14
DRIFT FLUX MODELS
14.1 INTRODUCTION
In this chapter we consider a class of models of multiphase flows in which
the relative motion between the phases is governed by a particular subset of
the flow parameters. The members of this subset are called
drift flux models
and were first developed by Zuber (see, for example, Zuber and Findlay
1965) and Wallis (1969) among others. To define the subset consider the
one-dimensional flow of a mixture of the two components,
A
and
B
.From
the definitions 1.4, 1.5 and 1.14, the volumetric fluxes of the two components,
j
A
and
j
B
, are related to the total volumetric flux,
j
, the drift flux,
j
AB
,
and the volume fraction,
α
=
α
A
=1
−
α
B
,by
j
A
=
αj
+
j
AB
;
j
B
=(1
−
α
)
j
−
j
AB
(14.1)
Frequently, it is necessary to determine the basic kinematics of such a flow,
for example by determining
α
given
j
A
and
j
B
.Todosoitisclearlynec-
essary to determine the drift flux,
j
AB
, and, in general, one must consider
the dynamics, the forces on the individual phases in order to determine
the relative motion. In some cases, this will require the introduction and
simultaneous solution of momentum and energy equations, a problem that
rapidly becomes mathematically complicated. There exists, however, a class
of problems in which the dominant relative motion is caused by an external
force such as gravity and therefore, to a reasonably good approximation,
is a simple function only of the magnitude of that external force (say the
acceleration due to gravity,
g
), of the volume fraction,
α
,andofthephysi-
cal properties of the components (densities,
ρ
A
and
ρ
B
, and viscosities,
μ
A
and
μ
B
). The drift flux models were designed for these circumstances. If the
relative velocity,
u
AB
, and, therefore, the drift flux,
j
AB
=
α
(1
−
α
)
u
AB
,are
known functions of
α
and the fluid properties, then it is clear that the so-
331
lution to the types of kinematic problems described above, follow directly
from equations 14.1. Often this solution is achieved graphically as described
in the next section.
Drift flux models are particularly useful in the study of sedimentation,
fluidized beds or other flows in which the relative motion is primarily con-
trolled by buoyancy forces and the fluid drag. Then, as described in section
2.4.4, the relative velocity,
u
AB
, is usually a decreasing function of the vol-
ume fraction and this function can often be represented by a relation of the
form
u
AB
=
u
AB
0
(1
−
α
)
b
−
1
;
j
AB
=
u
AB
0
α
(1
−
α
)
b
(14.2)
where
u
AB
0
is the terminal velocity of a single particle of the disperse phase,
A
,as
α
→
0and
b
is some constant of order 2 or 3 as mentioned in section
2.4.4. Then, given
u
AB
0
and
b
the kinematic problem is complete.
Of course, many multiphase flows cannot be approximated by a drift flux
model. Most separated flows can not, since, in such flows, the relative motion
is intimately connected with the pressure and velocity gradients in the two
phases. But a sufficient number of useful flows can be analysed using these
methods. The drift flux methods also allow demonstration of a number of
fundamental phenomena that are common to a wide class of multiphase
flows and whose essential components are retained by the equations given
above.
14.2 DRIFT FLUX METHOD
The solution to equations 14.1 given the form of the drift flux function,
j
AB
(
α
), is most conveniently displayed in the graphical form shown in figure
14.1. Since equations 14.1 imply
j
AB
=(1
−
α
)
j
A
−
αj
B
(14.3)
and since the right hand side of this equation can be plotted as the straight,
dashed line in figure 14.1, it follows that the solution (the values of
α
and
j
AB
) is given by the intersection of this line and the known
j
AB
(
α
)curve.
We shall refer to this as the operating point,
OP
. Note that the straight,
dashed line is most readily identified by the intercepts with the vertical axes
at
α
=0 and
α
=1. The
α
= 0 intercept will be the value of
j
A
and the
α
= 1 intercept will be the value of
−
j
B
.
To explore some of the details of flows modeled in this way, we shall con-
sider several specific applications in the sections that follow. In the process
332
Figure 14.1.
Basic graphical schematic or chart of the drift flux model.
we shall identify several phenomena that have broader relevance than the
specific examples under consideration.
14.3 EXAMPLES OF DRIFT FLUX ANALYSES
14.3.1 Vertical pipe flow
Consider first the vertical pipe flow of two generic components,
A
and
B
.
For ease of visualization, we consider
that vertically upward is the positive direction so that all fluxes and velocities in
the upward direction are positive
that
A
is the less dense component and, as a memory aid, we will call
A
the
gas and denote it by
A
=
G
. Correspondingly, the denser component
B
will be
termed the liquid and denoted by
B
=
L
.
that, for convenience,
α
=
α
G
=1
−
α
L
.
However, any other choice of components or relative densities are readily
accommodated in this example by simple changes in these conventions. We
shall examine the range of phenomena exhibited in such a flow by the some-
what artificial device of fixing the gas flux,
j
G
, and varying the liquid flux,
j
L
. Note that in this context equation 14.3 becomes
j
GL
=(1
−
α
)
j
G
−
αj
L
(14.4)
Consider, first, the case of downward or negative gas flux as shown on
the left in figure 14.2. When the liquid flux is also downward the operating
333
Figure 14.2.
Drift flux charts for the vertical flows of gas-liquid mixtures.
Left: for downward gas flux. Right: for upward gas flux.
point,
OP
, is usually well defined as illustrated by CASE A in figure 14.2.
However, as one might anticipate, it is impossible to have an upward flux of
liquid with a downward flux of gas and this is illustrated by the fact that
CASE B has no intersection point and no solution.
The case of upward or positive gas flux, shown on the right in figure 14.2,
is more interesting. For downward liquid flux (CASE C) there is usually just
one, unambiguous, operating point,
OP
. However, for small upward liquid
fluxes (CASE D) we see that there are two possible solutions or operat-
ing points,
OP
1and
OP
2. Without more information, we have no way of
knowing which of these will be manifest in a particular application. In math-
ematical terms, these two operating points are known as
conjugate states
.
Later we shall see that structures known as
kinematic shocks
or expansion
waves may exist and allow transition of the flow from one conjugate state to
the other. In many ways, the situation is analogous to gasdynamic flows in
pipes where the conjugate states are a subsonic flow and a supersonic flow
or to open channel flows where the conjugate states are a subcritical flow
and a supercritical flow. The structure and propagation of kinematic waves
and shocks are will be discussed later in chapter 16.
One further phenomenon manifests itself if we continue to increase the
downward flux of liquid while maintaining the same upward flux of gas.
As shown on the right in figure 14.2, we reach a limiting condition (CASE
F) at which the dashed line becomes tangent to the drift flux curve at the
operating point,
OPF
. We have reached the maximum downward liquid
flux that will allow that fixed upward gas flux to move through the liquid.
This is known as a
flooded
condition and the point
OPF
is known as the
334
A
B
CONSTANT
α
FLOODING
CONDITIONS
FLOW NOT
POSSIBLE
FLOW NOT
POSSIBLE
j
j
G
L
Figure 14.3.
Flooding envelope in a flow pattern diagram.
flooding point. As the reader might anticipate, flooding is quite analogous
to choking and might have been better named choking to be consistent with
the analogous phenomena that occur in gasdynamics and in open-channel
flow.
It is clear that there exists a family of flooding conditions that we shall
denote by
j
Lf
and
j
Gf
. Each member of this family corresponds to a different
tangent to the drift flux curve and each has a different volume fraction,
α
.
Indeed, simple geometric considerations allow one to construct the family
of flooding conditions in terms of the parameter,
α
, assuming that the drift
flux function,
j
GL
(
α
), is known:
j
Gf
=
j
GL
−
α
dj
GL
dα
;
j
Lf
=
−
j
GL
−
(1
−
α
)
dj
GL
dα
(14.5)
Often, these conditions are displayed in a flow regime diagram (see chapter
7) in which the gas flux is plotted against the liquid flux. An example is
shown in figure 14.3. In such a graph it follows from the basic relation 14.4
(and the assumption that
j
GL
is a function only of
α
)thatacontourof
constant void fraction,
α
, will be a straight line similar to the dashed lines
in figure 14.3. The slope of each of these dashed lines is
α/
(1
−
α
), the
intercept with the
j
G
axis is
j
GL
/
(1
−
α
) and the intercept with the
j
L
axis
is
−
j
GL
/α
. It is then easy to see that these dashed lines form an envelope,
AB
, that defines the flooding conditions in this flow regime diagram. No flow
is possible in the fourth quadrant and above and to the left of the flooding
envelope. Note that the end points,
A
and
B
, may yield useful information.
In the case of the drift flux given by equation 14.2, the points
A
and
B
are
335
given respectively by
(
j
G
)
A
=
u
GL
0
(1
−
b
)
1
−
b
/b
b
;(
j
L
)
B
=
−
u
GL
0
(14.6)
Finally we note that since, in mathematical terms, the flooding curve in
figure 14.3 is simply a mapping of the drift flux curve in figure 14.2, it is
clear that one can construct one from the other and vice-versa. Indeed, one
of the most convenient experimental methods to determine the drift flux
curve is to perform experiments at fixed void fractions and construct the
dashed curves in figure 14.3. These then determine the flooding envelope
from which the drift flux curve can be obtained.
14.3.2 Fluidized bed
As a second example of the use of the drift flux method, we explore a sim-
ple model of a fluidized bed. The circumstances are depicted in figure 14.4.
An initially packed bed of solid, granular material (component,
A
=
S
)is
trapped in a vertical pipe or container. An upward liquid or gas flow (com-
ponent,
B
=
L
) that is less dense than the solid is introduced through the
porous base on which the solid material initially rests. We explore the se-
quence of events as the upward volume flow rate of the gas or liquid is
gradually increased from zero. To do so it is first necessary to establish the
drift flux chart that would pertain if the particles were freely suspended in
the fluid. An example was given earlier in figure 2.8 and a typical graph of
j
SL
(
α
) is shown in figure 14.5 where upward fluxes and velocities are defined
Figure 14.4.
Schematic of a fluidized bed.
336
Figure 14.5.
Drift flux chart for a fluidized bed.
as positive so that
j
SL
is negative. In the case of suspensions of solids, the
curve must terminate at the maximum packing solids fraction,
α
m
.
At zero fluid flow rate, the operating point is
OPA
, figure 14.5. At very
small fluid flow rates,
j
L
, we may construct the dashed line labeled CASE
B; since this does not intersect the drift flux curve, the bed remains in its
packed state and the operating point remains at
α
=
α
m
,point
OPB
of
figure 14.5. On the other hand, at higher flow rates such as that represented
by CASE D the flow is sufficient to fluidize and expand the bed so that
the volume fraction is smaller than
α
m
. The critical condition, CASE C, at
which the bed is just on the verge of fluidization is created when the liquid
flux takes the first critical fluidization value, (
j
L
)
C
1
,where
(
j
L
)
C
1
=
j
SL
(
α
m
)
/
(1
−
α
m
)
(14.7)
As the liquid flux is increased beyond (
j
L
)
C
1
the bed continues to expand
as the volume fraction,
α
, decreases. However, the process terminates when
α
→
0, shown as the CASE E in figure 14.5. This occurs at a second critical
337
liquid flux, (
j
L
)
C
2
,givenby
(
j
L
)
C
2
=
−
dj
SL
dα
α
=0
(14.8)
At this critical condition the velocity of the particles relative to the fluid
cannot maintain the position of the particles and they will be blown away.
This is known as the
limit of fluidization
.
Consequently we see that the drift flux chart provides a convenient device
for visualizing the overall properties of a fluidized bed. However, it should be
noted that there are many details of the particle motions in a fluidized bed
that have not been included in the present description and require much
more detailed study. Many of these detailed processes directly affect the
form of the drift flux curve and therefore the overall behavior of the bed.
14.3.3 Pool boiling crisis
As a third and quite different example of the application of the drift flux
method, we examine the two-phase flow associated with pool boiling, the
background and notation for which were given in section 6.2.1. Our purpose
here is to demonstrate the basic elements of two possible approaches to the
prediction of boiling crisis. Specifically, we follow the approach taken by Zu-
ber, Tribius and Westwater (1961) who demonstrated that the phenomenon
of boiling crisis (the transition from nucleate boiling to film boiling) can be
visualized as a flooding phenomenon.
In the first analysis we consider the nucleate boiling process depicted in
figure 14.6 and described in section 6.2.1. Using that information we can
construct a drift flux chart for this flow as shown in figure 14.7.
It follows that, as illustrated in the figure, the operating point is given by
Figure 14.6.
Nucleate boiling.
338
Figure 14.7.
Drift flux chart for boiling.
the intersection of the drift flux curve,
j
VL
(
α
), with the dashed line
j
VL
=
̇
q
ρ
V
L
1
−
α
1
−
ρ
V
ρ
L
≈
̇
q
ρ
V
L
(1
−
α
)
(14.9)
where the second expression is accurate when
ρ
V
/ρ
L
1asisfrequently
the case. It also follows that this flow has a maximum heat flux given by the
flooding condition sketched in figure 14.7. If the drift flux took the common
form given by equation 14.2 and if
ρ
V
/ρ
L
1 it follows that the maximum
heat flux, ̇
q
c
1
, is given simply by
̇
q
c
1
ρ
V
L
=
Ku
VL
0
(14.10)
where, as before,
u
VL
0
, is the terminal velocity of individual bubbles rising
alone and
K
is a constant of order unity. Specifically,
K
=
1
b
1
−
1
b
b
−
1
(14.11)
so that, for
b
=2,
K
=1
/
4 and, for
b
=3,
K
=4
/
27.
It remains to determine
u
VL
0
for which a prerequisite is knowledge of the
typical radius of the bubbles,
R
. Several estimates of these characteristic
quantities are possible. For purposes of an example, we shall assume that
the radius is determined at the moment at which the bubble leaves the wall.
If this occurs when the characteristic buoyancy force,
4
3
πR
3
g
(
ρ
L
−
ρ
V
), is
339
balanced by the typical surface tension force, 2
πSR
, then an appropriate
estimate of the radius of the bubbles is
R
=
3
S
2
g
(
ρ
L
−
ρ
V
)
1
2
(14.12)
Moreover, if the terminal velocity,
u
VL
0
, is given by a balance between the
same buoyancy force and a drag force given by
C
D
πR
2
ρ
L
u
2
VL
0
/
2thenan
appropriate estimate of
u
VL
0
is
u
VL
0
=
8
Rg
(
ρ
L
−
ρ
V
)
3
ρ
L
C
D
1
2
(14.13)
Using these relations in the expression 14.10 for the critical heat flux, ̇
q
c
1
,
leads to
̇
q
c
1
=
C
1
ρ
V
L
Sg
(
ρ
L
−
ρ
V
)
ρ
2
L
1
4
(14.14)
where
C
1
is some constant of order unity. We shall delay comment on the
relation of this maximum heat flux to the critical heat flux, ̇
q
c
,andon
the specifics of the expression 14.14 until the second model calculation is
completed.
A second approach to the problem would be to visualize that the flow
near the wall is primarily within a vapor layer, but that droplets of water
are formed at the vapor/liquid interface and drop through this vapor layer
to impinge on the wall and therefore cool it (figure 14.8). Then, the flow
within the vapor film consists of water droplets falling downward through
an upward vapor flow rather than the previously envisaged vapor bubbles
rising through a downward liquid flow. Up to and including equation 14.11,
the analytical results for the two models are identical since no reference
was made to the flow pattern. However, equations 14.12 and 14.13 must
Figure 14.8.
Sketch of the conditions close to film boiling.
340
be re-evaluated for this second model. Zuber
et al.
(1961) visualized that
the size of the water droplets formed at the vapor/liquid interface would be
approximately equal to the most unstable wavelength,
λ
, associated with
this Rayleigh-Taylor unstable surface (see section 7.5.1, equation 7.22) so
that
R
≈
λ
∝
S
g
(
ρ
L
−
ρ
V
)
1
2
(14.15)
Note that, apart from a constant of order unity, this droplet size is func-
tionally identical to the vapor bubble size given by equation 14.12. This is
reassuring and suggests that both are measures of the
grain size
in this com-
plicated, high void fraction flow. The next step is to evaluate the drift flux
for this droplet flow or, more explicitly, the appropriate expression for
u
VL
0
.
Balancing the typical net gravitational force,
4
3
πR
3
g
(
ρ
L
−
ρ
V
) (identical to
that of the previous bubbly flow), with a characteristic drag force given by
C
D
πR
2
ρ
V
u
2
VL
0
/
2 (which differs from the previous bubbly flow analysis only
in that
ρ
V
has replaced
ρ
L
)leadsto
u
VL
0
=
8
Rg
(
ρ
L
−
ρ
V
)
3
ρ
V
C
D
1
2
(14.16)
Then, substituting equations 14.15 and 14.16 into equation 14.10 leads to a
critical heat flux, ̇
q
c
2
,givenby
̇
q
c
2
=
C
2
ρ
V
L
Sg
(
ρ
L
−
ρ
V
)
ρ
2
V
1
4
(14.17)
where
C
2
is some constant of order unity.
The two model calculations presented above (and leading, respectively, to
critical heat fluxes given by equations 14.14 and 14.17) allow the following
interpretation of the pool boiling crisis. The first model shows that the
bubbly flow associated with nucleate bo
iling will reach a critical state at
a heat flux given by ̇
q
c
1
at which the flow will tend to form a vapor film.
However, this film is unstable and vapor droplets will continue to be detached
and fall through the film to wet and cool the surface. As the heat flux
is further increased a second critical heat flux given by ̇
q
c
2
=(
ρ
L
/ρ
V
)
1
2
̇
q
c
1
occurs beyond which it is no longer possible for the water droplets to reach
the surface. Thus, this second value, ̇
q
c
2
, will more closely predict the true
boiling crisis limit. Then, the analysis leads to a dimensionless critical heat
341
Figure 14.9.
Data on the dimensionless critical heat flux, ( ̇
q
c
)
nd
(or
C
2
), plotted against the Haberman-Morton number,
Hm
=
gμ
4
L
(1
−
ρ
V
/ρ
L
)
/ρ
L
S
3
,forwater(+),pentane(
×
), ethanol (
), benzene (
),
heptane(
) and propane (
∗
) at various pressures and temperatures.
Adapted from Borishanski (1956) and Zuber
et al.
(1961).
flux, ( ̇
q
c
)
nd
, from equation 14.17 given by
( ̇
q
c
)
nd
=
̇
q
c
ρ
V
L
Sg
(
ρ
L
−
ρ
V
)
ρ
2
V
−
1
4
=
C
2
(14.18)
Kutateladze (1948) had earlier developed a similar expression using dimen-
sional analysis and experimental data; Zuber
et al.
(1961) placed it on a
firm analytical foundation.
Borishanski (1956), Kutateladze (1952), Zuber
et al.
(1961) and others
have examined the experimental data on critical heat flux in order to deter-
mine the value of ( ̇
q
c
)
nd
(or
C
2
)thatbestfitsthedata.Zuber
et al.
(1961)
estimate that value to be in the range 0
.
12
→
0
.
15 though Rohsenow and
Hartnett (1973) judge that 0
.
18 agrees well with most data. Figure 14.9
shows that the values from a wide range of experiments with fluids includ-
ing water, benzene, ethanol, pentane, heptane and propane all lie within
the 0
.
10
→
0
.
20. In that figure ( ̇
q
C
)
nd
(or
C
2
) is presented as a function of
the Haberman-Morton number,
Hm
=
gμ
4
L
(1
−
ρ
V
/ρ
L
)
/ρ
L
S
3
, since, as was
342
seen in section 3.2.1, the appropriate type and size of bubble that is likely
to form in a given liquid will be governed by
Hm
.
Lienhard and Sun (1970) showed that the correlation could be extended
from a simple horizontal plate to more complex geometries such as heated
horizontal tubes. However, if the typical dimension of that geometry (say
the tube diameter,
d
) is smaller than
λ
(equation 14.15) then that dimension
should replace
λ
in the above analysis. Clearly this leads to an alternative
correlation in which ( ̇
q
c
)
nd
is a function of
d
; explicitly Lienhard and Sun
recommend
( ̇
q
c
)
nd
=0
.
061
/K
∗
where
K
∗
=
d/
S
g
(
ρ
L
−
ρ
V
)
1
2
(14.19)
(the constant, 0
.
061, was determined from experimental data) and that the
result 14.19 should be employed when
K
∗
<
2
.
3. For very small values of
K
∗
(less than 0
.
24) there is no nucleate boiling regime and film boiling occurs
as soon as boiling starts.
For useful reviews of the extensive literature on the critical heat flux in
boiling, the reader is referred to Rohsenow and Hartnet (1973), Collier and
Thome (1994), Hsu and Graham (1976) and Whalley (1987).
14.4 CORRECTIONS FOR PIPE FLOWS
Before leaving this discussion of the use of drift flux methods in steady flow,
we note that, in many practical applications, the vertical flows under consid-
eration are contained in a pipe. Consequently, instead of being invariant in
the horizontal direction as assumed above, the flows may involve significant
void fraction and velocity profiles over the pipe cross-section. Therefore, the
linear relation, equation 14.3, used in the simple drift flux method to find the
operating point, must be corrected to account for these profile variations. As
described in section 1.4.3, Zuber and Findlay (1965) developed corrections
using the profile parameter,
C
0
(equation 1.84), and suggest that in these
circumstances equation 14.3 should be replaced by
j
AB
=[1
−
C
0
α
]
j
A
−
C
0
α
j
B
(14.20)
where the overbar represents an average over the cross-section of the pipe.
343