase change they have some gross

differences and it is prudent to discuss them separately. In this course only boiling will be

discussed briefly.

Boiling

Boiling can be broadly classified as follows:

i) Pool boiling, and

ii) Flow boiling

Through flow boiling presents various intriguing aspects of two phase flow, a discussion of pool

boiling is essential for a better appraisal of the process of boiling. Pool boiling refers to a process

where liquid evaporates from the surface of a heater submerged in a stagnant pool of liquid. The

growth and departure of bubbles as well as the density difference in the liquid pool may induce a

local motion but there is no bulk movement of the fluid in pool boiling. In pool boiling different

regimes of boiling can be observed with the change of the temperature of the heater surface while

the temperature of the pool liquid is kept constant, generally at the saturation temperature of the

liquid. If the heater wall temperature is denoted by Tw (Tw - Tsat) indicates the excess of wall

temperature with respect to the saturation temperature (Tsat) of the liquid pool. The physics of the

boiling process is conventionally described through the relationship between the boiling heat flux

(q"w, W/m2) and the wall superheat (Tw - Tsat).

This relationship was first described by Nukiyama (1934) through a unique experiment. If the

temperature of the heater is controlled, then one can generate a typical curve known as pool

boiling curve as shown in fig. 1. As the temperature of the heater is gradually increased from the

pool temperature (Tsat), it starts dissipating heat by natural conviction (Region I) with the increase

of heater temperature heat flux also increases. At point A nucleation of vapour bubble starts. This

phenomenon is known as "Onset of nucleate boiling" (ONB). Up to point B, denoted as region II

in fig.1, vaporization takes place from discrete locations or nucleation sites. With the increase of

temperature newer nucleation sites become active and the frequencies of bubble for motion from

the existing sites also increase. This region shows a rapid rise in heat flux with the increase in

temperature Region II is often referred as the region of partial nucleate boiling as the heater

surface is partly covered by the vapour bubbles. With further increase in heater temperature, the

frequency of departure of the vapour bubbles increases so much that they sometimes forms vapour

columns or upward jets (Region II). Region III (from B to C) exhibits a very high heat flux and is

known as fully developed nucleate boiling.

Figure 1: Different regimes of pool boiling

The maximum heat flux in nucleate boiling is observed at point C. This point is commonly

referred as critical heat flux. One may think that up to this point the heater receives a continuous

replenishment of water and vapour phase does not cover it for a long period. If the heater

temperature is increased beyond the critical heat flux (CHF) point vapour phase cannot leave the

heater surface as fast as it is formed. Part of the heater surface is covered by the vapour patches

which decrease the rate of heat transfer from those parts. As a result a decrease in heat flux is

observed as heater temperature is increased beyond the point of CHF (From C to D). This typical

mode of boiling is also known as transition boiling (Region IV). The trend of decrease in heat flux

with the increase in temperature is very in counter intuitive and is in general not preferred in

industrial practice where a high rate of heat transfer is often sought for. Therefore, in engineering

the heater temperature is restricted below that of CHF point. In transition boiling the heater

surface is covered by gradually increasing vapour film with the increase of temperature and at

point D the entire surface is covered with vapor film indicating the inception of film boiling. Point

D denotes the minimum heat flux in film boiling. From point D onwards, the increase in

temperature again increases the heat flux. This constitutes the last regime of pool boiling, namely

film boiling. As the name suggests here the entire heater surface is covered with a vapour blanket

or film of vapour. This reduces the heat transfer co‐efficient substantially with respect to that

obtained in the nucleate boiling regime. However, heat flux increases with temperature as higher

temperature enhances the transport processes. Further, due to the large temperature of the heater

radiation heat transfer also becomes prominent. The excessively high surface temperature of the

heater surface often prohibits the operation of industrial equipment in this region.

If the heat flux of heater is controlled instead of its temperature, a somewhat different sequence of

phenomena is observed as shown in q"w - (Tw - Tsat) relationship in figure 2. If the heater power

is gradually increased, heater temperature true increases and the heater exhibits nucleate boiling

till the CHF point is reached. Beyond CHF, a slight increase in heater power shifts the boiling

regime to film boiling, bypassing the transition boiling completely. This is also associated with a

substantial temperature rise almost instantaneously. Such a temperature rise is often detrimental

for the heater.

In the some arrangement one may start the experiment from the film boiling regime and reduces

the heater power gradually. The film boiling curve will be followed till the minimum heat flux

point. Decrease of heat flux beyond this point will be shift the boiling to the low heat flux nucleate

boiling region bypassing the transition boiling region.

These sequences are depicted in fig.2. As in most of the industrial situations the heaters are heat

flux regulated extreme care needs to be taken to ensure the operation well within nucleate boiling

region to avoid excessive temperature rise of the heater surface and its failure.

Figure 2: Pool boiling for heating with heat flux control

Heat Transfer in different regimes of boiling Nucleate boiling

Heat transfer during nucleate boiling is most important from the applications point of view. For a

proper assessment of heat transfer during nucleate boiling it is necessary to understand the basics

of nucleation. Most of the time we encounter boiling from a solid surface which is initiated by

heterogeneous bubble nucleation contrary to homogeneous nucleation which occur in the bulk of

the liquid and requires a substantially high degree of superheat, heterogeneous nucleation occurs

from discrete sites of the solid surface at a temperature only moderately higher than the saturation

temperature of the bulk liquid. These sites are known as known as nucleation sites. In any

commercial material the surface contains, cracks, pits, crevices, these act as the nucleation sites.

Depending on the geometrical shape and size the nucleation sites get activated at different values

of wall superheat and gives rise to bubble nucleation. These cracks or pore entropy some gas or

vapour. A few molecule of the gas of vapour 'seeds' the inception of bubble nucleus. Once the

nucleus is formed the growth of the bubble takes places through evaporation due to heat transfer.

The growth of the bubble takes places in several stages and the bubble leaves the nucleation site

after its growth in complete. Figure 3 shows a bubble at the mouth of a cavity. The solid surface

is at a temperature (Tw) higher than that of the bulk of the liquid which is not at the saturation

temperature (Tsat). RC denotes the radius of the cavity at its mouth and RB is the radius of the

bubble. The bubble makes an angle}θ 0 with the solid surface. The fluid adjacent to the solid

surface is superheated and the height of the thermal boundary layer is given by δ .

When the bubble forms a hemisphere at the mouth of the cavity one gets RB = RC and its

corresponds the largest excess pressure needed for the bubble to remain in equilibrium.

Figure 3: Bubble nucleation at the mouth of a cavity

It is assumed that the bubble is surrounded by liquid of uniform temperature. The pressure

difference between the vapour and the liquid phase can be expressed as

Where σ is the surface tension of the liquid. Applying Clausius - Clapeyron relationship one gets

The above equation is for uniform temperature of the liquid. However, on a heated wall the

temperature need not be uniform. The bubble formation on heated wall requires that the wall

superheat (TW - Tsat ) should be larger than that given by the above equation.

The bubble nucleation criteria was forwarded by Hsu (1962) based on the following criteria.

1. Bubble embryo is surrounded by a superheated liquid layer whose temperature is higher than

that of the bubble interior.

2. Bubbles are not hemispherical, but elongated.

3. The temperature profile in the thermal boundary layer ( δ in figure 3) is approximately

linear.

4. Bubble height (YB ) and the bubble radius are related to the cavity radius (Rc).

So the temperature inside the bubble is given by

From the contact angle of the bubble with the solid surface C1 and C2 can be estimated (Gihassian,

2007). Based on the above developments one can find the limits of cavity radius (minimum and

maximum) suitable for bubble nucleation at a given condition of wall temperature, liquid property

etc.

Cavities whose radius is within the above limits will only be activated for nucleation.

Based on some modification of the above relationship other criteria for the wall superheat

required for bubble growth is given as follows:

Bubble Growth :‐

Isolated bubbles on a heated surface pass through nucleation, growth and departure. These

processes are almost cyclic if the thermal‐hydraulic conditions are maintained.

After the nucleation the bubbles undergo a period of growth for a time tgr known as growth period.

The bubble grows due to evaporation. However, there are two postulates regarding the path of

evaporation. According to the first school of thought bubble growth is due to evaporation from the

liquid around the bubble. The alternative theory states that between the growing bubble and the

heated surface there is a thin layer of liquid known as micro layer. The growth of the bubble is

primarily due to evaporation from the micro layer. The thickness of the micro layer increases from

the stem of the bubble (could be of the order of molecular dimension) to the periphery of the

bubble (could be micron level). The average thickness of the micro layer can be obtained from the

following approximate relationship.

Where γf is the kinematics viscosity of the liquid and C is an empirical constant C≈1.0. The

bubbles are in general submerged within the thermal boundary layer. When a bubble leaves the

heated surface it disrupts the boundary layer and colder liquid rushes to the surface replenishing

the hot liquid. A new thermal boundary layer will form. Below this another nucleus will develop

from the same nucleation site and will start growing after a wailing period of twt. So the cycle

needs a time tgr + twt for generation, growth and release of a bubble.

The diameter of a bubble during departure is given by the following relationship

Where θ, the contact angle is measured in degree. The heat transfer in the isolated bubble region

can be obtained from a simple energy balance:

Where fB is the frequency of bubbling and is given by, The difficulty of applying

the above equation lies in estimating the number of nucleation sites per unit area N.

The above discussion for nucleate boiling heat transfer holds good for isolated bubble region and

does not apply for the entire nucleate boiling region. There are many other correlations for the

fully developed nucleate boiling region. Rohsenow correlation is the most renown amongst them.

Adopting the general form of forced corrective heat transfer Rohsenow proposed that the Nusselt

number (Na) in boiling is given by

Where h – heat transfer co‐efficient in boilingλL – suitable characteristic length

Kf – thermal conductivity of the fluid -

Re – equivalent Reynolds' number -

Prf – Prandtle number of the liquid

n,m – empirical constants

In the above equation Csf is another empirical parameter whose value depends on the solid fluid

combination and also on the surface condition.

Rohsenow argued that for fully developed nucleate boiling the physical length scale and the pool

temperature does not have much effect. Instead, Laplace length scale (λL) can be used.

The characteristic velocity (V) can be defined as

where q''w is the wall heat flux, ρf and hfg are the liquid density and the latent heat of vaporization

respectively. Besides the heat transfer co‐efficient was defined based on the saturation

temperature of the liquid,

Even if the bulk of the fluid is not at saturation temperature, the above equation represents the

transport of thermal energy during fully developed nucleate boiling.

Substituting all the above equations one gets

In the above equation Cp, μ and K denotes the specific heat, viscosity and thermal conductivity

and the subscript f denotes the liquid phase. σ is the surface tension, g is the acceleration due to

gravity andΔρ is the difference of density between liquid and the vapour phase.

Generally, η is taken as 0.33; m = 0 for water and 0.7 for other fluids. Value of Csf for some

common fluid‐solid combination is given in the table below. From the table it may be noted that

the value of Csf varies between 0.003 to 0.0154. Considering the complexity of the phenomenon

of boiling the accuracy

Table 1: Value of different empirical parameters for pool boiling

Surface combination

Cst

m+1

Water‐nickel

0.006

1.0

Water‐platinum 0.013

1.0

Water‐emery polished

copper 0.0128

1.0

Water‐brass

0.006

1.0

Water‐ground and

polished stainless 0.008

1.0

Water‐Teflon pitted

stainless steel 0.0058

1.0

Water‐chemically

etched stainless steel 0.0133

1.0

Water‐mechanically

polished stainless steel 0.0132

1.0

Water‐emery polished,

paraffin treated copper 0.0147

1.0

CCl4‐emery polished

copper 0.007

1.7

Benzene‐chromium

0.01

1.7

n‐Pentane‐chromium

0.015

1.7

n‐Pentane‐emery

polished copper 0.0154

1.7

n‐Pentane‐emery

polished nickel 0.0127

1.7

Ethyl

alcohol‐chromium 0.0027

1.7

Isopropyl

alcohol‐copper 0.0025

1.7

35% K2CO3‐copper

0.0054

1.7

50% K2CO3‐ copper

0.0027

1.7

n‐Butyl

alcohol‐copper 0.0030

1.7

Critical Heat Flux (CHF)

The termination of nucleate boiling is marked by a very high heat flux. The point is known as

maximum heat flux or critical heat flux. It denotes the point up to which nucleates boiling can

occur. It also denotes a point up to which the heater can be operated without burn out, as the

heater temperature may rise high very fast if the heat flux is increased beyond the critical heat

flux. Therefore it is very important to know the value of it for different operating conditions and

for different system configurations.

CHF has been predicted from the hydrodynamic theory of instability for flat horizontal surface by

Zuber and his co‐workers (1959, 1963). At the advanced stage of nucleate boiling bubble

formation takes place from most of the portions of the flat horizontal upward facing surface. The

vapour phase leaves the surface in the term of discrete jets forming a regular array on the surface

as shown in figure. According to Zuber the regular array is a square grid with the side length of

This is the fastest growing wave length of Taylor instability. The rising jets are further assumed to

have a critical velocity Ug, related to Helmholtz instability.

λH is the neutral wave length of the rising jets of radious Rj

The critical heat flux can be estimated from the latent heat leaving with the jets per unit heater

area.

Zuber further assumed that

Then one gets the critical heat flux as

It needs to be mentioned that the same mathematical form of the critical heat flux was obtained by

other researchers through a different approach of analysis or simply by dimensional analysis. The

above equation is reasonably successful in predicating critical heat flux.

Transition boiling

Transition boiling is the least investigated regime of boiling. As it unstable, it cannot be

conveniently by established in a simple experiment. Theories related to transition boiling are also

limited. Surface properties plays a very important role on the transport processes of this typical

boiling phase. Moreover, researchers are of the opinion that for transition boiling unique

temperature heat flux curve is not possible due to hysteresis. The process may follow different

paths depending on the system, operating conditions and the point of initiation.

Film Boiling

Through film boiling is not a preferred mode of heat transfer it is often encountered during

quenching of metals, in cryogenic systems and in nuclear power generation during various

accident conditions. The analysis of film boiling is relatively simple compared to that of the other

modes of boiling. Below, film billing from a vertical surface is briefly analysed. The analysis is

based on the following assumptions and the phenomenon is represented by figure 4.

Figure 4: Film boiling over an isothermal vertical wall

• The vertical surface is isothermal

• The surrounding liquid is in saturated state and in stagnant condition.

• The flow of vapour over the plate is laminar.

• The interface between the two phases is smooth and is in thermodynamic equilibrium

• Heat transfer through the thin film of vapour is due to conduction.

• The phenomenon is two dimensional, no variation normal to the plane of the figure.

The momentum balance for the vapour phase gives

Where, V is the velocity of the vapour phase at any location. The above equation is subjected to

the following boundary conditions :

The vapour flow rate per unit width of the vertical surface is given by:

Finally, one can establish an energy balance as follows:

The film thickness δ can be obtained as

From the above equation one can determine the local Nusselt number. Integrating over a plate

length L the average Nusselt number can be determined as follows :

From this background the analysis of flow boiling can be developed.

Parametric Measurement of Two Phase Flow

In spite of the extensive volume of past research activity, two phase flow is not yet an area in

which theoretical prediction of flow parameters is generally possible. Indeed, this situation is

likely to persist for the foreseeable future. Thus, the role of experiment and parametric

measurement is particularly important. The techniques of measurement for single phase flow are

well established. Based on these techniques, various meters and instruments have been developed

which are successfully employed for industrial measurement as well as for R&D activities.

Unfortunately, these instruments cannot be directly used for multiphase flow measurement. Most

of the problems in multiphase flow measurements arise from the fact that the parameters

characterizing it are many times larger than those in single phase flows. In single phase flow, the

flow regimes encountered are laminar, turbulent and a transition region between them. In

multiphase flow, numerous flow regimes are possible depending on flow geometry (size and

shape) and orientation (vertical, horizontal and inclined), flow direction in vertical or inclined

flows (up or down), phase flow rates and properties (density, viscosity, surface tension) as

discussed in Chapter-2. In addition the slip between the phases causes a difference in the in-situ

and inlet composition of the multiphase mixture, As a result, the methods of flow measurement

conventionally adopted for single phase flow are grossly inadequate.

This has given rise to the development of a number of techniques especially suited for the

measurement of two phase flow parameters. In the limited scope of this discussion, it is not

possible to consider the principles of measurements of all the parameters. However, void fraction

and flow pattern are two parameters of unique importance. Information regarding these

parameters is essential for the design and optimization of the components, control and monitoring

of the equipment, overall efficiency of the process and safety of the plant. Knowledge of these

two parameters is often used as the input for the measurement of other variables. In this chapter,

different techniques for measurement of void fraction and flow pattern is described. The

description is primarily based on gas–liquid two phase flow though reference to other types of

two-phase flow is made from time to time. Prior to the discussion of measurement of the

aforementioned parameters, we would describe the challenges involved in measuring pressure

drop of two phase flow just to emphasise the complexities involved in measurement of even

simple parameters under multiphase flow situations.

10.1 Measurement of pressure drop

This parameter is of interest since it governs the pumping power required to circulate two phase

fluids through the system and it governs the circulation rate in case of natural circulation. It is also

important in several flow metering applications like venturimeters and orifice meter. In two phase

flow, measurement of pressure drop requires special considerations as has been discussed below.

The scheme of the measurement is explained in Figure 10.1.

Fig. 10.1. Void fraction estimation by pressure drop measurement

Making a pressure balance at Section A one gets,

…(10.1)

Rearranging we have,

…(10.2)

If p1= p2, the manometric difference is

...(10.3)

This indicates an offset in the manometer which depends on (a) distance between tappings (b)

density of the line fluid (ρc).Further in absence of flow through the tube, considering no

acceleration pressure drop

…(10.4)

he is the head loss due to friction. ρt is the mixture density and is given in terms of volume average

void fraction,

…(10.5)

Equating equations (10.2) and (10.4), we get the manometric difference as:

…(10.6)

Neglecting head loss due to friction

...(10.7)

Which shows that for zero differential in the manometer, ρt = ρc or line fluid has the same density

as the fluid in the tube. While this is an expected situation in single phase flows, the case is not so

simple when two phase flow occurs in the pipeline because the lines, under these conditions, can

always be filled with a two phase mixture of unknown and variable composition and from eqn

(10.2), it is very important to know the composition and density of fluid within connection lines

(ρc). Or it is mandatory to control the manometer lines in such a way that they are filled by a

single phase fluid and there is no ingress of the second fluid in them. In case of gas-liquid flow, it

is generally filled up with the fluid corresponding to either gas or liquid phase flowing in the pipe.

Usually it is the continuous phase which fills the lines.

To summarise, the additional difficulties in measurement of pressure drop in two phase flow

are as follows:

1. Possible ambiguities in content of lines joining tapping points to measuring device

2. Pressure drop fluctuations can be quite large

3. Added problems in heated systems particularly when they are Joule heated

The methods for pressure drop measurement are same as those adopted in single phase

flows, viz

1. Fluid – fluid manometers

2. Subtraction of signals from two locally mounted pressure transducers

3. Differential pressure transducers

For fluid ‐fluid manometers, water‐mercury manometers or inverted water manometers are

adopted depending on the pressure range. For greater sensitivity water‐carbon tetrachloride or

water‐kerosene manometers are used. Using inverted liquid‐gas manometer with liquid filled

tapping lines is more accurate as compared to liquid mercury manometer. In special cases where

gas filled pressure lines can be employed, inclined manometers/micro manometers are used.

The problem for any fluid‐fluid manometer is that the content of the line can be two phase

by a variety of mechanisms namely,

1. Changes in pressure drop and consequent movement in manometer can cause two phase

mixture to enter into tapping lines from flow passage. This can have disastrous results eg.

Mercury from manometers entering metal flow system. To overcome this difficulty, large

diameter catch pots are introduced in the tapping lines for two liquids to meet.

2. Condensation or evaporation can occur in lines particularly as a result of rapid changes in

system pressure, for example generation of vapor bubble in liquid filled lines following

depressurization. For liquid‐vapor systems and vapor filled lines, an evaporator just downstream

of tapping points evaporates any liquid entering the line. This is particularly useful for low latent

heat liquids like cryogenic fluids but the technique is not common since the rate of evaporation is

rather slow. Similarly a condenser can be installed in liquid filled lines if low latent heat liquids

are used as the test fluids.

3. Pressure fluctuations can cause a pumping action leading to gas ingress into liquid filled lines

or vice versa. For example, if lines are filled up with liquid phase, gas/ vapor ingress can occur by

(a) Changes in pressure drop and movement of manometric fluid allowing two phase mixture to

enter any one of the pressure tappings. (b) Flashing in the lines after rapid depressurization

(changes in system pressure) (c) Pressure fluctuation causing pumping action leading to gas

ingress into tappings. Similarly for gas/vapor filled lines, liquid ingress can occur by (a) Changes

in pressure drop and movement of manometric fluid (b) Pressure fluctuation leading to liquid

pumping into lines (c) Vapor condensation in lines. Further, for liquid filled lines, the

performance can be improved by using a balanced liquid purge system as shown inFig. 10.2. It

may be noted that the lines have to be transparent to check gas locks if

any. Alternatively, compressibility of fluid in line can be checked by using acoustic methods.

4. The additional disadvantage of manometers is that it is not suitable for transient measurements.

One has to use transducers for this purpose.

Although the consequences of liquid ingress are same as gas ingress, liquid ingress is more severe

and likely than gas ingress due to the following reasons:

1. Compressibility of fluid in gas filled lines causes much worse pumping action by pressure

fluctuation.

2. Tendency of liquid phase to wet the channel wall causes capillary effects at gas‐ liquid

interface where line enters channel. This is particularly significant for small diameter lines.

Thus on the whole, gas filled lines are less satisfactory. The only advantage is the low offset value

at zero Δp (eqn 10.3) and thus greater accuracy of pressure measurements possible.

Fig. 10.2: Purging system in pressure measurement

An alternative to manometers particularly when a rapid response is required is to use a pair of wall

mounted pressure transducers mounted locally at the points between which the pressure drop is to

be measured, the signals of which are electronically subtracted to obtain the required pressure

drop. The principle of measurement is shown schematically in Fig. 10.3.

An alternative to manometers particularly when a rapid response is required is to use a pair of wall

mounted pressure transducers mounted locally at the points between which the pressure drop is to

be measured, the signals of which are electronically subtracted to obtain the required pressure

drop. The principle of measurement is shown schematically in Fig. 10.3.

Among the different types of transducers namely potentiometric, strain gauge, capacitive,

reluctance, inductive, eddy current and piezoelectric, the capacitance and piezoelecric type are

suited for measurements using signal subtraction. A comparative study of the performance of the

two types of transducers is given in Table 10.1.

Table 10.1: Types of transducers particularly suitable for measurement using signal

subtraction

The advantages of transducers are:

1. Fast response ‐

2. Enables study of fluctuations in pressure drop

3. Avoids ambiguity in line content

The disadvantages are:

1. Signals from two separate instruments are measured and subtracted and this obviously increases

errors. In order to avoid this, differential pressure transducers are adopted. However, this is

unavoidable if rapidly fluctuating pressure drop are to be measured. In that case, special care is

required to calibrate transducers and to ensure that the output is properly converted to the required

pressure drop.

2. Furtherer although the amount of fluid between flow passage and transducer is rather small, the

volume of the tapping line and the fluid adjacent to the diaphragm should be kept at a minimum in

order to avoid reductions in frequency response.

3. Both capacitance and piezoelectric transducers are limited as to operating temperatures and

need to be cooled for higher operating temperatures.

Fig. 10.3: Mounting for absolute pressure transducers for measuring two‐ phase pressure drop

In order to minimize the error in measurement, a single differential pressure (DP) transducer is

adopted. Here the two pressure tapping lines are attached to two chambers separated by a

diaphragm. The movement of the diaphragm indicates the differential pressure. Capacitance and

piezoelectric transducers are unsuitable for differential pressure operation, rather the reluctance

type and the strain gauge type transducers are most often used. A comparative study of the two

types is provided in table 10.2.

Table 10.2: Types of transducers particularly suitable for differential pressure operations

Since only a small movement of the diaphragm occurs, the problem of fluids moving into

and out of lines as in manometers is minimized. Nevertheless, DP transducers have their set

of problems, namely

1. Since differential pressure transducers are operated with tapping lines, all problems associated

with ambiguity in tapping line content also apply in this case and it is desirable to keep the lines

full of liquid. Nevertheless, since the diaphragm movement is quite small, the amount of test fluid

passing into the lines following a change in pressure drop is much less as compared to

manometers.

2. Slight fluctuations in pressure drop due to rig vibration, etc. can alter the readings

3. The pressure differential to be measured has to be higher than the offset corresponding to zero

pressure difference. If one is measuring pressure differentials that are small compared to the

offset, the accuracy is limited since a differential pressure transducer must be chosen with a range

at least equal to the offset value. This can be minimized by using a compensating manometer as

shown in Fig. 10.4.

For pressure transducers or DP cells, periodic calibration is essential. This can be used for

fluctuating pressure measurement although influence of lines on frequency response is critical.

For rapid response, the only feasible method is to use subtraction of signals from two wall

mounted pressure transducers mounted locally at tapping points.

In conclusion we summarise the criteria for selection of a suitable technique to measure pressure

drop

1. Is a rapid response needed eg. In the study of pressure drop fluctuation?

Then locally mounted pressure transducers are mandatory. For fluctuating measurements, choice

of response time and temperature and whether direct access of the fluid to the transducer is

feasible indicates choice of transducer type.

2. Is electrical (automatically recorded) output required?

For this, some form of differential pressure transducer is necessary or else liquid‐ liquid

manometers are adopted

3. Are tapping lines to be kept full of gas or liquid?

Better to keep tapping lines full of liquid and have a liquid purging system, but more accurate

measurements need gas filled lines if line content is completely unambiguous. In practice, this is

not achievable and highly accurate measurements are not possible.

4. Is channel fluid condensable or evaporable?

Under this condition, unambiguity in tapping line content can be minimised by either

condensation or evaporation near pressure tapping.

5. Is high accuracy required?

For liquid‐ liquid manometers this can be ensured by using inverted liquid gas manometer with

liquid filled tapping lines rather than liquid mercury manometer. In special cases where gas filled

pressure lines can be employed, inclined manometers/micro manometers are used.

Fig 10.4 : Compensating CCl4 Water Manometer for DP Transducer

10.2 Measurement of in-situ composition:

The insitu composition is a unique property which can be used to differentiate two phase flow

from single phase flow phenomena. Let us, assume gas–liquid two phase flow is taking place

through a flow channel of volume V under steady state condition. In this space, the individual

volumes of gas and liquid phases are Vg and Ve respectively, so that

…(10.8)

The insitu composition in this case is expressed as void fraction (α) and defined as

…(10.9)

It may be noted that the above equation gives the instantaneous value of void fraction. If a

sufficiently large volume is taken and average values over a suitable duration is considered,

average void fraction can be defined as

…(10.10)

can be taken as a characteristic of the two phase flow phenomena. The importance of

can be realized easily. All average properties of the mixture like density, viscosity, specific heat

enthalpy, etc. are direct functions of void fraction. Moreover, the flow regime and the interfacial

area are also dependant on this. The knowledge of void fraction is particularly necessary for the

processing of costly components. In a nuclear reactor, the absorption of neutron depends directly

on it. Therefore, it is not surprising that in two phase flow, most of the efforts for instrumentation

has been made for the measurement of void fraction. Before describing the method of void

fraction measurement, it is prudent to recall the discussion and the various definitions of void

fraction discussed in Chapter-4.

The measurement techniques for different types of void fraction are described pictorially in fig.

10.5. In this figure in (a) denotes the scheme of measuring volume averaged void fraction. The

probing is done over a volume of the conduit. If measurement is done for a cross sectional area as

shown in (b), one gets area averaged void fraction. In this area, if measurement is done along any

chord length (not necessarily along the diameter) chordal average is obtained. If the probing area

is much smaller compared to the cross sectional area (fig.10.5b) the obtained measurement

approaches local (point) value.

Fig. 10.5: Different schemes of void fraction measurement

2.2 Mechanical Techniques:

2.2.1 Direct Volume Measurement by Quick Closing Valve Technique –

This is the most widely used method for measuring holdup for adiabatic gas-liquid as well as

vapor-liquid flows. It has been used by a majority of the researchers in the past. In this method,

the flowing two phase mixture is instantaneously trapped between a pair of valves placed at the

beginning and end of the test section. The relative proportion of the two phases is then obtained

by noting the fraction of the channel volume occupied by either of them after gravity separation of

the phases. The main drawbacks of this method include a finite time requirement for closure of the

valves. The closing down of the system for each measurement and bringing the system back to the

steady state which might require considerable time between successive runs. This is also not

suitable for transient situations. Figure 10.6 depicts the arrangement

Fig. 10.6: Volumetric void fraction by quick closing valves

2.2.2 Measurement of void fraction by pressure drop:

Volume average void fraction or the overall void fraction for a section of conduits can be

estimated from the differential pressure of the section. The scheme of the measurement is

explained in Figure 10.1 and the void fraction in terms of the manometric levels is obtained as:

…(10.15)

From the above equations, the overall void fraction may be determined if he is known. The head

loss due to friction is neglected in a number of cases or it can be determined approximately using

a suitable correlation. The above method is simple and is well suited for situations where

i) Frictional pressure drop is small,

ii) Acceleration pressure drop is small,

iii) There is no drastic change in void fraction along the channel length.

It is mandatory to ensure that the manometer lines are filled up by a single phase fluid. In case of

gasliquid flow, it is generally filled up with the fluid, which forms the continuous phase. This

method is also suitable for transient cases if the manometer is replaced by a pressure transducer.

This technique is applied for gas-liquid, solid-liquid and gas-solid flows.

10.2.4 Radiation absorption and scattering methods :

Attenuation of a beam of gamma rays is one of the widely used techniques of void fraction

measurement. Attenuation of gamma rays passing through the two phase medium occurs by three

distinct processes.

1. Photoelectric effect,

2. Production and absorption of positron-electron pair,

3. Compton effect.

Details about these effects are elaborated in Hewitt (1978).

The absorption of a collimated beam of initial intensity Io (photons / m2s) is described by a

exponential relationship.

…(10.16)

where μ is the linear absorption coefficient and z is the distance traveled. Absorption coefficient

on the other hand depends on the density of the medium.

Estimation of void fraction, can be made purely from a theoretical point of view. However, that

needs gamma spectrometry for separating out the secondary photons produced in the process. This

is expensive. In most of the cases, in-situ calibration is done by measuring intensity of the

absorbed beams for tubes full of liquid (IL) and full of gas (IG) respectively. If the intensity for a

two phase system is I, then void fraction is given by

…(10.17)

This method void fraction measurement is very versatile and can be used for a wide range of

applications containing two or more phases. However, there are some inherent sources of error

and difficulties in this measurement.

1. The system is in general costly and needs specially trained manpower for its operation.

2. There are difficulties of handling high-energy radiation.

3. Normal fluctuations of photons gives rise to an error which can be minimized by increasing

strong sources and using long counting times. This makes the process unsuitable for transient

measurement

4. If the phase interface is parallel to the beam void fraction is given by

…(10.18)

This induces some error in the measurement.

5. Presence of the metal wall also induces some error when averaging is done.

6. Gamma ray absorption also has some limitations for liquid-liquid systems.

Some of the above shortcomings can be overcome by the use of calibration, multiple sources, etc.

In specific applications, neutron attenuation and the use of x-ray produce good results. However,

availability of neutron source in the former case and safety in the later are major concerns.

10.2.5 Optical Techniques:

A large number of measurement systems have been developed using optical techniques. Local and

global as well as steady state and transient measurements are possible by suitable design of the

instrumentation system. Some of the important optical methods are described below.

10.2.5.1 Photographic techniques:

This is suitable for flow regimes where the two phases are not thoroughly mixed like stratified

flow. By suitable arrangement, one can use the technique for complex gas-liquid regimes like

annularmist flow.Figure 10.7 describes one such arrangement. In this arrangement a section of

the tube is illuminated. Photograph is taken from the top of the conduit, which is closed by a

transparent cover. The camera is focused on the illuminated portion of the conduit. Photographic

technique provides a good visual appraisal at the flow phenomena.

Fig. 10.7: Photographic technique for void fraction measurement

The photographic technique got a boostup in recent times due to the availability of laser sources

and digital camera at lower prices. Using simple optical arrangement thin light sheet can be

generated from a laser beam. By this, illumination of a particular cross section is possible. This

gives better information regarding phase distribution. The arrangement is shown in Figure 10.8.

Digital photography has also made the post processing of photographed information easier. With

the videographic recording, the transient phenomena can be analysed while applying image

analysis a better quantification of the flow regime is possible.

Fig. 10.8: Lighting by laser sheet

10.5.2 Local measurements by Intrusive Probes :

Optical probes of small dimension, as shown in Figure 10.9 have been developed for local void

fraction measurement. It uses the principle that incident light, passing down the optical fibre, is

totally reflected and returned back when the probe in a gas phase. Therefore, it is capable of

detecting minute void and suitable for local measurement.

Fig. 10.9: Optical probe for phase detection

An innovative design based on the combination of optical and isokinetic sampling technique is

illustrated in Figure 10.10. In this probe, a small volume of the two phase mixture is

isokinetically sucked and passed through a capillary tube. Inside the tube, the size of the bubble is

determined with the help of two light sources. This gives an estimation of the size distribution and

flow rate of the gas phase (Steevers et. al. 1995).

Fig. 10.10: Operation of photo suction probe

10.5.3 Fluorescence Technique:

Instantaneous and highly localized, measurements of film thickness can be obtained using the

fluorescence technique. Blue light from a mercury vapor lamp is passed through a microscope

illuminator and focused in a conical beam into the liquid film. The circulating water in the

apparatus contains fluorescein dye-stuff and the incident beam excites a green fluorescence in the

film. The fluorescent illumination is separated in a spectrometer and its intensity is a direct

measure of the liquid film thickness.

Scattering technique forms the basis of different instruments for continuous drop size distribution.

A light beam passing through dispersion will have extinction proportional to the effective

superficial area. This is true for both transparent dispersed phases and opaque ones. Provided the

received beam is exactly in line with the transmitted one, the transparent dispersed elements will

scatter the incident beam and behave as though it is opaque with regard to the received beam. This

method of light scattering is suitable for both bubbly and drop flow. Figure 10.11 illustrate the

use of light scattering technique for mist annular flow.

Fig. 10.11: Scattering technique

Some of the recent developments in optical technique are reported in Mayinger and Feldmann

(2001).

10.6 Impedance Method:

As the electrical impedance of a two phase mixture is a function of concentration, measurement of

impedance can form a basis for the estimation of void fraction. Several instruments for the

measurement of void fraction and associated parameters have been developed based on

impedance technique. Impedance technique has the following advantages:

1. It is a low cost technique.

2. It is suitable for transient measurement.

3. Large variations of electrode design are possible making the method appropriate for different

flow situations and geometry.

4. Both intrusive and non-intrusive measurements are possible.

5. Point measurement as well as global measurement can be made by suitable design of the

probe.

6. Same principle (sometimes the same probe) may be used for the measurement of associated

parameters like,

a) flow regime identification,

b) bubble size and frequency,

c) bubble velocity.

An impedance probe can operate either in resistance (conductance) or in capacitance mode. If the

liquid phase is the continuous one and electrically conducting, then the probe is used in the

resistive mode. If the gas phase is continuous or the liquid phase is non-conducting, then the probe

is used in capacitance mode. A large variety of probe design is possible some of them are

described in the next lecture.

2.6.1 Separated flow of gas and liquid:

In annular flow, stratified flow, film flow, the gas and liquid phases are separated by a well

defined interface and generally the liquid phase does not contain any gas bubble in dispersed

condition. In such a situation, impedance probe gives a good estimate of the liquid film thickness.

The film thickness may be obtained either from direct calibration or from a theoretical analysis.

The theoretical analysis will be described shortly. Some of the probe geometry suitable for the

measurement of film thickness is shown in Figure 10.12.

Fig. 10.12: Probes for local film thickness

The same principle can be used for two phase flow measurement through a conduit. For conduits

of circular section arc electrode probes, parallel wire probe or ring electrode probe may be used.

The methodology of measurement can be explained based on the work of Gupta et. al. (1997) in

which a pair of arc electrode probe were used to measure the liquid height in stratified gas-liquid

system. In Figure 10.13 stratified flow in a circular tube is shown. Flush mounted arc electrode

probes may be used for the estimation of liquid fraction.

Fig. 10.13: Arc electrode probe

From electrostatics one gets,

…(10.19)

…(10.20)

…(10.21)

From Ohm's Law,

.... (10.22)

and … (10.23)

where V is the electric potential, J the current density, E the electric field, I the current, σ the

conductivity of the liquid and R the resistance.

Finally, the current I can be expressed as

…(10.24)

From the above relationships, one can calculate the resistance for a given liquid height and liquids

conductivity. Figure 10.15 is the circuit for the probe. It may be noted that the probes induces 3-D

effect due to fringing from the end of the probes as shown in Figure 10.14.

However, it is difficult to take care of the fringing effects by the theory. This difficulty can be

avoided using guard electrodes as shown in the figure. A probable measurement scheme is shown

in fig. 10.15. This arrangement renders the electrical field two dimensional along a small axial

length.

Fig. 10.14: Fringing Effect

However, it is difficult to take care of the fringing effects by the theory. This difficulty can be

avoided using guard electrodes as shown in the figure. A probable measurement scheme is shown

in fig. 10.15. This arrangement renders the electrical field two dimensional along a small axial

length.

Fig. 10.15: Measuring circuit with guard electrodes

The resistance can be determined from the following formula:

…(10.25)

and related to the void fraction.

Impedance probes can also be used when the phases are well mixed. When the phases are well

mixed from Maxwell's theory, one gets

…(10.26)

where Ac is the admittance of the gauge when immersed in the liquid phase alone. εG and εL are

the conductivities of the gas and liquid phases if conductivity is dominating. On the other hand,

one should use the dielectric constants if capacity is important. The above equation is suitable for

bubbly flow. Accordingly arrangement for volumetric measurement of void fraction can be made

as shown in figure 10.16.

Fig. 10.16: Grid electrode probe for volumetric measurement

The arrangement (a) depicts ring type of electrodes. The void fraction between two rings can be

estimated. The arrangement is suitable for bubbly, slug, annular and stratified flow. A pair of grid

electrodes is shown in (b). It is well suited for measuring the volumetric void fraction in bubbly

flow. People have also used this probe for slurry flow.

For liquid droplet flow through a gas, one gets

…(10.27)

Another probe configuration widely used in gas‐ liquid flow is shown in fig. 10.17 (a). It uses two

thin parallel conducting wires spanning the cross section. This arrangement is suitable for the

measurement in annular, stratified and slug flow. In case of curved interface in stratified flow

multiple wires may be used as shown in fig. 10.17(b).

Fig. 10.17: Wire probes

Conductivity principle can be used for local measurement. For this purpose, endoscopic probes

with single or multiple needle electrodes as shown in Figure 10.18 are used.

Fig. 10.18: Single and multiple needle contact probes

The needle electrode which is a fine needle with only its uninsulated tip exposed to the

two‐ phase mixture forms the heart of the measurement scheme. The other electrode is so large

that always some part of it is in contact with the conducting phase of the mixture. If the small tip

of the probe comes in contact with the non‐ conducting phase (say gas bubble) the circuit is

broken and there is a change in signal level. Theoretically, the probe should produce a square

wave signal, if the probe tip is infinitesimally small and the interface is infinitesimally thin. But in

practice the bubble deforms as it approaches the probe tip as shown in fig. 10.18 (b). Also finite

time is needed for wetting the probe tip. Hence the signal is often distorted and careful signal

processing is needed for deriving qualitative information. Fig. 10.18 (c) shows double needle

probe, which may be used for determining the bubble velocity. Using multiple probes, as shown

in fig. 10.18 (d) bubble geometry can be determined. Fig. 10.19 shows the typical nature of the

signal from a needle probe.

Fig. 10.19: Signals from needle probe

Most of the conductivity probes developed so far are suitable for flat surface or circular tubes.

Some efforts have been made to develop probes for annular geometry. Parallel ring type probes on

the outer surface of the inner tube and inner surface of outer tube have been used. Das et.al. (

2000 ) have used a unique parallel plate type probe. The probe is made from fine strips cut from

double sided printed circuit board (PCB) and can be supported only from the outer tube due to its

inherent rigidity. Das et.al. (1998,2002) have also used it for the estimation of bubble velocity and

bubble shape.

One of the major drawbacks of the conductivity probes is the change of conductivity with

temperature and impurity. As the probe is exposed to a conducting medium there is

zlectrochemical interaction(McNaughtan et.al. 1999) between the two. This is commonly known

as double layer formation. As it is difficult to model the double layer effect it makes the

quantitative prediction difficult. However, conductivity probes are extensively used for qualitative

prediction like flow regime identification (Das et.al. 1999a, 1999b).

In addition, the dependence of void fraction on admittance ratio is a function of the flow pattern as

shown in Fig. 10.20. As a result, it is necessary to have apriori knowledge of the prevailing flow

pattern for an accurate estimate of void fraction.

Fig 10.20

Some of the difficulties of conductivity probes may be avoided using the probe in the capacitance

mode. This is suitable for non conducting fluids and the probes can also be mounted outside the

conduit wall (it need not contact the two phase mixture). Some of the popular probe

geometries (Sami et. al. 1980) are shown in fig. 10.21. However, the dielectric constants of the

two phases do not vary widely and there are stray capacitance effects. These make the

measurement challenging.

Fig. 10.21: Different types of capacitance probe

10.2.7 Other methods:

Apart from the above techniques, hot wire principle, micro thermocouple, microwave and

radiowave attenuation as well as NMR technique and ultrasonic probes are also used for void

fraction measurement. A summary of the other techniques is provided below.

Accoustic techniques

The velocity of sound in a two phase mixture is highly sensitive to void fraction. Sound Velocity

measurement as a means to determine void fraction is frequently proposed particularly when other

methods are difficult and unsuitable. The main limitation of this technique is that sound velocity

depends on void size and void fraction and can also vary with sound frequency.

Measurements of average phase velocity defined as

Average phase velocity = phase volumetric flux per unit area/phase volume fraction

or

If j1 is known and u1 can be determined, can be estimated. One method of measuring u1 is by

radioactive tracer technique.

Electromagnetic flow metering

Principle of this method is an independent measurement of average liquid velocity u1 from

which can be calculated. The technique can be used with calibration over a wide range of flow

patterns. This is particularly useful for liquid metal systems although it can be used for air‐ water

and steam‐ water systems.

Optical methods

Based on photographs of two phase flow and their analysis them to measure total gas volume.

Alternatively, the gas content can also be measured using light scattering. It is tedious and has

limited application.

Microwave absorption

Applicable to organic reactor - coolant void measurement. This technique also measures fluid

density and gives measure of for H2 vapor - liquid mixture and water content measurement in

margarine.

Link between pressure & flow oscillations

Used to measure in Na liquid - vapor systems. Pressure fluctuations induced by a pump cause

flow fluctuations. Relationship between pressure fluctuations and flow fluctuations depends on

.

Infra red absorption methods

Show considerable promise at high void fractions. They can be used to measure component

concentration in air/steam/water mixture and is particularly useful for high void fractions where

most of the techniques fail.

Neutron noise analysis

In case of nuclear reactors, it is possible to deduce from correlation analysis of signals from

neutron detectors.

NMR (Nuclear Magnetic Resonance) :

Nuclei possessing a spin angular momentum are introduced in a magnetic field. They take up

specific orientations. In NMR experiments the amount of radio – frequency energy required for

reorientation is measured. In flowing systems, it is important to ensure that relaxation time for

reorientation is small as compared to residence time in measurement region. This can be achieved

by doping liquid with paramagnetic salt (CuSO4 if liquid is water).

2.8 Tomographic Imaging :

As local and spatial fluctuations are inherent to any two phase flow phenomenon, none of the

methods described above can give the instantaneous value of void fraction over a volume

correctly. Moreover, they can supply only a partial distorted picture regarding the flow regime as

some sort of averaging is done in all the above measurements. To overcome this limitation,

scientists have adopted tomographic imaging technique for scanning the entire flow passage.

Tomographic imaging of a flow passage may be obtained by a variety of basic void measurement

techniques like impedance, optical, radiation attenuation, etc. The basic principle of all these

systems is the same though the detail arrangement may vary. In Figure 10.22, the typical

arrangement for a tomographic measurement with conductivity or capacitance probe is shown.

Fig. 10.22: Tomographic imaging system

In any of these methods, a large number of miniature sensors / probes are used. Two of the probes

are energized in turn. A probable sequence could be 1-2, 1-3, ……….1-N; 2-3, 2-4,………..2-1

and so on. This allows collection of high accuracy local signals from a three dimensional probing

volume. By using tomographic techniques, the physical property of interest is recovered from the

observations integrated along the different paths of probe measurement for each plane (slice) of

the measurement volume. The three dimensional information is then reconstructed. Theoretical

background as well as hardware details of different tomographic systems may be obtained from

the classical book by Beck and Williams (1995).

To summarise, some of these techniques are to be improved substantially so that they can give

reliable prediction in an industrial atmosphere. The status of two phase flow measurement is still

in its early development stage. The tomographic measurement shows enough promise as one can

measure velocity, temperature and composition using the same principle. However, more research

is needed to produce rugged instrument within affordable price.

10.3 Estimation of Flow pattern:

Most of the techniques are based on observing the time, amplitude and positional variation of

voids. The commonly used techniques are as follows:

1.(a) Visual methods - It is the simplest technique and is based on visual observation of the void

distribution. It is accurate for low flow rates of transparent fluids through transparent conduit

walls. For non transparent pipes, visualization is effected through glass windows placed at

intervals in the pipe. The disadvantages are (a) suitable visual access needs to be facilitated for

low speed flows and (b) For higher speed flows with waves/bubbles moving at relatively constant

velocity, the applicability of visual observation is limited and can be extended by using scanning

device.

(b) Photographic Techniques ‐ At high flow rates visualization often leads to obscure images

and is often supplemented with photography. The photographic methods are limited by size of the

field of view. Further, only instantaneous local behavior can be observed and this restricts

observation of axial variation of voids. For non transparent fluids, X‐ ray photography is adopted.

Moving picture photography is less successful because field of view is smaller than that necessary

to get resolution required for the observations. The general unreliability of photographic measures

has led to a search for other techniques for flow pattern identification.

The general problems of visualization techniques are

a. The descriptions of certain flow pattern are arbitrary and this calls for a strong element of

subjectivity in deducing flow patterns

b. A large amount of information is produced particularly in photography and it is difficult to

analyze and interpret it.

c.The complex interfacial structures give multiple reflection and refraction that obscure the view

particularly of the central region of the channel.

d.The transition between flow patterns is often a gradual process and it is difficult to pinpoint the

regime boundaries.

2.) X‐ Radiography :While in photography, there is a complex series of interactions between

light and interfaces, the image in this case depends on absorption of photons. X‐ ray machines

with very short duration pulses are used. X‐ ray fluoroscopy is used directly for visual

observations but the time resolution is often insufficient. It is particularly useful in visualising

flows in non‐ transparent channels including cases in which there is a heat flux. The major

limitations are the problems in handling radiation and insufficient resolution.

3) Multi beam densitometry (Fig 10.23) can be adopted for transient flow measurements. For

horizontal tubes where the flow is highly asymmetric, multi beam X‐ ray gives an idea about the

distribution of voids. It is useful for transiently varying flows in nuclear reactor safety and has a

good time response. Although the technique is very promising, it has the general problems of

radiation absorption techniques pertaining to high cost and need for careful installation as well as

operation to ensure safety of personnel. Further it needs to be tested at high mass velocity where

all major problems of delineation occur. It also needs a high degree of in‐ house electronics and it

is difficult to demarcate flow distributions for complex flows or high mass velocities

Fig. 10.23 An arrangement for multi beam gamma densitometry

3.Methods based on for pressure measurements :

As flow pattern changes with a systematic variation of gas/liquid rate, the slope of time

averaged pressure gradient curve as a function of phase flow rates is different for different

patterns. However, the results are usually descriptive and not very useful for detecting flow

regimes since slope changes can be related to flow pattern transitions only thorough visual

observations.

For vertical systems, the pressure gradient obtained from two wall mounted

pressure transducers located axially apart is strongly dominated by gravity. So, for bubbly flow, it

fluctuates about while for annular flow it is approximately close to

If , this can be used as a diagnostic tool.

A third technique based on measurements of pressure fluctuations is the spectral analysis

of wall pressure fluctuations since the fluctuations of wall pressure is a function of the manner in

which the two phases are distributed in the pipe and their velocities i.e the flow pattern.

Mathematically

Where, stands for the time variation of pressure fluctuation, is the measured time

dependent pressure and is the time average pressure.

The auto correlation function for the fluctuation is:

And the power spectral density function which is the Fourier transform of the autocorrelation is:

The analysis yields the three basic spectra shown in Fig 10.24. From the figure, curve A with

peaking at f=0 and decay at increasing f signifies separated flow patterns namely the stratified or

the annular flow configuration with low entrainment rates. Curve B denotes intermittent patterns

which can be either slug or elongated bubble where the peak displaced from f=0 gives a measure

of the mean frequency of slug passage and curve C which is marked by absence of any peak

denotes the dispersed or the distributed flow patterns (gas bubbles in liquid or liquid drops in

gases) where the fluctuation is controlled by passage of successive elements of dispersion.

Fig.10.24 Curves obtained from PSDF analysis of different flow patterns

The major disadvantage of this technique is that

The identification of flow regimes is not always clear. In reality, we do not obtain well

defined curves shown above. Usually a superposition of spectra as shown in Fig 10.25 is obtained

where curve D denotes an annular dispersed distribution with the peak at low frequency depicting

the separated nature of annular flow while the spread out curve is indicative of dispersed

distribution. Similarly curve E depicts the high frequency wavy annular (annular + slug) flow.

Further, it does not discriminate clearly between the types of separated or dispersed flow.

It is noted that conditions most difficult to interpret are conditions most difficult to

interpret by visual means.

In addition pressure waves can reflect from outside of channel giving spurious signals.

Fig 10. 25: Superposed spectra

4.) Photon Attenuation Techniques ray. This technique has been discussed in the previous

section for measurement of void fraction. A single beam from a continuous X‐ ray source can be

used for steady state measurements. The beam is horizontal for vertical pipes and vertical for

horizontal systems. Apart from visualization of the random signals, the probability density

function (PDF) analysis of the time varying signal is often performed for a better appraisal of the

flow phenomenon. The technique can be explained as follows:measures the instantaneous

variation of void fraction based on attenuation of X‐ ray or

If the probability that the void fraction is less than some specific value is given by ,

then represents the probability per unit void fraction that the void fraction lies

between and . For this, in the void‐ time trace records, the void scale is broken into

equal increments of and time scale into equal increments of and time scale into equal

increments of During the total time interval T , if is seen in a total of times then

The ratio is the probability that lies within or mathematically

is the PDF of the particular record examined

Thus a probability density function can be seen as a "smoothed out" version of a histogram

depicting the distribution of amplitudes of the random signal. It may be noted that the time

interval over which the signal is recorded has an influence over the nature of PDF obtained. For

example, taken over the next time period may yield entirely different results depending

on the nature of flow field and the length of time over which the sample is obtained. For accurate

results, a number of records are considered and the PDF results are averaged. Mathematically,

If sufficient records are used for a statistically stationary process to cover a time interval large

compared with the longest significant period of fluctuation, the PDF curve becomes constant. The

result is then identical to a single PDF taken over the total period of time representing that used

for all k records.

The PDF curve estimates flow patterns from the position of the peaks. It is obvious that for bubbly

flow, a large count of at low void fraction and a small at higher values of is

obtained while the reverse (high local values at high mean void fraction) occurs during annular

flow. From the PDF curves depicting the different flow patterns as shown in Fig. 10.26 , it is

evident that a single peak at low void fraction denotes bubbly flow while a single peak at high

void fraction is indicative of annular flow. On the other hand, the slug flow pattern is

characterized by the presence of two peaks at low and high values of void fraction and can thus be

regarded as an intermediate between bubbly and annular flow

5.) The Impedance Technique: In this method, flow regimes are identified by conductivity

probes when there is a difference in electrical conductivity between the two phases and by

capacitance probes for different dielectric constant between them. Suitable design of probe to

identify holdup in horizontal and vertical pipes has already been discussed in the previous section.

For air‐ water systems, when the pipe is full of water, the electric circuit is closed through water

and the maximum voltage is detected at the output signal while in an empty pipe, the circuit is

open and the output signal is zero. One of the common techniques is the needle probe where the

current flowing from a needle facing directly into flow and a wall mounted electrode is

measured (Fig. 10.27a). The fluctuations in current as displayed on an oscilloscope or a PC

represents the prevailing flow pattern. For example, no contact between needle and wall indicates

a continuous gas core which is indicative of annular flow while high frequency interruptions of

current denote bubbly flow (Fig.10.27b). The main challenge in this technique is to pinpoint the

transition region e.g. distinguishing semi annular and annular flow, a situation difficult to predict

both by visual and contact methods. Usually a multiple wire conductance probe is used to identify

slug flow. The conductivity technique is an effective method for electrically conductivity liquids

while hot flux anemometry is adopted for non conducting fluids. The latter is an expensive

proposition. The PDF analysis of the random output is also adopted in this case for a more

objective identification technique.

Fig.10.27 Conductivity probe technique for evaluation of flow pattern in two‐ phase flow

6.) Electrochemical measurement of wall shear stress – The fluctuations in wall shear stress is

related to flow patterns eg a changing sign of wall shear stress denotes slug flow with the shear

being positive during the passage of Taylor bubble and negative during subsequent passage of

liquid slug. This is particularly effective in heated channels where most of the previous techniques

fail.

In conclusion, it may be noted that measurement of other parameters like temperature, pressure,

heat flux, heat and mass transfer coefficient and wall shear stress in two phase flow are equally

challenging. Some techniques suitable for the above measurement can be found in Hewitt (1978).