Static Liquid Holdup in Packed Beds of Spherical

Particles
A. E. Saez, M. M. YCpez, C. Cabrera, and E. M. Soria
Departamento de Termodinamica y Fen6menos de Transferencia, Universidad Sim6n Bolivar, Caracas 1086-A, Venezuela

The static liquid holdup in a packed bed is defined as the

volume fraction of liquid, referred to total bed volume, that
remains in the bed after complete draining. The liquid that
remains in the bed is prevented from draining by a balance
between gravitational and surface tension forces.
The static liquid holdup is commonly used as a parameter
in the characterization of the hydrodynamics of gas-liquid flow
through packed beds. Shulman et al. (1955) indicated that,
under gas-liquid flow conditions, the liquid holdup in a packed
bed could be visualized as made up of static and dynamic
contributions. Afterwards, several investigators have used this
idea in the development of empirical correlations for calculating pressure drops and liquid holdups in trickling flow
through packed beds. In this line of thought, Specchia and
Baldi (1977) correlated the two-phase pressure drop through
the bed in terms of an Ergun-type equation whose characteristic
constants were evaluated in terms of the pressure drop through
the wetted packing, that is, when the liquid present was
that corresponding to the static holdup. The liquid holdup that
is usually correlated to the gas and liquid flow rates is usually
the dynamic holdup (Specchia and Baldi, 1977; Saez and Carbonell, 1985). This is a very convenient representation since it
can be easily forced to yield a zero value for the case of no
flow.
In the literature, the static liquid holdup has been empirically
correlated as a function of the Eotvos number:

.. pgP
Eo=where p is the density of the liquid phase, g is the acceleration
of gravity, a is the gas-liquid interfacial tension, and I is a
characteristic length that represents the geometry of the packing particles. The choice of characteristic length differs in the
available correlations. Charpentier et al. (1968) proposed that
I be chosen as the nominal particle diameter, I = d . Saez and
Carbonell (1985) selected a parameter proportional to the hydraulic diameter as the characteristic length:

I=-- 4 E
1-

where E is the porosity of the packed bed and d, is the equivalent

particle diameter, defined as the diameter of the sphere that
has the same volume-to-surface ratio of the particle. Dombrowski and Brownell (1954) used a characteristic length proportional to the square root of the absolute permeability of
the bed.
Turner and Hewitt (1959) developed a correlation based on
the experimental measures of liquid volumes held at menisci
between spheres. Their results, however, were limited to large
particles (with diameters over 1 cm), which correspond to Eotvos numbers larger than the usual range encountered in packed
beds. Recently, Reddy et al. (1990) used the experimental data
reported by Turner and Hewitt to propose a model with the
objective of exploring the distribution of liquid flow in tricklebed reactors. The model used a geometric variable related to
the geometry of the meniscus formed between the particles as
an adjustable parameter.
The static liquid holdup is also relevant in the modeling of
mixing and transport processes in gas-liquid, flowthrough
packed beds, in which an amount of liquid equal to the static
liquid holdup is commonly considered as a stagnant region
under gas-liquid flow conditions. The dynamic response of
perturbations in tracer concentrations has been used to determine the static holdup from dispersion experiments (Bennett
and Goodridge, 1970), although recent results suggest that the
static holdup obtained from such methods differs appreciably
from that obtained from draining experiments (Schubert et
al., 1986).
In this work, we use dimensional analysis to establish the
independent parameters that affect the static liquid holdup.
We also present experimental data to explore the dependence
of the static holdup on the Eotvos number; through comparison with previous theoretical works, we analyze the mechanisms responsible for the effect of Eotvos number on the static
liquid holdup.

Experimental Study
Correspondence concerning this work should be addressed to A. E. Saez.

AIChE Journal

The static liquid holdup was measured in this work for three

November 1991 Vol. 37, No. 11

1733

Table 1. Physical Properties of Liquids

Liquids

Water
Methanol
Kerosene

(kdm3)
1,000
192
808

(mN/m)
73.1
22.7
25.0

different liquids (Table 1) and packings of spherical glass beads

with narrow size distributions of diameters ranging from 0.5
to 4.0 mm. The experiments were performed at atmospheric
pressure and 20C in a cylindrical packed bed with a diameter
of 7 . 5 cm and a packing height of 50 cm. Before the experiments, the glass beads were carefully cleaned with an acid
solution to ensure perfect wetting properties.
The volume of liquid required to fill the packing section of
the bed (V,) can be used t o calculate the porosity of the medium, E = V J V , where Vis the total volume of packing section.
In all the packings employed, the porosity was always with the
range 0.35-0.39. After the bed was completely filled with liquid, it was allowed to drain for 60 minutes by opening a valve
located at the bottom of the column. This period of time was
determined to be more than enough to ensure a complete draining for the conditions employed in this work. The volume of
liquid drained (V,) was then used to calculate the static liquid
holdup, es = ( Vh- Vd)/V. Each experiment was repeated ten
times to assess the reproducibility of the measures. We found
that the maximum difference between each value of the drained
volume and its mean over the ten experiments rarely exceeded
5%.
In the draining process, two regimes of flow are observed.
The first period of draining is observed starting from the bed
completely filled with liquid, in which most of the liquid drains
forming a front that displaces downward. It is followed by the
second period, in which the liquid flows through liquid films
that form over the particles. The second period ends when the
films break over the particles, leaving the static liquid holdup
in the bed as menisci formed between particles. In this work,
we analyzed the effect of the draining rate on the static liquid
holdup by performing experiments with the same systems at
different draining rates. The draining rate was controlled by
means of the valve located at the bottom of the bed. We found,
after extensive experimentation, that the differences observed
between static liquid holdups at different draining rates were
within the reproducibility limit. This indicates that the draining
rate has no effect on the static liquid holdup. This result is
due to the fact that the draining rate controls the motion of
the front of liquid in the first period of draining, but has no
apparent effect on the second period, in which the flow of
liquid through the films is determined exclusively by the gravitational pull. It is worthwhile to mention that Turner and
Hewitt (1959) reported a dependence of the static holdup on
the draining rate for beds of large particles. This indicates that
the existence of a dynamic effect was not observed in the
experiments performed in this work.

Analysis and Results

Consider the distribution of a liquid phase under static conditions in an unconsolidated porous medium. The main purpose of the following discussion is to establish, through
dimensional analysis, the parameters that affect the liquid distribution.
1734

The pressure distribution in the liquid phase under static

conditions is given by:

where V, represents the liquid phase, and g is the acceleration

of gravity vector. The boundary condition at the gas-liquid
interface (A,,) is the equation of Young-LaPlace:

P,-P,=u

(<+--)
1

at A,,

(4)

where the pressure of the gas phase (PJ will be considered

uniform; r , and r, are the local principal radii of curvature of
the gas-liquid interface, which can be related to the shape of
the interface.
An additional condition can be imposed at the contact line
formed by the intersection between the gas-liquid interface and
the gas-solid interface (C). Under static conditions, it is appropriate to use the contact angle (4) as a geometric constraint
on the shape of the gas-liquid interface at the contact line as
follows:

n,.nf = cos 4 at C

(5)

where n, and n,are unit vectors normal to the gas-solid interface

and the gas-liquid interface, respectively, pointing into the gas
phase.
In addition, to specify completely the liquid pressure field,
the value of P, should be specified at a point within V,:

PI=P,,o at a point in V,

(6)

Consider the following dimensionless variables,

(7)

Equations 3 , 4 and 6 can be expressed in dimensionless form

as follows:

V * P b = u ,in V,

i0

p*=--

(r;-+- r;)

(9)
at A,,

P* =P$ at a point in V,

(11)

where u, is a unit vector parallel to g.

Once the local geometry of the solid phase (that is, the
location of the solid surface) is known and a value of P$ is
specified, Eqs. 9 to 11 and Eq. 5 can be solved to find the
shape and location of the gas-liquid interface (Ax,).The volume
of liquid can be determined by integration and the liquid holdup
can then be evaluated. The formulation presented indicates
that the liquid holdup in the medium is governed by the following relation:

November 1991 Vol. 37, No. 11

AIChE Journal

E,= E,

(P:, EO, 4, local geometry)

(12)

The procedure outlined above has been applied in previous

works to determine the shape of menisci formed at the contact
point between parallel cylinders (SBez and Carbonell, 1987)
and vertically aligned spheres (Saez and Carbonell, 1990). For
specified values of EO and 4 and a given local geometry, Eq.
12 indicates that the liquid holdup depends on the dimensionless datum pressure, Po*.This means that there are many physically feasible menisci. There is, however, a limit to the size
of the meniscus that can be attained. In some cases, as the size
of the meniscus increases, a point is reached at which the
meniscus becomes unstable. For aligned cylinders (Saez and
Carbonell, 1987), the stability condition yields a value of Po*
below which no stable menisci can be formed. In other cases,
stable menisci with relatively large volumes can be formed, but
there is a geometrical constraint owing to the shape of the
particles that does not allow the formation of larger menisci.
This geometrical limit is also related to a value of P: below
which menisci cannot be physically feasible. This condition
yields menisci of maximum possible volume for the case of
touching spheres (Saez and Carbonell, 1990).
At the end of the draining process from a packed bed, it
seems reasonable to assume that when the breakage of the
liquid films occurs, the amount of liquid that remains in the
resulting menisci is the possible maximum amount that can
reach a stable configuration. From this point of view, the static
liquid holdup in a packed bed can be conceived as the value
obtained from Eq. 12 for the limiting value of Po* previously
described. The resulting relation can then be expressed as follows:
E,

= t,

(EO, 4, local geometry)

(13)

Equation 13 indicates explicitly that the static liquid holdup

should be affected only by the Eotvos number, the contact
angle, and the local geometry of the solid surface. For the case
of parallel cylinders and menisci between vertically aligned
spheres, the representation given by Eq. 13 has been rigorously
evaluated in previous works (Saez and Carbonell, 1987, 1990).
The large differences observed between those two simple geometries for the same values of EO and 4 suggest that the local
geometry has a strong effect on the static holdup. This reflects
the fact that the meniscus of maximum volume is very sensitive
to the geometrical constraint imposed by the geometry of the
solid particles. In addition, the effect of the contact angle is
appreciable, and it does not exhibit monotonous trends: depending on the values of 4 and Eo, an increase in the contact
angle might result in a decrease or an increase in the static
holdup.
For real packed beds, finding the relation expressed by Eq.
13 seems to be a formidable problem at this point, owing to
the high complexity of the local geometry. Empirical relations
such as those proposed by Charpentier et al. (1968) and Saez
and Carbonell (1985) appear to be the only realistic way to
assess the quantitative description of static holdup. Those correlations, however, have been developed by using experimental
data reported in the literature that correspond to packings of
different shapes, thus altering the local geometry of the solid
surface. The use of various characteristic lengths in the correlations has not been enough to characterize the differences
AIChE Journal

Methanol

Kerosene

- Correlation

in local geometry exhibited by the packings used. Besides, the

contact angle has not been reported in the experiments, so that
the data used might correspond to various values of this parameter. These two facts might explain the large scatter observed in the comparison of the correlations with the
experimental data.
In the experiments performed in this work, we used randomly-packed beds of spherical particles, and possible impurities at the solid surface (glass) were removed to ensure that
the contact angle is close to + = O . Under these controlled
conditions, Eq. 13 indicates a direct relation between the static
holdup and the Eotvos number. Figure 1 shows the experimental data gathered in this work for three different liquids
and various particle diameters. The plot suggests a unique
dependence between E, and EO, regardless of the liquid used.
The data are well fitted by the following equation, which is
also plotted in Figure 1,

0.11
1+EO

E, =-

When Figure 1 is compared with the correlations developed

in the literature (see, for example, Charpentier et al., 1968;
Saez and Carbonell, 1985; Wammes et al., 1990), several important observations can be made.
First of all, the observed scatter of the data is appreciably
smaller in Figure 1, which is a result of the controlled conditions
used in this work. The fact that liquids of different viscosities
are in agreement with a single representation is consistent with
the observation made previously regarding the absence of dynamic effects on the static holdup.
Second, the experimental data obtained in this work suggest
a limiting value of ~ , = 0 . 1 1for very small EO (when gravitational effects become negligible), whereas a value of 0.05 has
been traditionally reported to be the maximum static holdup
(Charpentier et al., 1968;Wammes et al., 1990). This difference
is a result of the fact that we have reached smaller Eotvos
numbers than those reported in the literature (reported data
fall in the range EO>0.3). It is important to point out that,
if the particles are small enough, a point is reached for which
the liquid retained in the bed is not the product of pendular
menisci formed at the contact point between the particles, but
a consequence of the formation of liquid blobs that engulf
several particles. Essentially, a static holdup equal to the porosity can be obtained by this mechanism of liquid retention.

November 1991 Vol. 37, No. 11

1735

1
0 Experimental

- Theory

force, combined with the influence of the local geometry of

the packing particles.
The results suggest that a general empirical relation for calculating the static liquid holdup in packed beds should be
established based on the representation in Eq. 13. The other
two parameters that have not been usually taken into account
(contact angle and local geometry) can have a very strong
influence on the static holdup and, therefore, should not be
disregarded.

01
01

10

Eo

Figure 2. Static holdups: randomly-packed beds of

spheres vs. vertically-aligned touching
spheres.

Our analysis is restricted to particles large enough that the

latter mechanism is not the controlling one. Even with this fact
in mind, our work indicates that static holdups larger than
0.05 can be obtained by menisci retention.
Third, our static holdup data are more sensitive to the Eotvos
number than those previously reported. For instance, Charpentier et al.s representation suggest a value of the static
holdup around 0.3 for EO=lO, whereas our measurements
indicate appreciably lower values (Figure 1). This difference
might be a result of the effects of local geometry and contact
angle.
In Figure 2 we compare all the experimental data obtained
in this work with relation 13 as obtained theoretically by Saez
and Carbonell (1990) for menisci between vertically aligned
spheres for 1$=0. It can be seen that the theory greatly overestimates the experimental data. This is a consequence of the
difference in local geometry between both situations. Note
that, in real packed beds, the touching point between any
couple of spheres is not vertically aligned, as in the case of the
theory. However, the representation in Fig. 2 shows that both
the theory and the experimental data exhibit the same trend
with regards to variations in the Eotvos numbers. This fact
leads us to think that the mechanism responsible for the static
liquid holdup in both cases is similar: the static holdup is a
consequence of the destabilizing effect of the gravitational

1736

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Manuscript received May 30, 1991.

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AlChE Journal