cabook.DVI - chap1.pdf

INTRODUCTION TO MULTIPHASE FLOW

1.1 INTRODUCTION

1.1.1 Scope

In the context of this book, the term

multiphase flow

is used to refer to

any fluid flow consisting of more than one phase or component. For brevity

and because they are covered in other texts, we exclude those circumstances

in which the components are well mixed above the molecular level. Conse-

quently, the flows considered here have some level of phase or component

separation at a scale well above the molecular level. This still leaves an

enormous spectrum of different multiphase flows. One could classify them

according to the state of the different phases or components and therefore

refer to gas/solids flows, or liquid/solids flows or gas/particle flows or bubbly

flows and so on; many texts exist that limit their attention in this way. Some

treatises are defined in terms of a specific type of fluid flow and deal with

low Reynolds number suspension flows, dusty gas dynamics and so on. Oth-

ers focus attention on a specific application such as slurry flows, cavitating

flows, aerosols, debris flows, fluidized beds and so on; again there are many

such texts. In this book we attempt to identify the basic fluid mechanical

phenomena and to illustrate those phenomena with examples from a broad

range of applications and types of flow.

Parenthetically, it is valuable to reflect on the diverse and ubiquitous chal-

lenges of multiphase flow. Virtually every processing technology must deal

with multiphase flow, from cavitating pumps and turbines to electropho-

tographic processes to papermaking to the pellet form of almost all raw

plastics. The amount of granular material, coal, grain, ore, etc. that is trans-

ported every year is enormous and, at many stages, that material is required

to flow. Clearly the ability to predict the fluid flow behavior of these pro-

cesses is central to the efficiency and effectiveness of those processes. For

example, the effective flow of toner is a major factor in the quality and speed

of electrophotographic printers. Multiphase flows are also a ubiquitous fea-

ture of our environment whether one considers rain, snow, fog, avalanches,

mud slides, sediment transport, debris flows, and countless other natural

phenomena to say nothing of what happens beyond our planet. Very critical

biological and medical flows are also multiphase, from blood flow to semen

the bends

to lithotripsy to laser surgery cavitation and so on. No single

list can adequately illustrate the diversity and ubiquity; consequently any

attempt at a comprehensive treatment of multiphase flows is flawed unless

it focuses on common phenomenological themes and avoids the temptation

to digress into lists of observations.

Two general topologies of multiphase flow can be usefully identified at

the outset, namely

disperse flows

and

separated flows

.By

disperse flows

we mean those consisting of finite particles, drops or bubbles (the disperse

phase) distributed in a connected volume of the continuous phase. On the

other hand

separated flows

consist of two or more continuous streams of

different fluids separated by interfaces.

1.1.2 Multiphase flow models

A persistent theme throughout the study of multiphase flows is the need to

model and predict the detailed behavior of those flows and the phenomena

that they manifest. There are three ways in which such models are explored:

(1) experimentally, through laboratory-sized models equipped with appro-

priate instrumentation, (2) theoretically, using mathematical equations and

models for the flow, and (3) computationally, using the power and size of

modern computers to address the complexity of the flow. Clearly there are

some applications in which full-scale laboratory models are possible. But,

in many instances, the laboratory model must have a very different scale

than the prototype and then a reliable theoretical or computational model

is essential for confident extrapolation to the scale of the prototype. There

are also cases in which a laboratory model is impossible for a wide variety

of reasons.

Consequently, the predictive capability and physical

understanding must

rely heavily on theoretical and/or computational models and here the com-

plexity of most multiphase flows presents a major hurdle. It may be possible

at some distant time in the future to code the Navier-Stokes equations for

each of the phases or components and to compute every detail of a multi-

phase flow, the motion of all the fluid around and inside every particle or

drop, the position of every interface. But the computer power and speed

required to do this is far beyond present capability for most of the flows

that are commonly experienced. When one or both of the phases becomes

turbulent (as often happens) the magnitude of the challenge becomes truly

astronomical. Therefore, simplifications are essential in realistic models of

most multiphase flows.

In disperse flows two types of models are prevalent,

trajectory models

and

two-fluid models

. In trajectory models, the motion of the disperse phase is

assessed by following either the motion of the actual particles or the motion

of larger, representative

particles

. The details of the flow around each of the

particles are subsumed into assumed drag, lift and moment forces acting on

and altering the trajectory of those particles. The thermal history of the

particles can also be tracked if it is appropriate to do so. Trajectory mod-

els have been very useful in studies of the rheology of granular flows (see

chapter 13) primarily because the effects of the interstitial fluid are small. In

the alternative approach,

two-fluid models

, the disperse phase is treated as

a second continuous phase intermingled and interacting with the continuous

phase. Effective conservation equations (of mass, momentum and energy) are

developed for the two fluid flows; these included interaction terms modeling

the exchange of mass, momentum and energy between the two flows. These

equations are then solved either theoretically or computationally. Thus, the

two-fluid models neglect the discrete nature of the disperse phase and ap-

proximate its effects upon the continuous phase. Inherent in this approach,

are averaging processes necessary to characterize the properties of the dis-

perse phase; these involve significant difficulties. The boundary conditions

appropriate in two-fluid models also pose difficult modeling issues.

In contrast, separated flows present many fewer issues. In theory one must

solve the single phase fluid flow equations in the two streams, coupling them

through appropriate kinematic and dynamic conditions at the interface. Free

streamline theory (see, for example, Birkhoff and Zarantonello 1957, Tulin

1964, Woods 1961, Wu 1972) is an example of a su

ccessful implementation

of such a strategy though the interface conditions used in that context are

particularly simple.

In the first part of this book, the basic tools for both trajectory and

two-fluid models are developed and discussed. In the remainder of this first

chapter, a basic notation for multiphase flow is developed and this leads

naturally into a description of the mass, momentum and energy equations

applicable to multiphase flows, and, in particular, in two-fluid models. In

chapters 2, 3 and 4, we examine the dynamics of individual particles, drops

and bubbles. In chapter 7 we address the different topologies of multiphase

flows and, in the subsequent chapters, we examine phenomena in which

particle

interactions and the particle-fluid interactions modify the flow.

1.1.3 Multiphase flow notation

The notation that will be used is close to the standard described by Wallis

(1969). It has however been slightly modified to permit more ready adop-

tion to the Cartesian tensor form. In particular the subscripts that can be

attached to a property will consist of a group of uppercase subscripts fol-

lowed by lowercase subscripts. The lower case subscripts (

,etc.)are

used in the conventional manner to denote vector or tensor components. A

single uppercase subscript (

) will refer to the property of a specific phase

or component. In some contexts generic subscripts

A, B

will be used

for generality. However, other letters such as

(continuous phase),

(disperse phase),

(liquid),

(gas),

(vapor) or

(solid) will be used for clarity in other contexts. Finally two upper-

case subscripts will imply the difference between the two properties for the

two single uppercase subscripts.

Specific properties frequently used are as follows.

Volumetric fluxes

(vol-

ume flow per unit area) of individual components will be denoted by

(

= 1, 2 or 3 in three dimensional flow). These are sometimes referred to as

superficial component velocities. The

total volumetric flux

is then given

...

(1.1)

Mass fluxes

are similarly denoted by

. Thus if the densities

of individual components are denoted by

,ρ

it follows that

;

(1.2)

Velocities of the specific phases are denoted by

or, in general, by

. The relative velocity between the two phases

and

will be denoted

ABi

such that

−

ABi

(1.3)

The volume fraction of a component or phase is denoted by

and, in

the case of two components or phases,

and

, it follows that

−

. Though this is clearly a well defined property for any finite volume in

the flow, there are some substantial problems associated with assigning a

value to an infinitesimal volume or point in the flow. Provided these can

be resolved, it follows that the volumetric flux of a component,

,andits

velocity are related by

(1.4)

and that

...

(1.5)

Two other fractional properties are only relevant in the context of one-

dimensional flows. The

volumetric quality

, is the ratio of the volumetric

flux of the component,

, to the total volumetric flux, i.e.

(1.6)

where the index

has been dropped from

and

because

is only used in

the context of one-dimensional flows and the

refer to cross-sectionally

averaged quantities.

The

mass fraction

, of a phase or component,

, is simply given by

/ρ

(see equation 1.8 for

). On the other hand the

mass quality

is often referred to simply as

the quality

and is the ratio of the mass flux of

component,

, to the total mass flux, or

(1.7)

Furthermore, when only two components or phases are present it is often

redundant to use subscripts on the volume fraction and the qualities since

−

,β

−

and

−X

. Thus unsubscripted quanti-

ties

and

will often be used in these circumstances.

It is clear that a multiphase mixture has certain

mixture

properties of

which the most readily evaluated is the

mixture

density denoted by

and

given by

(1.8)

On the other hand the specific enthalpy,

, and specific entropy,

,being

defined as per unit mass rather than per unit volume are weighted according

ρh

;

ρs

(1.9)

Other properties such as the

mixture

viscosity or thermal conductivity can-

not be reliably obtained from such simple weighted means.

Aside from the relative velocities between phases that were described ear-

lier, there are two other measures of relative motion that are frequently

used. The

drift velocity

of a component is defined as the velocity of that

component in a frame of reference moving at a velocity equal to the total

volumetric flux,

, and is therefore given by,

NJi

,where

NJi

−

(1.10)

Even more frequent use will be made of the

drift flux

of a component which

is defined as the volumetric flux of a component in the frame of reference

moving at

. Denoted by

NJi

this is given by

NJi

−

(

−

NJi

(1.11)

It is particularly important to notice that the sum of all the drift fluxes must

be zero since from equation 1.11

NJi

−

= 0

(1.12)

When only two phases or components,

and

, are present it follows that

AJi

−

BJi

and hence it is convenient to denote both of these drift fluxes

by the vector

ABi

where

ABi

AJi

−

BJi

(1.13)

Moreover it follows from 1.11 that

ABi

−

)

ABi

(1.14)

and hence the drift flux,

ABi

and the relative velocity,

ABi

,aresimply

related.

Finally, it is clear that certain basic relations follow from the above def-

initions and it is convenient to identify these here for later use. First the

relations between the volume and mass qualities that follow from equations

1.6 and 1.7 only involve ratios of the densities of the components:

;

(1.15)

On the other hand the relation between the volume fraction and the volume

quality necessarily involves some measure of the relative motion between

the phases (or components). The following useful results for two-phase (or

two-component) one-dimensional flows can readily be obtained from 1.11

and 1.6

;

−

(1.16)

which demonstrate the importance of the drift flux as a measure of the

relative motion.

1.1.4 Size distribution functions

In many multiphase flow contexts we shall make the simplifying assumption

that all the disperse phase particles (bubbles, droplets or solid particles)

have the same size. However in many natural and technological processes it

is necessary to consider the distribution of particle size. One fundamental

measure of this is the size distribution function,

(

), defined such that

the number of particles in a unit volume of the multiphase mixture with

volume between

and

(

)

. For convenience, it is often assumed

that the particles size can be represented by a single linear dimension (for

example, the diameter,

,orradius,

, in the case of spherical particles) so

that alternative size distribution functions,

(

)or

(

), may be used.

Examples of size distribution functions based on radius are shown in figures

1.1 and 1.2.

Often such information is presented in the form of cumulative number

distributions. For example the cumulative distribution,

∗

(

∗

), defined as

∗

(

∗

(

)

(1.17)

is the total number of particles of volume less than

∗

. Examples of cumu-

lative distributions (in this case for coal slurries) are shown in figure 1.3.

In these disperse flows, the evaluation of global quantities or characteris-

tics of the disperse phase will clearly require integration over the full range

of particle sizes using the size distribution function. For example, the volume

fraction of the disperse phase,

,isgivenby

∞

(

)

∞

(

)

(1.18)

where the last expression clearly applies to spherical particles. Other prop-

erties of the disperse phase or of the interactions between the disperse and

continuous phases can involve other moments of the size distribution func-

tion (see, for example, Friedlander 1977). This leads to a series of mean

Figure 1.1.

Measured size distribution functions for small bubbles in three

different water tunnels (Peterson

et al.

1975, Gates and Bacon 1978, Katz

1978) and in the ocean off Los Angeles, Calif. (O’Hern

et al.

1985).

Figure 1.2.

Size distribution functions for bubbles in freshly poured Guin-

ness and after five minutes. Adapted from Kawaguchi and Maeda (2003).

Figure 1.3.

Cumulative size distributions for various coal slurries.

Adapted from Shook and Roco (1991).

diameters (or sizes in the case of non-spherical particles) of the form,

where

∞

(

)

∞

(

)

−

(1.19)

A commonly used example is the

mass mean

diameter,

. On the other

hand processes that are controlled by particle surface area would be char-

acterized by the

surface area mean

diameter,

. The surface area mean

diameter would be important, for example, in determining the exchange of

heat between the phases or the rates of chemical interaction at the disperse

phase surface. Another measure of the average size that proves useful in

characterizing many disperse particulates is the Sauter mean diameter,

This is a measure of the ratio of the particle volume to the particle sur-

face area and, as such, is often used in characterizing particulates (see, for

example, chapter 14).

1.2 EQUATIONS OF MOTION

1.2.1 Averaging

In the section 1.1.3 it was implicitly assumed that there existed an

infinites-

imal

volume of dimension,

, such that

was not only very much smaller

than the typical distance over which the flow properties varied significantly

but also very much larger than the size of the individual phase elements (the

disperse phase particles, drops or bubbles). The first condition is necessary

in order to define derivatives of the flow properties within the flow field.

The second is necessary in order that each

averaging

volume (of volume

)

contain representative samples of each of the components or phases. In the

sections that follow (sections 1.2.2 to 1.2.9), we proceed to develop the ef-

fective differential equations of motion for multiphase flow assuming that

these conditions hold.

However, one of the more difficult hurdles in treating multiphase flows,

is that the above two conditions are rarely both satisfied. As a consequence

the averaging volumes contain a finite number of finite-sized particles and

therefore flow properties such as the continuous phase velocity vary signifi-

cantly from point to point within these averaging volumes. These variations

pose the challenge of how to define appropriate average quantities in the

averaging volume. Moreover, the gradients of those averaged flow properties

appear in the equations of motion that follow and the mean of the gradient

is not necessarily equal to the gradient of the mean. These difficulties will

be addressed in section 1.4 after we have explored the basic structure of the

equations in the absence of such complications.

1.2.2 Continuum equations for conservation of mass

Consider now the construction of the effective differential equations of mo-

tion for a disperse multiphase flow (such as might be used in a two-fluid

model) assuming that an appropriate elemental volume can be identified.

For convenience this elemental volume is chosen to be a

unit

cube with

edges parallel to the

directions. The mass flow of component

through one of the faces perpendicular to the

direction is given by

and therefore the net outflow of mass of component

from the cube is given

bythedivergenceof

∂

(

)

∂x

(1.20)

The rate of increase of the mass of component

stored in the elemental

volume is

∂

(

)

/∂t

and hence conservation of mass of component

requires that

∂

∂t

(

∂

(

)

∂x

(1.21)

where

is the rate of transfer of mass to the phase

from the other phases

per unit total volume. Such mass exchange would result from a phase change

or chemical reaction. This is the first of several phase interaction terms that

will be identified and, for ease of reference, the quantities

will termed

the

mass interaction terms

Clearly there will be a continuity equation like 1.21 for each phase or

component present in the flow. They will referred to as the Individual Phase

Continuity Equations (IPCE). However, since mass as a whole must be con-

served whatever phase changes or chemical reactions are happening it follows

that

= 0

(1.22)

and hence the sum of all the IPCEs results in a Combined Phase Continuity

Equation (CPCE) that does not involve

∂

∂t

∂

∂x

= 0

(1.23)

or using equations 1.4 and 1.8:

∂ρ

∂t

∂

∂x

= 0

(1.24)

Notice that only under the conditions of

zero

relative velocity in which

does this reduce to the Mixture Continuity Equation (MCE) which is

identical to that for an equivalent single phase flow of density

∂ρ

∂t

∂

∂x

(

ρu

) = 0

(1.25)

We also record that for one-dimensional duct flow the individual phase

continuity equation 1.21 becomes

∂

∂t

(

∂

∂x

(

Aρ

(1.26)

where

is measured along the duct,

(

) is the cross-sectional area,

,α

are cross-sectionally averaged quantities and

is the rate of transfer

of mass to the phase

per unit length of the duct. The sum over the

constituents yields the combined phase continuity equation

∂ρ

∂t

∂

∂x

= 0

(1.27)

When all the phases travel at the same speed,

, this reduces to

∂ρ

∂t

∂

∂x

(

ρAu

) = 0

(1.28)

Finally we should make note of the form of the equations when the two

components or species are intermingled rather than separated since we will

analyze several situations with gases diffusing through one another. Then

both components occupy the entire volume and the void fractions are effec-

tively unity so that the continuity equation 1.21 becomes:

∂ρ

∂t

∂

(

)

∂x

(1.29)

1.2.3 Disperse phase number continuity

Complementary to the equations of conservation of mass are the equations

governing the conservation of the number of bubbles, drops, particles, etc.

that constitute a disperse phase. If no such

particles

are created or destroyed

within the elemental volume and if the number of

particles

of the disperse

component,

, per unit total volume is denoted by

, it follows that

∂n

∂t

∂

∂x

(

) = 0

(1.30)

This will be referred to as the Disperse Phase Number Equation (DPNE).

If the volume of the particles of component

is denoted by

it follows

that

(1.31)

and substituting this into equation 1.21 one obtains

∂

∂t

(

∂

∂x

(

(1.32)

Expanding this equation using equation 1.30 leads to the following relation

for

∂

(

)

∂t

∂

(

)

∂x

(

)

(1.33)

where

denotes the Lagrangian derivative following the disperse

phase. This demonstrates a result that could, admittedly, be assumed, a

priori. Namely that the rate of transfer of mass to the component

in each

particle,

, is equal to the Lagrangian rate of increase of mass,

of each particle.

It is sometimes convenient in the study of bubbly flows to write the bubble

number conservation equation in terms of a population,

, of bubbles per

unit

liquid

volume rather than the number per unit total volume,

.Note

that if the bubble volume is

and the volume fraction is

then

−

)

;

(1 +

ηv

)

;

(1 +

ηv

)

(1.34)

and the bubble number conservation equation can be written as

∂u

∂x

−

(1 +

ηv

)

ηv

(1.35)

If the number population,

, is assumed uniform and constant (which re-

quires neglect of slip and the assumption of liquid incompressibility) then

equation 1.35 can be written as

∂u

∂x

ηv

(1.36)

In other words the divergence of the velocity field is directly related to the

Lagrangian rate of change in the volume of the bubbles.

1.2.4 Fick’s law

We digress briefly to complete the kinematics of two interdiffusing gases.

Equation 1.29 represented the conservation of mass for the two gases in

these circumstances. The kinematics are then completed by a statement of

Fick’s Law which governs the interdiffusion. For the gas,

,thislawis

−

ρD

∂

∂x

(1.37)

where

is the diffusivity.

1.2.5 Continuum equations for conservation of momentum

Continuing with the development of the differential equations, the next step

is to apply the momentum principle to the elemental volume. Prior to do-

ing so we make some minor modifications to that control volume in order

to avoid some potential difficulties. Specifically we deform the bounding

surfaces so that they never cut through disperse phase particles but every-

where are within the continuous phase. Since it is already assumed that the

dimensions of the particles are very small compared with the dimensions of

the control volume, the required modification is correspondingly small. It

is possible to proceed without this modification but several complications

arise. For example, if the boundaries cut through particles, it would then be

necessary to determine what fraction of the control volume surface is acted

upon by tractions within each of the phases and to face the difficulty of de-

termining the tractions within the particles. Moreover, we shall later need to

evaluate the interacting force between the phases within the control volume

and this is complicated by the issue of dealing with the parts of particles

intersected by the boundary.

Now proceeding to the application of the momentum theorem for either

thedisperse(

) or continuous phase (

), the flux of momentum

of the

component in the

direction through a side perpendicular to the

direction is

and hence the net flux of momentum (in the

di-

rection) out of the elemental volume is

∂

(

)

/∂x

.Therateof

increase of momentum of component

in the

direction within the ele-

mental volume is

∂

(

)

/∂t

. Thus using the momentum conservation

principle, the net force in the

direction acting on the component

in the

control volume (of unit volume),

,mustbegivenby

∂

∂t

(

∂

∂x

(

)

(1.38)

It is more difficult to construct the forces,

in order to complete the

equations of motion. We must include body forces acting within the control

volume, the force due to the pressure and viscous stresses on the exterior of

the control volume, and, most particularly, the force that each component

imposes on the other components within the control volume.

The first contribution is that due to an external force field on the compo-

nent

within the control volume. In the case of gravitational forces, this is

clearly given by

(1.39)

where

is the component of the gravitational acceleration in the

direction

(the direction of

is considered vertically downward).

The second contribution, namely that due to the tractions on the control

volume, differs for the two phases because of the small deformation discussed

above. It is zero for the disperse phase. For the continuous phase we define

the stress tensor,

Cki

, so that the contribution from the surface tractions

to the force on that phase is

∂σ

Cki

∂x

(1.40)

For future purposes it is also convenient to decompose

Cki

into a pressure,

, and a deviatoric stress,

Cki

−

pδ

Cki

(1.41)

where

is the Kronecker delta such that

=1for

and

=0for

The third contribution to

is the force (per unit total volume) imposed

the component

the other components within the control volume.

We write this as

so that the Individual Phase Momentum Equation

(IPME) becomes

∂

∂t

(

∂

∂x

(

)

−

∂p

∂x

−

∂σ

Cki

∂x

(1.42)

where

= 0 for the disperse phase and

= 1 for the continuous phase.

Thus we identify the second of the interaction terms, namely the

force

interaction

. Note that, as in the case of the mass interaction

,it

must follow that

= 0

(1.43)

In disperse flows it is often useful to separate

into two components, one

due to the pressure gradient in the continuous phase,

−

∂p/∂x

,andthe

remainder,

, due to other effects such as the relative motion between

the phases. Then

−F

−

∂p

∂x

(1.44)

The IPME 1.42 are frequently used in a form in which the terms on the

left hand side are expanded and use is made of the continuity equation

1.21. In single phase flow this yields a Lagrangian time derivative of the

velocity on the left hand side. In the present case the use of the continuity

equation results in the appearance of the mass interaction,

. Specifically,

one obtains

∂u

∂t

∂u

∂x

−I

−

∂p

∂x

−

∂σ

Cki

∂x

(1.45)

Viewed from a Lagrangian perspective, the left hand side is the normal rate

of increase of the momentum of the component

;theterm

is the

rate of increase of the momentum in the component

due to the gain of

mass by that phase.

If the momentum equations 1.42 for each of the components are added

together the resulting Combined Phase Momentum Equation (CPME) be-

comes

∂

∂t

∂

∂x

ρg

−

∂p

∂x

∂σ

Cki

∂x

(1.46)

Note that this equation 1.46 will only reduce to the equation of motion

for a single phase flow in the absence of relative motion,

.Note

also that, in the absence of any motion (when the deviatoric stress is zero),

equation 1.46 yields the appropriate hydrostatic pressure gradient

∂p/∂x

ρg

based on the mixture density,

Another useful limit is the case of uniform and constant sedimentation

of the disperse component (volume fraction,

−

) through the

continuous phase under the influence of gravity. Then equation 1.42 yields

αρ

∂σ

Cki

∂x

+(1

−

)

(1.47)

But

−F

and, in this case, the deviatoric part of the continuous

phase stress should be zero (since the flow is a simple uniform stream) so

that

Ckj

−

. It follows from equation 1.47 that

−F

−

αρ

and

∂p/∂x

ρg

(1.48)

or, in words, the pressure gradient is hydrostatic.

Finally, note that the equivalent one-dimensional or duct flow form of the

IPME is

∂

∂t

(

∂

∂x

Aρ

−

∂p

∂x

Pτ

(1.49)

where, in the usual pipe flow notation,

(

) is the perimeter of the cross-

section and

is the wall shear stress. In this equation,

is the force

imposed on the component

in the

direction by the other components

per unit length of the duct. A sum over the constituents yields the combined

phase momentum equation for duct flow, namely

∂

∂t

∂

∂x

−

∂p

∂x

−

Pτ

ρg

(1.50)

and, when all phases travel at the same velocity,

, this reduces to

∂

∂t

(

ρu

∂

∂x

Aρu

−

∂p

∂x

−

Pτ

ρg

(1.51)

1.2.6 Disperse phase momentum equation

At this point we should consider the relation between the equation of mo-

tion for an individual particle of the disperse phase and the Disperse Phase

Momentum Equation (DPME) delineated in the last section. This relation

is analogous to that between the number continuity equation and the Dis-

perse Phase Continuity Equation (DPCE). The construction of the equation

of motion for an individual particle in an infinite fluid medium will be dis-

cussed at some length in chapter 2. It is sufficient at this point to recognize

that we may write Newton’s equation of motion for an individual particle

of volume

in the form

(

(1.52)

where

is the Lagrangian time derivative following the particle so

that

≡

∂

∂t

∂

∂x

(1.53)

and

is the force that the surrounding continuous phase imparts to the

particle in the direction

.Notethat

will include not only the force due

to the velocity and acceleration of the particle relative to the fluid but also

the

buoyancy

forces due to pressure gradients within the continuous phase.

Expanding 1.52 and using the expression 1.33 for the mass interaction,

one obtains the following form of the DPME:

∂u

∂t

∂u

∂x

(1.54)

Now examine the implication of this relation when considered alongside

the IPME 1.45 for the disperse phase. Setting

in equation 1.45,

expanding and comparing the result with equation 1.54 (using the continuity

equation 1.21) one observes that

(1.55)

Hence the appropriate force interaction term in the disperse phase momen-

tum equation is simply the sum of the fluid forces acting on the individual

particles in a unit volume, namely

. As an example note that the

steady, uniform sedimentation interaction force

given by equation 1.48,

when substituted into equation 1.55, leads to the result

−

or,

in words, a fluid force on an individual particle that precisely balances the

weight of the particle.

1.2.7 Comments on disperse phase interaction

In the last section the relation between the force interaction term,

and the force,

, acting on an individual particle of the disperse phase was

established. In chapter 2 we include extensive discussions of the forces acting

on a single particle moving in a infinite fluid. Various forms of the fluid force,

,acting

the particle are presented (for example, equations 2.47, 2.49,

2.50, 2.67, 2.71, 3.20) in terms of (a) the particle velocity,

,(b)the

fluid velocity

that would have existed at the center of the particle

in the latter’s absence and (c) the relative velocity

−

Downstream of some disturbance that creates a relative velocity,

,the

drag will tend to reduce that difference. It is useful to characterize the rate

of equalization of the particle (mass,

, and radius,

) and fluid velocities

by defining a velocity

relaxation

time,

. For example, it is common in

dealing with gas flows laden with small droplets or particles to assume that

the equation of motion can be approximated by just two terms, namely the

particle inertia and a Stokes drag, which for a spherical particle is 6

πμ

(see section 2.2.2). It follows that the relative velocity decays exponentially

with a time constant,

,givenby

πRμ

(1.56)

This is known as the velocity relaxation time. A more complete treatment

that includes other parametric cases and other fluid mechanical effects is

contained in sections 2.4.1 and 2.4.2.

There are many issues with the equation of motion for the disperse phase

that have yet to be addressed. Many of these are delayed until section 1.4

and others are addressed later in the book, for example in sections 2.3.2,

2.4.3 and 2.4.4.

1.2.8 Equations for conservation of energy

The third fundamental conservation principle that is ut

ilized in developing

the basic equations of fluid mechanics is the principle of conservation of

energy. Even in single phase flow the general statement of this principle is

complicated when energy transfer processes such as heat conduction and

viscous dissipation are included in the analysis. Fortunately it is frequently

possible to show that some of these complexities have a negligible effect on

the results. For example, one almost always neglects viscous and heat con-

duction effects in preliminary analyses of gas dynamic flows. In the context

of multiphase flows the complexities involved in a general statement of en-

ergy conservation are so numerous that it is of little value to attempt such

generality. Thus we shall only present a simplified version that neglects, for

example, viscous heating and the global conduction of heat (though not the

heat transfer from one phase to another).

However these limitations are often minor compared with other difficul-

ties that arise in constructing an energy equation for multiphase flows. In

single-phase flows it is usually adequate to assume that the fluid is in an

equilibrium thermodynamic state at all points in the flow and that an appro-

priate thermodynamic constraint (for example,

constant

and

locally uniform

entropy or temperature) may be used to relate the pressure, density, tem-

perature, entropy, etc. In many multiphase flows the different phases and/or

components are often

not

in equilibrium and consequently thermodynamic

equilibrium arguments that might be appropriate for single phase flows are

no longer valid. Under those circumstances it is important to evaluate the

heat and mass transfer occuring between the phases and/or components;

discussion on this is delayed until the next section 1.2.9.

In single phase flow application of the principle of energy conservation

to the control volume (CV) uses the following statement of the first law of

thermodynamics:

Rate of heat addition to the CV,

+ Rate of work done on the CV,

Net flux of total internal energy out of CV

+ Rate of increase of total internal energy in CV

In chemically non-reacting flows the total internal energy per unit mass,

∗

is the sum of the internal energy,

, the kinetic energy

are the

velocity components) and the potential energy

(where

is a coordinate

measured in the vertically upward direction):

∗

(1.57)

Consequently the energy equation in single phase flow becomes

∂

∂t

(

ρe

∗

∂

∂x

(

ρe

∗

W−

∂

∂x

(

)

(1.58)

where

is the stress tensor. Then if there is no heat addition to (

=0)

or external work done on (

= 0) the CV and if the flow is steady with no

viscous effects (no deviatoric stresses), the energy equation for single phase

flow becomes

∂

∂x

ρu

∗

∂

∂x

{

ρu

∗

}

= 0

(1.59)

where

∗

p/ρ

is the total enthalpy per unit mass. Thus, when the

total enthalpy of the incoming flow is uniform,

∗

is constant everywhere.

Now examine the task of constructing an energy equation for each of the

components or phases in a multiphase flow. First, it is necessary to define a

total internal energy density,

∗

, for each component

such that

∗

(1.60)

Then an appropriate statement of the first law of thermodynamics for each

phase (the individual phase energy equation, IPEE) is as follows:

Rate of heat addition to

from outside CV,

+ Rate of work done to

by the exterior surroundings,

+Rateofheattransferto

within the CV,

+ Rate of work done to

by other components in CV,

Rate of increase of total kinetic energy of

in CV

+ Net flux of total internal energy of

out of the CV

where each of the terms is conveniently evaluated for a unit total volume.

First note that the last two terms can be written as

∂

∂t

(

∗

∂

∂x

(

∗

)

(1.61)

Turning then to the upper part of the equation, the first term due to external

heating and to conduction of heat from the surroundings into the control

volume is left as

. The second term contains two contributions: (i) minus

the rate of work done by the stresses acting on the component

on the

surface of the control volume and (ii) the rate of external

shaft work

done on the component

. In evaluating the first of these, we make the same

modification to the control volume as was discussed in the context of the

momentum equation; specifically we make small deformations to the control

volume so that its boundaries lie wholly within the continuous phase. Then

using the continuous phase stress tensor,

Cij

,asdefinedinequation1.41

the expressions for

become:

∂

∂x

(

Cij

)and

(1.62)

The individual phase energy equation may then be written as

∂

∂t

(

∗

∂

∂x

(

∗

∂

∂x

(

Cij

)

(1.63)

Note that the two terms involving internal exchange of energy between the

phases may be combined into an

energy interaction

term given by

. It follows that

and

(1.64)

Moreover, the work done terms,

, may clearly be related to the inter-

action forces,

. In a two-phase system with one disperse phase:

−QI

and

−WI

−

and

−E

(1.65)

As with the continuity and momentum equations, the individual phase

energy equations can be summed to obtain the combined phase energy equa-

tion (CPEE). Then, denoting the total rate of external heat added (per unit

total volume) by

and the total rate of external shaft work done (per unit

total volume) by

where

and

(1.66)

the CPEE becomes

∂

∂t

∗

∂

∂x

−

Cij

∗

(1.67)

When the left hand sides of the individual or combined phase equations,

1.63 and 1.67, are expanded and use is made of the continuity equation 1.21

and the momentum equation 1.42 (in the absence of deviatoric stresses),

the results are known as the

thermodynamic

forms of the energy equations.

Using the expressions 1.65 and the relation

+ constant

(1.68)

between the internal energy,

, the specific heat at constant volume,

and the temperature,

, of each phase, the thermodynamic form of the

IPEE can be written as

∂T

∂t

∂T

∂x

Cij

∂u

∂x

(

−

)

−

(

∗

−

)

(1.69)

and, summing these, the thermodynamic form of the CPEE is

∂T

∂t

∂T

∂x

Cij

∂u

∂x

−F

(

−

)

−I

(

∗

−

∗

(1.70)

In equations 1.69 and 1.70, it has been assumed that the specific heats,

can be assumed to be constant and uniform.

Finally we note that the one-dimensional duct flow version of the IPEE,

equation 1.63, is

∂

∂t

(

∗

∂

∂x

(

Aρ

∗

−

∂

∂x

(

)

(1.71)

where

is the rate of external heat addition to the component

per

unit length of the duct,

is the rate of external work done on component

per unit length of the duct,

is the rate of energy transferred to the

component

from the other phases per unit length of the duct and

is the

pressure in the continuous phase neglecting deviatoric stresses. The CPEE,

equation 1.67, becomes

∂

∂t

∗

∂

∂x

Aρ

∗

W−

∂

∂x

(

)

(1.72)

where

is the total rate of external heat addition to the flow per unit

length of the duct and

is the total rate of external work done on the

flow per unit length of the duct.

1.2.9 Heat transfer between separated phases

In the preceding section, the rate of heat transfer,

, to each phase,

from the other phases was left undefined. Now we address the functional

form of this rate of heat transfer in the illustrative case of a two-phase flow

consisting of a disperse solid particle or liquid droplet phase and a gaseous

continuous phase.

In section 1.2.7, we defined a relaxation time that typifies the natural at-

tenuation of velocity differences between the phases. In an analogous man-

ner, the temperatures of the phases might be different downstream of a flow

disturbance and consequently there would be a second

relaxation

time as-

sociated with the equilibration of temperatures through the process of heat

transfer between the phases. This temperature relaxation time is denoted

and can be obtained by equating the rate of heat transfer from the

continuous phase to the particle with the rate of increase of heat stored

in the particle. The heat transfer to the particle can occur as a result of

conduction, convection or radiation and there are practical flows in which

each of these mechanisms are important. For simplicity, we shall neglect the

radiation component. Then, if the relative motion between the particle and

the gas is sufficiently small, the only contributing mechanism is conduction

and it will be limited by the thermal conductivity,

, of the gas (since

the thermal conductivity of the particle is usually much greater). Then the

rate of heat transfer to a particle (radius

) will be given approximately by

πRk

(

−

)where

and

are representative temperatures of the

gas and particle respectively.

Now we add in the component of heat transfer by the convection caused

byrelativemotion.TodosowedefinetheNusseltnumber,

,astwicethe

ratio of the rate of heat transfer with convection to that without convection.

Then the rate of heat transfer becomes

times the above result for con-

duction. Typically, the Nusselt number is a function of both the Reynolds

number of the relative motion,

WR/ν

(where

is the typical mag-

nitude of (

−

)), and the Prandtl number,

.One

frequently used expression for

(see Ranz and Marshall 1952) is

=2+0

(1.73)

and, of course, this reduces to the pure conduction result,

= 2, when the

second term on the right hand side is small.

Assuming that the particle temperature has a roughly uniform value of

, it follows that

πRk

(

−

)

(1.74)

where the material derivative,

D/Dt

, follows the particle. This provides the

equation that must be solved for

namely

(

−

)

(1.75)

where

(1.76)

Clearly

represents a typical time for equilibration of the temperatures in

the two phases, and is referred to as the

temperature relaxation time

The above construction of the temperature relaxation time and the equa-

tion for the particle temperature represents perhaps the simplest formulation

that retains the essential ingredients. Many other effects may become impor-

tant and require modification of the equations. Examples are the rarefied gas

effects and turbulence effects. Moreover, the above was based on a uniform

particle temperature and steady state heat transfer correlations; in many

flows heat transfer to the particles is highly transient and a more accurate

heat transfer model is required. For a discussion of these effects the reader

is referred to Rudinger (1969) and Crowe

et al.

(1998).

1.3 INTERACTION WITH TURBULENCE

1.3.1 Particles and turbulence

Turbulent flows of a single Newtonian fluid, even those of quite simple ex-

ternal geometry such as a fully-developed pipe flow, are very complex and

their solution at high Reynolds numbers requires the use of empirical mod-

els to represent the unsteady motions. It is self-evident that the addition of

particles to such a flow will result in;

1. complex unsteady motions of the particles that may result in non-uniform spatial

distribution of the particles and, perhaps, particle segregation. It can also result

in particle agglomeration or in particle fission, especially if the particles are

bubbles or droplets.

2. modifications of the turbulence itself caused by the presence and motions of the

particles. One can visualize that the turbulence could be damped by the presence

of particles, or it could be enhanced by the wakes and other flow disturbances

that the motion of the particles may introduce.

In the last twenty five years, a start has been made in the understanding

of these complicated issues, though many aspects remain to be understood.

The advent of laser Doppler velocimetry resulted in the first measurements

of these effects; and the development of direct numerical simulation allowed

the first calculations of these complex flows, albeit at rather low Reynolds

numbers. Here we will be confined to a brief summary of these complex

issues. The reader is referred to the early review of Hetsroni (1989) and the

text by Crowe

et al.

(1998) for a summary of the current

understanding.

To set the stage, recall that turbulence is conveniently characterized at

any point in the flow by the Kolmogorov length and time scales,

and

given by

and

(1.77)

where

is the kinematic viscosity and

is the mean rate of dissipation per

unit mass of fluid. Since

is proportional to

where

and

are the

typical velocity and dimension of the flow, it follows that

λ/

∝

−

and

Uτ/

∝

−

(1.78)

and the difficulties in resolving the flow either by measurement or by com-

putation increase as

increases.

Gore and Crowe (1989) collected data from a wide range of turbulent

pipe and jet flows (all combinations of gas, liquid and solid flows, volume

fractions from 2

−

to 0

2, density ratios from 0

001 to 7500, Reynolds

numbers from 8000 to 100

000) and constructed figure 1.4 which plots the

fractional change in the turbulence intensity (defined as the rms fluctuating

velocity) as a result of the introduction of the disperse phase against the

ratio of the particle size to the turbulent length scale,

. They judge

that the most appropriate turbulent length scale,

, is the size of the most

energetic eddy. Single phase experiments indicate that

is about 0

2times

the radius in a pipe flow and 0

039 times the distance from the exit in a jet

flow. To explain figure 1.4 Gore and Crowe argue that when the particles

are small compared with the turbulent length scale, they tend to follow the

turbulent fluid motions and in doing so absorb energy from them thus re-

ducing the turbulent energy. It appears that the turbulence reduction is a

strong function of Stokes number,

πRμτ

, the ratio of the particle