Brennen Ch.E. Fundamentals of Multiphase Flow

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Figure 6.6. Sketch for the ﬁlm boiling analysis.

on the vapor element will be its weight ρ

gδdy and the shear stress at the

wall that we will estimate to be given roughly by µ

u/δ. Then if the vapor

momentum ﬂuxes are neglected the balance of forces on the vapor element

yields

u =

(ρ

− ρ

)gδ

(6.11)

Substituting this expression for u into equation 6.10 and solving for δ(y)

assuming that the origin of y is chosen to be the origin or virtual origin of

the vapor layer where δ = 0 we obtain the following expression for δ(y)

δ(y)=



∆Tµ

3ρ

(ρ

− ρ

)gL



(6.12)

This deﬁnes the geometry of the ﬁlm.

We can then evaluate the heat ﬂux ˙q(y) per unit surface area of the plate;

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the local heat transfer coeﬃcient, ˙q/∆T becomes

˙q(y)

∆T



3ρ

(ρ

− ρ

)gLk

4∆Tµ



−

(6.13)

Note that this is singular at y = 0. It also follows by integration that the

overall heat transfer coeﬃcient for a plate extending from y =0toy =  is







(ρ

− ρ

)gLk

∆Tµ





(6.14)

This characterizes the ﬁlm boiling heat transfer coeﬃcients in the upper

right of ﬁgure 6.1. Though many features of the ﬂow have been neglected this

relation gives good agreement with the experimental observations (Westwa-

ter 1958). Other geometrical arrangements such as heated circular pipes on

which ﬁlm boiling is occurring will have a similar functional dependence on

the properties of the vapor and liquid (Collier and Thome 1994, Whalley

1987).

6.4 CONDENSATION

The spectrum of ﬂow processes associated with condensation on a solid

surface are almost a mirror image of those involved in boiling. Thus drop

condensation on the underside of a cooled horizontal plate or on a vertical

surface is very analogous to nucleate boiling. The phenomenon is most ap-

parent as the misting up of windows or mirrors. When the population of

droplets becomes large they run together to form condensation ﬁlms, the

dominant form of condensation in most industrial contexts. Because of the

close parallels with boiling ﬂows, it would be superﬂuous to repeated the

analyses for condensation ﬂows. However, in the next section we include the

speciﬁcs of one example, namely ﬁlm condensation on a vertical surface. For

more detail on condensation ﬂows the reader is referred to the reviews by

Butterworth (1977).

6.4.1 Film condensation

The circumstance of ﬁlm condensation on a vertical plate as sketched in

ﬁgure 6.7 allows an analysis that is precisely parallel to that for ﬁlm boiling

detailed in section 6.3.1. The obvious result is a ﬁlm thickness, δ(y)(where

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Figure 6.7. Sketch for the ﬁlm condensation analysis.

y is now measured vertically downward) given by

δ(y)=



(−∆T )µ

3ρ

(ρ

− ρ

)gL



(6.15)

a local heat transfer coeﬃcient given by

˙q(y)

∆T



3ρ

(ρ

− ρ

)gLk

4(−∆T )µ



−

(6.16)

and the following overall heat transfer coeﬃcient for a plate of length :







(ρ

− ρ

)gLk

(−∆T )µ





(6.17)

Clearly the details of ﬁlm condensation will be diﬀerent for diﬀerent geo-

metric conﬁgurations of the solid surface (inclined walls, horizontal tubes,

161

etc.) and for laminar or turbulent liquid ﬁlms. For such details, the reader

is referred to the valuable review by Collier and Thome (1994).

162

FLOW PATTERNS

7.1 INTRODUCTION

From a practical engineering point of view one of the major design diﬃ-

culties in dealing with multiphase ﬂow is that the mass, momentum, and

energy transfer rates and processes can be quite sensitive to the geometric

distribution or topology of the components within the ﬂow. For example, the

geometry may strongly eﬀect the interfacial area available for mass, momen-

tum or energy exchange between the phases. Moreover, the ﬂow within each

phase or component will clearly depend on that geometric distribution. Thus

we recognize that there is a complicated two-way coupling between the ﬂow

in each of the phases or components and the geometry of the ﬂow (as well as

the rates of change of that geometry). The complexity of this two-way cou-

pling presents a major challenge in the study of multiphase ﬂows and there

is much that remains to be done before even a superﬁcial understanding is

achieved.

An appropriate starting point is a phenomenological description of the

geometric distributions or ﬂow patterns thatareobservedincommonmulti-

phase ﬂows. This chapter describes the ﬂow patterns observed in horizontal

and vertical pipes and identiﬁes a number of the instabilities that lead to

transition from one ﬂow pattern to another.

7.2 TOPOLOGIES OF MULTIPHASE FLOW

7.2.1 Multiphase ﬂow patterns

A particular type of geometric distribution of the components is called a ﬂow

pattern or ﬂow regime and many of the names given to these ﬂow patterns

(such as annular ﬂow or bubbly ﬂow) are now quite standard. Usually the

ﬂow patterns are recognized by visual inspection, though other means such

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as analysis of the spectral content of the unsteady pressures or the ﬂuctu-

ations in the volume fraction have been devised for those circumstances in

which visual information is diﬃcult to obtain (Jones and Zuber, 1974).

For some of the simpler ﬂows, such as those in vertical or horizontal pipes,

a substantial number of investigations have been conducted to determine

the dependence of the ﬂow pattern on component volume ﬂuxes, (j

on volume fraction and on the ﬂuid properties such as density, viscosity,

and surface tension. The results are often displayed in the form of a ﬂow

regime map that identiﬁes the ﬂow patterns occurring in various parts of a

parameter space deﬁned by the component ﬂow rates. The ﬂow rates used

may be the volume ﬂuxes, mass ﬂuxes, momentum ﬂuxes, or other similar

quantities depending on the author. Perhaps the most widely used of these

ﬂow pattern maps is that for horizontal gas/liquid ﬂow constructed by Baker

(1954). Summaries of these ﬂow pattern studies and the various empirical

laws extracted from them are a common feature in reviews of multiphase

ﬂow (see, for example, Wallis 1969 or Weisman 1983).

The boundaries between the various ﬂow patterns in a ﬂow pattern map

occur because a regime becomes unstable as the boundary is approached

and growth of this instability causes transition to another ﬂow pattern. Like

the laminar-to-turbulent transition in single phase ﬂow, these multiphase

transitions can be rather unpredictable since they may depend on otherwise

minor features of the ﬂow, such as the roughness of the walls or the entrance

conditions. Hence, the ﬂow pattern boundaries are not distinctive lines but

more poorly deﬁned transition zones.

But there are other serious diﬃculties with most of the existing literature

on ﬂow pattern maps. One of the basic ﬂuid mechanical problems is that

these maps are often dimensional and therefore apply only to the speciﬁc

pipe sizes and ﬂuids employed by the investigator. A number of investiga-

tors (for example Baker 1954, Schicht 1969 or Weisman and Kang 1981)

have attempted to ﬁnd generalized coordinates that would allow the map to

cover diﬀerent ﬂuids and pipes of diﬀerent sizes. However, such generaliza-

tions can only have limited value because several transitions are represented

in most ﬂow pattern maps and the corresponding instabilities are governed

by diﬀerent sets of ﬂuid properties. For example, one transition might occur

at a critical Weber number, whereas another boundary may be character-

ized by a particular Reynolds number. Hence, even for the simplest duct

geometries, there exist no universal, dimensionless ﬂow pattern maps that

incorporate the full, parametric dependence of the boundaries on the ﬂuid

characteristics.

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Beyond these diﬃculties there are a number of other troublesome ques-

tions. In single phase ﬂow it is well established that an entrance length of 30

to 50 diameters is necessary to establish fully developed turbulent pipe ﬂow.

The corresponding entrance lengths for multiphase ﬂow patterns are less well

established and it is quite possible that some of the reported experimental

observations are for temporary or developing ﬂow patterns. Moreover, the

implicit assumption is often made that there exists a unique ﬂow pattern

for given ﬂuids with given ﬂow rates. It is by no means certain that this is

the case. Indeed, in chapter 16, we shall see that even very simple models

of multiphase ﬂow can lead to conjugate states. Consequently, there may

be several possible ﬂow patterns whose occurence may depend on the ini-

tial conditions, speciﬁcally on the manner in which the multiphase ﬂow is

generated.

In summary, there remain many challenges associated with a fundamental

understanding of ﬂow patterns in multiphase ﬂow and considerable research

is necessary before reliable design tools become available. In this chapter

we shall concentrate on some of the qualitative features of the boundaries

between ﬂow patterns and on the underlying instabilities that give rise to

those transitions.

7.2.2 Examples of ﬂow regime maps

Despite the issues and reservations discussed in the preceding section it is

useful to provide some examples of ﬂow regime maps along with the deﬁ-

nitions that help distinguish the various regimes. We choose to select the

ﬁrst examples from the ﬂows of mixtures of gas and liquid in horizontal

and vertical tubes, mostly because these ﬂows are of considerable industrial

interest. However, many other types of ﬂow regime maps could be used as

examples and some appear elsewhere in this book; examples are the ﬂow

regimes described in the next section and those for granular ﬂows indicated

in ﬁgure 13.5.

We begin with gas/liquid ﬂows in horizontal pipes (see, for example, Hub-

bard and Dukler 1966, Wallis 1969, Weisman 1983). Figure 7.1 shows the

occurence of diﬀerent ﬂow regimes for the ﬂow of an air/water mixture in a

horizontal, 5.1cm diameter pipe where the regimes are distinguished visually

using the deﬁnitions in ﬁgure 7.2. The experimentally observed transition

regions are shown by the hatched areas in ﬁgure 7.1. The solid lines repre-

sent theoretical predictions some of which are discussed later in this chapter.

Note that in a mass ﬂux map like this the ratio of the ordinate to the abscissa

is X /(1 −X) and therefore the mass quality, X , is known at every point in

165

the map. There are many industrial processes in which the mass quality is

a key ﬂow parameter and therefore mass ﬂux maps are often preferred.

Other examples of ﬂow regime maps for horizontal air/water ﬂow (by

diﬀerent investigators) are shown in ﬁgures 7.3 and 7.4. These maps plot

the volumetric ﬂuxes rather than the mass ﬂuxes but since the densities

of the liquid and gas in these experiments are relatively constant, there is

a rough equivalence. Note that in a volumetric ﬂux map the ratio of the

ordinate to the abscissa is β/(1 − β)and therefore the volumetric quality, β,

is known at every point in the map.

Figure 7.4 shows how the boundaries were observed to change with pipe

diameter. Moreover, ﬁgures 7.1 and 7.4 appear to correspond fairly closely.

Note that both show well-mixed regimes occuring above some critical liquid

ﬂux and above some critical gas ﬂux; we expand further on this in section

7.3.1.

Figure 7.1. Flow regime map for the horizontal ﬂow of an air/water mix-

ture in a 5.1cm diameter pipe with ﬂow regimes as deﬁned in ﬁgure 7.2.

Hatched regions are observed regime boundaries, lines are theoretical pre-

dictions. Adapted from Weisman (1983).

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Figure 7.2. Sketches of ﬂow regimes for ﬂow of air/water mixtures in a

horizontal, 5.1cm diameter pipe. Adapted from Weisman (1983).

Figure 7.3. A ﬂow regime map for the ﬂow of an air/water mixture in a

horizontal, 2.5cm diameter pipe at 25

◦

C and 1bar. Solid lines and points

are experimental observations of the transition conditions while the hatched

zones represent theoretical predictions. From Mandhane et al. (1974).

167

Figure 7.4. Same as ﬁgure 7.3 but showing changes in the ﬂow regime

boundaries for various pipe diameters: 1.25cm (dotted lines), 2.5cm (solid

lines), 5cm (dash-dot lines) and 30cm (dashed lines). From Mandhane et

al. (1974).

7.2.3 Slurry ﬂow regimes

As a further example, consider the ﬂow regimes manifest by slurry

(solid/liquid mixture) ﬂow in a horizontal pipeline. When the particles are

small so that their settling velocity is much less than the turbulent mixing

velocities in the ﬂuid and when the volume fraction of solids is low or moder-

ate, the ﬂow will be well-mixed. This is termed the homogeneous ﬂow regime

(ﬁgure 7.5) and typically only occurs in practical slurry pipelines when all

the particle sizes are of the order of tens of microns or less. When somewhat

larger particles are present, vertical gradients will occur in the concentra-

tion and the regime is termed heterogeneous; moreover the larger particles

will tend to sediment faster and so a vertical size gradient will also occur.

The limit of this heterogeneous ﬂow regime occurs when the particles form

a packed bed in the bottom of the pipe. When a packed bed develops, the

ﬂow regime is known as a saltation ﬂow. In a saltation ﬂow, solid material

may be transported in two ways, either because the bed moves en masse or

because material in suspension above the bed is carried along by the sus-

pending ﬂuid. Further analyses of these ﬂow regimes, their transitions and

their pressure gradients are included in sections 8.2.1, 8.2.2 and 8.2.3. For

further detail, the reader is referred to Shook and Roco (1991), Zandi and

Govatos (1967), and Zandi (1971).

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