Nuclear Engineering and Design 95 (1986) 275-283
North-Holland, Amsterdam
SOME
ASPECTS
275
OF TWO PHASE FLOW THROUGH
POROUS
MEDIA
V.K. DHIR
Mechanical, Aerospace and Nuclear Engineering Department, School of Engineering and Applied Science, University of California,
Los Angeles. CA 90024, USA
In this paper results of experiments conducted on two phase flow through porous layers formed of non heated glass
particles (nominal diameter 3-19 mm) are reviewed. The porous layers were 30-70 cm deep and were formed in a 21 cm inside
diameter plexiglass pipe. Experiments were conducted with water and air in both co and counter current modes. Data for the
average void fraction and two phase friction pressure drop were taken. A flow regime map for two phase flow through porous
media, a semi-theoretical model for void fraction, an empirical correlation for counter current flooding and empirical
correlations for relative permeability multipliers under co-flow conditions are reported. The interaction between the overlying
liquid layer and the porous layer under counter current flow conditions is discussed.
1. Introduction
An understanding of the physical processes governing two phase flow and heat transfer characteristics of
porous media is essential before models to assess the
coolability of a degraded core can be implemented in
system codes currently being used in light water reactor
safety evaluation. For example, for a fixed pressure
drop across the core, the flow rate of coolant undergoing phase change in the core will depend on the particle
size and porosity characterizing the core and relative
permeabilities of liquid and vapor phases. The relative
permeabilities in turn will depend on the void fraction.
If an impervious blockage forms in the lower portion of
the core, the coolability of the core will depend on the
counter current flooding limitations in the core. In the
present work hydrodynamic characteristics of two phase
flow through porous media are established. Specifically,
data and semi-theoretical and empirical correlations for
the void fraction, relative permeability multipliers and
counter current flooding in porous layers composed of
large diameter particles (3-19 mm) are reported. A two
phase flow regime map for porous layers is also developed.
•
I
2. Experimental apparatus and procedure
The experiments were designed to obtain pertinent
data on two phase flow parameters for co and counter
current flow through porous media. In all of the experi-
ments, the porous layers were formed of glass particles
while water and air were used as the continuous and
discontinuous phases, respectively. A brief description
of the experiments is given in the following text, but
complete experimental details can be found in ref. [1].
2.1. Experimental apparatus
A schematic diagram of the experimental setup for
co-flow experiments is shown in fig. 1. The basic components of the setup are: a plexiglass test section, a
water reservoir, a main centrifugal pump, and a small
centrifugal pump to fluidize the bed. The instrumentation consists of water manometers, air and water flow
meters, and a pressure gage to measure the air back
pressure through the flow meter. Solenoid valves are
placed in the air and water inlet lines for rapid flow
cutoff. The test section, a 21 cm inside diameter and
1.69 m long plexiglass tube, is placed on an inlet header
with a perforated grid plate sandwiched in between.
U p o n the grid plate a 3.175 mm inside diameter copper
tubing wound in the form of a spiral is placed. The
copper tubing is connected to an air supply line and has
0.794 mm diameter holes drilled on its top side to
provide a uniform flow of air through the bed. Sixteen
pressure taps are located on the test section. The pressure taps are placed 15.24 cm apart in the upper part of
the test section but are located only 5.08 cm apart near
the bottom. Several water outlets are placed on the test
section diametrically across from the pressure taps. The
water outlets are also plexiglass tubes with 22.2 mm
inside diameter.
0 0 2 9 - 5 4 9 3 / 8 6 / $ 0 3 . 5 0 © E l s e v i e r S c i e n c e P u b l i s h e r s B.V.
( N o r t h - H o l l a n d Physics P u b l i s h i n g D i v i s i o n )
ILK. Dhir / Two phase flow through porous media
276
Spiral
Alumi
Wa
)ut
obtained by knowing the difference between total void
fraction and the active void fraction. The total void
fraction in the particulate bed was measured by cutting
off the gas and liquid flow rates and fluidizing the
particulate bed with flow from the seconda~' pump.
The secondary pump had an intake at the overlying
liquid layer and a discharge at the bottom of the particulate bed. During fluidization, any gas trapped m the
interstitials was released. After fluidization, the particulate bed was allowed to settle and the height of the
overlying liquid layer was noted.
2.3. Data reduction
The porosity of the particulate bed was determined
by using the following relationship
Mp
~=l-opA-- C-"
Gate Valve
Gate Valve
Fig. l. Schematic diagram of the experimental apparatus for
co-flow studies.
2.2. Procedure
where Mp is the mass of the particles forming a bed of
depth L, 0p is the material density of the particles, and
A is the cross-sectional area of the bed. Knowing the
hydrostatic head, Ap0, and height, H 0, of the free
surface of the overlying liquid layer when gas is bubbling through the layer, the average void fraction, ~0, in
the overlying liquid layer is determined as
Pl -- A p ' / g (
Clean, dried preweighed particles of a given size
were poured into the test section to a certain height.
After formation of the bed, a predetermined flow rate
of water was established through the porous layer. Air
was gradually added to the test section. Air flow rate
was increased in steps while keeping the water flow rate
constant. At each flow rate, readings of the manometers
connected to pressure taps placed along the bed were
noted. Generally, a liquid layer overlaid the particulate
bed. The gas leaving the particular bed bubbled through
this layer. This height as well as the hydrostatic head of
the layer were also noted for particular gas and liquid
flow rates. Thereafter, the water outlet and air inlet
were closed rapidly and the height acquired by the
liquid in a given time was noted. The time was chosen
such that any bubbles in transit will leave the particulate bed and the overlying liquid layer. This information was used to determine the average active void
fraction in the particulate bed.
During the course of the experiments, it was discovered that a certain number of bubbles got trapped in
the interstitials of the particles. These bubbles did not
move with the flow and constituted an inactive void
fraction in the particulate bed. This void fraction was
(1)
G=--
"
-- ~ )
(o,-og)
(2)
With the knowledge of the height, H I , the liquid layer
acquired in time t after the air flow rate was cut off and
making a volume balance, the average active void fraction. ~,c, in the particulate bed is calculated as
j , t - ao ( Ho - L ) - ( H, - Ho )
aac
(3)
( L
=
The average total void fraction in the particulate bed
was calculated by knowing the difference between the
overlying liquid layer heights H 0 and H z when both gas
and liquid flow rates were maintained and when both
flows were cut off the bed was fluidized and allowed to
resettle, respectively. Making a volume balance, the
total average void fraction, ~, in the particulate bed is
written as
=
(H 0 - H2) - a0(H 0 -
L)
(4)
~L
From eqs. (3) and (4), the average inactive void fraction,
ain in the particulate bed can be found as
ai =~-,~ac.
(5)
V.K. Dhir / Two phase flow through porous media
Knowing the total pressure gradient in the particular
bed, the frictional pressure gradient is calculated as
dPf
dP t
dzz ]~ ~z I - ( l - ~ a c ) p l g - ~ a c o g g "
Ogj2g, if the Reynolds member, Reg based on the gas
properties is much greater than 8 6 ( 1 - e), the mean
particle diameter is defined as
(6)
Only the active void fraction is used in the differential
hydrostatic head term since the frictional pressure
gradient through a particulate bed containing trapped
air with no gas or liquid flow must equal zero.
In the counter current experiments water was added
at the top while air was injected at the bottom. A
predetermined liquid flow rate was established and gas
flow rate was gradually increased in steps. In the experiment, two phase air-water layer 10-40 cm depth existed over the bed. Readings of the water manometers
and overlying layer height were noted at each incremental air flow rate. The volume of air in the bed and
overlying layer was found by simultaneously closing the
water inlet and exit lines and the air inlet line. The
effects of air compression on the air flow rate and
non-instantaneous closure of the solenoid valves on
overlying layer height drop were tested and accounted
for. Experiments were terminated at air flow rates at
which flooding was observed to occur. The onset of
flooding in the bed was discerned by noticing a rapid
rise in the height of the overlying layer and formation of
large gas pocket in the lower portion of the bed.
3. Discussionof results
In the following section the aspects of two phase
flow through porous media considered are:
(i) definition of mean particle diameter in mixture of
particles,
(ii) flow regime map,
(iii) void fraction relationship,
(iv) relative permeability multipliers,
(v) fluidization of the porous layer,
(vi) counter current flooding,
(vii) coupling between overlying liquid layer and the
porous layer.
3.1. Mean particle diameter
Since in two phase flow, the gas/vapor and liquid
superficial velocities can vary independently over a wide
range, no generally applicable simple expression for the
mean particle diameter in a mixture of various particle
sizes is possible [1]. However in the limiting situations,
simple expressions for the upper and lower values of the
mean particle diameter can be obtained. For pljl z <<
277
Xl
i=1
Here jg and Jt are the superficial velocities of gas and
liquid in the test section in the absence of the particles.
The gas Reynolds number is defined as
Reg = OgjgT)p/l~g,
(8)
and x i in eq. (7) denotes the volume fraction of particles of size i in the mixture. If however Reg << 86(1 c), the expression for the mean particle diameter is
written as
]
Op =
1/2
~ Xi
i=1 Op2t
(9)
Similarly for plj 2 > > pgjg2, eq. (7) is valid if Re l >> 86(1
- e) and eq. (9) if Re l << 86(1 - e), The liquid Reynolds
number Re t is defined in the same way as the gas
Reynolds number in eq. (8) except that now the liquid
superficial velocity and liquid properties are used. In
mixtures containing different fractions of particles varying in size from 1 to 19 mm and large range of superficial velocities of gas and liquid it is found that the
observed friction pressure drops compare well with those
predicted from the Kozeny-Carman equation when the
mean particle diameter given by eq. (9) is used.
3.2. Flow regime map
Visual observations of two phase flow through a bed
of clear glass particles revealed a change of flow pattern
from bubbly to slug to annular flow as the gas velocity
was increased with a fixed liquid velocity. During bubbly flow, discrete bubbles were seen to move through
the bed. These bubbles tended to be nearly spherical in
shape, although they became slightly oblong at higher
gas flow rates. Further increase in the gas flow rate
caused the discrete bubbles to merge and form extremely oblong bubbles or slugs. These slugs often
spanned the height of two to three particles. With
further increase in the superficial gas velocity, the slugs
merged and formed a continuous path for the gas. The
liquid was seen to be pushed to the particle surface
while the gas moved through the center of the pores. In
beds composed of particles greater than 6 mm in diameter, the average bubble size in bubbly flow was seen to
V.K. Dhir / Two phase flow through porous medm
278
remain constant at about 2.5 to 3.5 mm. For beds
composed of particles in the size range 1.6 to 6 ram, the
bubbles appeared to be larger than the pore size and
were observed to engulf the particles as they moved
upwards. Often the bubbles were seen to break up into
one large and a few smaller bubbles after hitting a
particle. The average bubble size was generally observed
to be somewhat smaller than that observed in beds of
larger particles. No evidence of channel formation in
these layers was found as observed in beds of particles
smaller than 1.6 mm [2].
Formation of an oscillating flow pattern prior to
transition from slug to annular flow was observed in
porous layers composed of 3.6 and 10 mm nominal
diameter particles. In the oscillating flow the liquid was
seen to travel upwards in the bed in the form of
discontinuous periodic waves. In the bed region between the two liquid wave fronts, annular gas flow was
observed in the pores of the particles. The period of
oscillation and the spacing between two wave fronts
was found to increase with jg but decrease with j¢. In
porous layers formed of particles 10 mm or larger in
size, transition from slug to annular flow occurred in
the absence of an oscillating flow. The boundaries of
flow regimes are correlated using dimensionless superficial velocities, j/* and Js* defined as
.....
h* =h g(p~_ o~)~d3
jg*
,
- [[ g(p,6Ps(!
e ~pg)
)-_ ~p¢3 ],/2
_jg
.
(10)
(11)
This non-dimensionalization results from the inertial
term in the Kozeny-Carman equation [3]. Thus for
flows in which viscous term dominates, a physical
meaning to the above scheme of non-dimensionalizing
the superficial velocities can not be advanced. Fig. 2
shows the flow regime map. The figure contains both co
and counter flow data. It is interesting to note that for
counter-current flow, the transition to the annular flow
corresponds to the flooding limit for jg* < 0.13. For
counter current flow with low jg* and high Jr*, the flow
may go directly from bubbly to annular regime.
i - - - r
i
.....
'
I
Counter-Current
i
t
%
Cn -Current
0.3
/
O 3ram
[] 6mm
10mm
15mm
Q 19mm
/
I/
/
/
t Jg Annular Flow
0.2 & ~ - " ~
~/
Flooding Limit
-J-J~
Slug Flow
).I
Annular F
~
0.8
Bubbly Flow
i
I
0.4
0.4
It"
f
I
0,8
Fig. 2. Flow regime map.
volume fraction occupied by the inactive voids was
found to be maximum for beds composed of 3 mm
diameter particles and decreased as the particle size was
increased or decreased. The inactive void fraction decreased with increase in superficial velocities of gas and
liquid and varied between 0.2 and 0.1 for 3 mm diameter particles. However for particles of 6 mm diameter
and larger the inactive void fraction was always found
to be less than 0.05.
The active void fraction, aac, is associated with the
bubbles in transit and it is this void fraction on which
most of the effort has been devoted to.
The cross sectionally averaged active void fraction,
~ac, in a fixed bed is defined as
volume occupied by active gas in a distance dz
pore volume in a distance dz
vg _A sdz
A~ dz
A~ dz
As.
(12)
A~
The average pore velocities of the two phases in terms
of the superficial velocities can be written as
-
Q,
Jl
(13)
Js
(14)
h
A~CI - ~CI ,
Jg
A~C~
Qs
,C~ '
3.3. Void fraction
Visual observations showed that certain volume of
gas got trapped in the interstitials of the particles. These
trapped air bubbles did not move with the flow and
thus constituted inactive voids in the porous layer. The
where QI and Qg are the volume fluxes of liquid and
gas respectively and Ct and Cs are the constants which
account for the increase in the local velocity as the
liquid has to flow along the periphery of the particles.
From geometrical consideration, the constants can easily
V.K. Dhir / Two phase flow through porous media
be found to be 2/7r for flow over spherical particles. In
single phase flow, generally a value of about 1 / v ~ has
been used [3]. Combining eqs. (13) and (14) and substituting for C / and Cg, average pore velocity of the
mixture is obtained as
), + )~ = ~( j, + j O / 2 , .
(15)
The local velocity of the gas phase averaged over the
cross-section of the bed can be written as
fig =;g/aac •
velocity of the bubble can be written as
u t = 2.9
02
sin 0.
(19)
The time, -r, taken by the bubble to travel a vertical
distance equal to diameter of the particles can be found
to be
"r =
(16)
Dp dO
f~-o,
JO~
2Ut "
(20)
Knowing, ~', the average terminal velocity in the vertical
direction is written as
After writing the identity
fig ~-j + fig - j ,
Otac
279
(17)
(jl+Jg) d- (2/'/7)~(Ug __)7)
(18)
In eq. (18), f g - j is simply the drift velocity [4]. The
drift velocity can be related to the terminal velocity, fit,
of a single bubble rising in porous layer saturated with
liquid. At this point a distinction needs to be made
between large and small particles.
As mentioned earlier, in beds composed of particles
larger than 6 mm in diameter the bubble size was
independent of particle size. Bubbles generally moved
along the periphery of the particles. Fig. 3 shows schematically typical path of a bubble. The local terminal
2, = Dp/l".
(21)
Substituting for ~" in eq. (21) from eq. (20) we get
[Üg(Pl--Og)]l/4 COSO1
fit = - 2 . 9
02
in(tan½01) .
From fig. 3, the angle 0, can be evaluated in terms of
the bubble diameter D b and the particle diameter D 0. If
we do so, eq. (22) becomes
= 5 8[ ° g ( P l - pg) ]1/4
t
2t
;i
[0 + vu/5.)
× ln(1 + 2 D p / D b)] -1.
PARTICLE
(22)
(23)
It is interesting to note that as Dp ~ 0, eq. (23) reduces
to the terminal velocity of a bubble rising in a pipe.
However, it should be mentioned that as the particle
size becomes small, the bubble dynamics is influenced
by the relative size of the pores and eq. (23) is not
directly applicable.
The velocity ft is related to the drift velocity, fg - j , as
fig - j = 7rf t / 2 .
(24)
The bubble size which remained nearly invariant in
bubbly flow in beds of large diameter particles was
found to be approximately given by
D b = 1.1 g ( o l -o- Og)
Db
.
(25)
If a dimensionless particle size is defined as
LIQUID
Fig. 3. bcnematic diagram of the path of a bubble around large
particles.
Op, _-- Op
g( Pl-o Pg)
(26)
280
V.K. Dhir / Two phase flow through porous media
substitution of eqs. (23), (24), (25) and (26) in eq. (181
yields an expression for the active void fraction as
(1 - / J ~ . ) dPt
•
dz
_
FD/ + /"Di .... pl~(1 - - i i ~ )
et dz
)i-ti-z
I~9)
~ac = jg/
( jg + j, ) + 5.8,
×[(1 +
)02
1.1/5;)ln(1
+ 1.82~p)]
"l
t
(27)
It should be stressed that eq. (27) is applicable only
when Dp >> 1.1.
In fig. 4, the active void fraction predicted from eq.
(26) is compared with the data. The solid lines show the
predictions. In plotting the data, non-dimensionalized
superficial velocities of gas and liquid are used. The
plotted data are for particles with nominal diameters of
10, 15 and 19 ram. It is seen that the comparison
between the predictions and the data is quite satisfactory noticing the magnitude of uncertainty in the measurements.
The two phase frictional pressure gradient in a porous layer can be considered to result from the interracial and the particle drag. If the two phases are assumed
to flow separately, the momentum equations for the
co-flowing gas and liquid can be written respectively as
rDg
Adz
I
rDi
A dz
Psg~ac'
I
I
dPf
z
(28)
I
0.6
dP t
dz
FDg
P g g ~ - ozg(1 -- ~ )
FDI
-- A d z + A d z "
(30)
Defining relative permeability multipliers to account
for the fractional area occupied by each phase and to
provide a correction to the particle drag experienced by
each phase if present alone, eq. (30) can be re-written
after replacing Fog and FDt as
dPf
dz
3. 4. Relative permeability multipliers
dP,
a~¢ dz
In writing the above equations the pressure gradient in
the gas and liquid is assumed to be the same. The total
pressure gradient is denoted by dPt/dz; whereas the
particle drag on gas, the particle drag on liquid and the
interfacial drag are denoted by FDg, FDt and FDI
respectively. If the two equations are added, we can
write the frictional pressure gradient as
150(1 - ~)2
K/~: Dp
+
.
1.75(1 - ( )
~/c Dp
150(1-t)2
1.75(1__- c) 0 .2
g ~e3~p2 /xgJs +
7/;¢3~p
gJg"
(31)
The measured two phase frictional pressure gradient
data taken with Dp >t 4.5 mm and mac up to 0.6 have
been used to develop correlations for the viscous multit
pliers ~/' and rg' and the inertial multipliers ~/~ and ~/s
of liquid and gas phases respectively. The dependence
of these parameters on ~ac is found as
~ = 0.01~a ° ' + 0.02 ~ac + 0.57~a~ + 0 . 4 0 ~ 2 ,
(32)
)); = 0.01~,c + 0.02fi~ + 0.57~, 3 + 0.40~, s,
(33)
~ = 0.05(1 - ~.o)o..~ + 0.10(1 - ~ac) + 0.50(1 -- ~.c) 3
0.4
+0.35(1 - ~ac) 5,
2:
"q) = 0 . 0 5 ( 1
+0.35(1 - ~.c)'.
0.2
£.E o ~ ( P ~ - P g ) I
0
-- mac ) q- 0 . 1 0 ( 1
P~ ] ~,.4
I
I
I
I
5
10
15
20
.
-
-
25
Fig. 4. Comparison of void fraction predictions with the data
for D_' > 2.4.
(34)
-- ~ a c ) 2 + 0 . 5 0 ( 1
-- ~ a c ) 3
(35)
Figs. 5 and 6 graphically show this dependence. All of
the available data were correlated within +_20% with
these parameters.
At this point it should be emphasized that the above
formulation is empirical and is not totally mechanistic.
It is visually observed that the discontinuous phase
V.K. Dhir / Two phase flow through porous media
1.0
I
~
I
written as
I
/
0.8
K;
0.6
\,'
K'Q, K' i
-dPf/dz
I
= (1 - ~ac) pt + ~acPg'
(36)
(37)
Knowing d P f / d z from eq. (31) and aac from eq. (27),
the superficial velocities of gas and liquid at which a
given bed will fluidize can be determined, After fluidization the pressure drop may decrease somewhat due to
formation of channels as the bed reforms.
/
/
O. 2
= g( pp - ~)(1 - , ) ,
where
/
0.4
281
/
3.6. Counter current flooding
0
0.2
i
.0
0.8
0.6
0.4
-
Fig. 5. Dependence of x~ and Kg on %c.
(gas) never directly interacts with the particles and as
such it only experiences the interfacial drag. The relative permeability multipliers obtained from co flow data
when used in a counter current flooding model give the
correct trend but generally result in over-prediction of
the flooding limits. To overcome this deficiency, interfacial drag needs to be modeled.
3.5. Fluidization
A particulate bed will fluidize when the frictional
pressure drop across the bed equals the pressure due to
the effective weight of the bed. This condition can be
1,0
I
I
I
I
/
III
o.8
0.6
'
~
_
1"1'o,II'l
0.4
0.2
0
0:2
0.4
0.6
(~ac
p
-
Fig. 6. Dependenceof ~1~and 7]g on a~.
0.8
.0
As stated earlier, the coolability of a degraded core
in case an impervious blockage forms at the inlet, will
depend on the rate at which water can flow into the
core and steam can flow out. These flow rates in turn
are determined by the countercurrent flooding limitation in the core. In the present work, the flooding limit
in porous layers composed of uniform size particles
varying in diameter from 3-19 mm and mixtures of
these particles was determined by setting a given flow
rate of water and gradually increasing the air flow rate.
As the gas flow rate reached approximately 50% of the
flooding limit, the overlying liquid layer height began
increasing at a very slow rate, indicating liquid ejection
from the bed as a result of the larger fraction of the
pore volume occupied by gas. Increasing the gas velocity to near the flooding limit, the overlying layer height
was seen to rise much more rapidly and the discontinuous (gas) phase tended to become continuous in small
pockets within the bed. With increase in the gas flow
rate slightly above the flooding point, the flow became
annular and the gas formed a big pocket in the outlet
header. The flooding limit, therefore, was observed to
coincide with the transition to the annular flow regime
in the range of liquid and gas superficial velocities
studied.
From the definition of the flooding limit stated
above, it is known that the liquid flow through the bed
is limited by the gas flow at the flooding point and
cannot be further increased. Opening the water drainage
valve, which usually increased the water drainage rate
until a lower equilibrium height was reached, should
have no effect on the water height at the flooding point.
The experimental determination of the flooding limit
was the gas velocity at which, upon opening the water
drainage valve and thus drastically reducing the resistance to water flow downstream of the bed, the water
layer height within the test section was seen to remain
bqK. Dhir / Two phase flow through porous medm
282
1.0
i
i
,\..~
C=0.875
0.8
"'\ ,."~
~
0.6
The dimensionless superficial velocities .h* and jg:" ha~c
been defined in eqs. (10) and (tl). Figs. 7 and 8 sho~s
comparison of the data and the correlation eq. (38). In
these figures the chain line shows the correlation pr~,-~
posed by Wallis [4] based on the earlier data of several
investigators. It is noted that the data for small particles
with nominal diameter of 3 mm tends to be low. A
plausible reason for this could be the different character
of flow over smaller particles. It should also be pointed
out that equation (38) may not hold in the limit of
jg* -~ 0 or/;* -* O.
i
Dp
O 3mm
D 6mm
O 10mm
A 15mm
O 19mm
"~N~NN~
0.4
0.2
C=0.775 " - - - / ' \ . \ ~
0
,
,
0.2
0.4
3.7. Coupling between the ooerlying liquid layer and the
porous layer
\,,\
,
0.6
0.8
1.0
.,1/2
Jg
Fig. 7. Flooding data for beds composed of uniform size
spherical particles.
constant or increase slightly. Decrease or increase of the
water height was indicated very accurately by pressure
readings within the overlying layer. This method allowed determination of the flooding point within an
uncertainty of + 15%.
All of the counter current flooding data obtained
with single size particles and mixtures of particles are
correlated as
~.1/2 +j~l/2 = 0.875.
1.0
0.8
I
(38)
I
I
1
Dp (ram)
~
o6,10
1200
i
I
"\'~"~ ' ~ .
i
i
i
I
tx 3'6'10'15'1~
800
LAYER
" ~--P=-.~oop,
dZ
0
t
I
--~
C=0.875
C=0.775 ~z"
s
II O V E R L Y I N G
POROUS BED
E
0.2
,I
T
D 6,15
A 6,10,15,19
'\..\~
J~/20"60.4
The flooding correlation given by eq. (38) represents
the limiting mean velocities in the bed in the absence of
any extraneous effects. The presence of an overlying
liquid layer alters the hydrodynamic conditions in the
top portion of the bed adjacent to this layer. Since the
pores in the bed occupy only a fraction of the total bed
cross-section, the void fraction must decrease as the gas
leaves the particulate bed and enters the overlying liquid
layer. The reduction in void fraction will occur gradually over a portion of the bed. This reduction in the void
fraction is equivalent to an acceleration of the gas and a
reduction in the particle drag on the liquid. As a result
of this adjustment of flow in the upper portion of the
bed, the absolute value of the pressure gradient increases from that in the lower portion of the bed to that
in the overlying liquid layer. The variation in pressure
within the top portion of the bed at the flooding limit is
given in fig. 9 for a bed composed of 10 mm nominal
-.475
pig
~,l
dZ
\.
I
J
I
0.2
0.4
0.6
\~.
N
0.8
1.0
I
0
I
400
I
I
800
I
1200
1600
.,1/2
lg
Fig. 8. Flooding data for beds composed of mixtures of various
size particles.
Z (ram)
Fig. 9. Variation of pressure near the top of particulate bed at
flooding limit.
V.K. Dhir / Two phase flow through porous media
diameter particles. In this case, all the gas was injected
at the bottom of the bed. It is noted that at about 90
mm below the top of the bed the pressure gradient
starts to deviate from that in the lower portion of the
bed. The region over which this deviation occurs is
termed as the coupling layer and the depth of this layer
measured from the top of the bed is denoted as h c.
Data obtained with other liquid and gas superficial
velocities showed a variation of the coupling depth
between 65 and 90 mm. The coupling height was found
to decrease with decrease in particle size.
For beds in which all the gas is injected at the
bottom and the bed depth is greater than the depth of
the coupling layer, coupling between the overlying liquid
layer and the bed does not affect the maximum possible
liquid and gas velocities through the bed. However for
shallow beds or beds in which gas or vapor is injected
axially as is the case in beds of volumetrically heated
particles, the presence of coupling between the overlying liquid layer and the bed can significantly increase
the flooding limit.
4. Conclusions
1. For certain range of flow rates of gas and liquid, it is
possible to obtain simple expressions for the mean
particle diameter in porous layers containing mixtures of several size particles.
283
2. Two phase flow regimes similar to those in tubes
have been observed to exist in porous layers of large
diameter particles.
3. Both inactive and active voids exist in porous layers.
The active void fraction depends on the particle size
and porosity. Correlation based on drift flux approach have been developed for -Dp
' > 2.4.
4. For two phase friction pressure drop under co flow
conditions, the relative permeability multipliers have
been found to depend on the active void fraction.
5. Under counter current flow conditions the coupling
between the overlying liquid layer and the bed can
influence the flooding conditions especially if the
beds are shallow a n d / o r gas is injected axially.
References
[1] J.S. Marshall and V.K. Dhir, Hydrodynamics of counter
current two phase flow through porous media, NUREG/
CR-3995 (1984).
[2] A.S. Naik and V.K. Dhir, Forced flow evaporative cooling
of a volumetrically heated porous layer, Int. J. Heat Mass
Transf. 25 (1982).
[3] M. Leva, Fluidization (McGraw Hill Book Co., New York,
1959).
[4] N. Zuber and J.A. Findlay, Average volumetric concentration in two phase flow systems, J. Heat Transf. (1965).
[5] G.W. Wallis, One Dimensional Two Phase Flow (McGraw
Hill Book Co., New York, 1969).