Colloids and Surfaces
A: Physicochemical and Engineering Aspects 204 (2002) 239– 250
www.elsevier.com/locate/colsurfa
Wicking flow in irregular capillaries
Thomas L. Staples *, Donna G. Shaffer
Superabsorbents Research & De6elopment, The Dow Chemical Company, 1603 Building, Midland, MI 48674, USA
Received 23 August 2001; accepted 7 December 2001
Abstract
Understanding capillary flow in porous media (wicking) is critical for predicting and improving the performance of
absorbent structures. Capillary flow in straight circular tubes has long been known to be quite well described by the
Lucas–Washburn equation. Experimental wicking results with beds of glass beads led us to investigate a minor
modification of this equation to describe more complex systems. These results, as well as the mathematical treatment,
are presented to describe capillary flow as a function of time in tubes irregularly shaped along their primary axis. The
cross-section of the model tubes remains circular. Variations of the diameter that are periodic along the length of the
tube are the focus of the study, and a sinusoidal description is argued to be sufficiently general to describe most
systems. The variables used to describe the system can be divided into two groups. Fluid and surface chemistry are
defined by the viscosity (p), surface tension (k), contact angle (q), and density (z). In the sinusoidal case, the capillary
itself is described by Dcap, the diameter at the largest portion of the tube which determines the limiting capillary
pressure, Dvis, the diameter of the throat which dominates the viscous drag, and u, ‘wavelength’ of the fluctuations.
The conclusions are: (1) as long as the wavelength of the fluctuation in diameter for the capillary is small relative to
the length of travel, the actual value of this wavelength does not matter in the time derivative of the flow. This is
typically the case for porous media in which the flow path dimensions fluctuate in the range of microns, and the
wicking distances are measured in centimeters. (2) As a result of this simplification, a two-parameter equation can be
written which yields the time t for the fluid front to reach a given distance L, analogous to the Lucas– Washburn
equation:
ln 1−
L
L
C 2D 2capzg
+
=−
t
32pLeq
Leq
Leq
The term Leq defines the equilibrium height and is a function of k, q, z, and Dcap. The rate is determined in addition
by p and Dvis (=C ×Dcap). © 2002 Published by Elsevier Science B.V.
Keywords: Wicking; Porous media; Lucas–Washburn equation; Capillary flow; Sinusoidal capillaries
1. Introduction
* Corresponding author. Tel.: +1-989-636-1082; fax: + 1989-636-6454.
Wicking is a factor affecting the flow of fluid in
porous media and quantitatively accounting for it
is important for modeling flow, our interest being
0927-7757/02/$ - see front matter © 2002 Published by Elsevier Science B.V.
PII: S 0 9 2 7 - 7 7 5 7 ( 0 1 ) 0 1 1 3 8 - 4
T.L. Staples, D.G. Shaffer / Colloids and Surfaces A: Physicochem. Eng. Aspects 204 (2002) 239–250
240
the field of absorbent structures. Aside from differences in surface wettability and the like, various substrates, e.g. packed beds, fiber mats, or
foams, may show different wicking behaviors due
to the shape and dimensions of their flow paths.
In this report we investigated the quantitative
treatment of just these geometrical aspects of the
wicking problem, prompted by experimental data
obtained with beds of glass beads.
Capillary rise in straight, vertical circular tubes
as a function of time was more or less completely
described by Lucas [1] and Washburn [2] and has
been conveniently summarized by Chatterjee [3].
The driving force, or ‘pull,’ is the capillary pressure as reflected in the Laplace relation, and the
‘drag’ retarding flow depends on viscosity
(Poiseuille Law) and the weight of the column of
fluid. The differential equation (Eq. (1)) is solved
to yield Eq. (2).
dL R 2 2k cos q
=
−zgL
dt 8p
R
ln 1 −
L
L
R 2zg
+
= −
t
Leq
Leq
8pLeq
(1)
(2)
where
Leq =
2k cos q
Rzg
(3)
In this expression, t is the time, L is the wetted
length of the capillary, the radius of which is
given by R. Fluid and surface chemistry are described by the viscosity (p), fluid surface tension
(k), contact angle (q), and density (z). The acceleration due to gravity is g. The term Leq denotes
the equilibrium height where the capillary pressure exactly balances the weight of the fluid
column. The equation (Eq. (2)) yields t as an
explicit function of L but can be pointwise computed to give L(t); this is usually termed the
Lucas –Washburn equation. For horizontal systems, in which the weight of the liquid column is
zero, or for any system at early times far from the
equilibrium height (L Leq), it can be shown that
L is proportional to t 1/2, and the resultant equation is often termed the Lucas equation.
Exploring the behavior predicted by this equation leads one to conclude that, for a given
elapsed time, there is an optimum tube diameter
leading to a maximum height of fluid. Although
smaller tubes will always lead to higher liquid
levels (i.e. larger Leq) eventually due to their
greater capillary suction pressure, they create
sufficient drag to slow the rise. Larger tubes, while
having less drag and hence allowing quicker approach to the equilibrium height, have a capillary
suction pressure too low for this equilibrium
height to be as great.
This can be illustrated using the classic high
speed photographic data from LeGrand and
Rense [4], as illustrated in Fig. 1, a double logarithmic plot of height (L) versus tube radius. The
curves are exact solutions of the Lucas–Washburn equation for specific elapsed times; the
straight line is the equilibrium height line; all are
calculated using literature values for parameters k,
q, p, and z. The data values used from Ref. [4] are
for the capillary rise of water in specified glass
tubes and show rather good agreement with the
predictions, although lagging at elapsed times less
than 300 ms. A maximum capillary rise calculated
for a 1-h test is 86.6 cm for a tube of radius 0.001
cm in this system. To the left of this maximum the
flow is viscous drag-limited; to the right the height
is capillary pressure-limited.
Our experimental interest was to extend this
observation to packed beds, seeking to determine
an optimum bead size for wicking height. The
resultant analysis followed as an explanation of
the experimental findings.
The engineering literature has much to say
about flow in porous media, especially the subset
labeled as periodically constricted tubes (PCT).
The general intent of these works is to determine
the streamlines and velocity profiles [5], from
which one can calculate, for example, reaction
rates [6] and filter efficiencies [7] in packed beds
and other porous media. The petroleum industry
is interested in high pressure forced flow of nonNewtonian fluids through porous rock for enhanced oil recovery [8]. Specifically for fibrous
systems, the PhD thesis of Mahale [9] provides
both an excellent set of references and useful
treatment, including data, for void formation.
Our objective here was a model useful for our
absorbent systems for which the flow front is the
T.L. Staples, D.G. Shaffer / Colloids and Surfaces A: Physicochem. Eng. Aspects 204 (2002) 239–250
241
Fig. 1. Maximum rise for water in vertical glass tubes.
primary interest, and we have not found that
specific system treated with adequate simplicity
elsewhere.
Two papers by Payatakes et al. [10,11], however, do deserve particular mention. They treat a
periodic circular channel with a cusp-shaped
profile, which rather accurately mimics a packed
bed of spheres. They are interested in the more
complex problem of calculating velocity profiles,
streamlines, and onset of turbulence for systems
in steady state. We will note later how their
conclusions differ from ours, but are not in conflict because of the difference in focus.
2. Experimental
2.1. Materials
Glass beads were obtained from Sigma–
Aldrich. They were screened to the indicated mesh
sizes and washed with deionized water and dried
in an oven. A 0.9% NaCl solution was used as the
wicking fluid because we were comparing these
data with swelling systems. The interpretation
here assumes there is little difference in saline and
water for glass substrates.
2.2. Equipment
The equipment and procedure used for measuring wicking have been described in a European
Patent [12]. The primary piece of equipment is an
array of V-shaped channels, essentially a corrugated metal sheet, clamped lengthwise in a rigid
frame with an opening at one end, which is covered by a 100-mesh screen. For the standard
wicking test 2.5 g of glass beads (1 g was used for
superabsorbent polymers with lower density) is
spread over 20 lineal cm of the trough; in these
examples more than 20 cm was used in some cases
but the standard depth was maintained by scaling
the sample weight. The multiple trough is set at
20° to the horizontal in a bath such that the open
T.L. Staples, D.G. Shaffer / Colloids and Surfaces A: Physicochem. Eng. Aspects 204 (2002) 239–250
242
Table 1
Wicking distance of 0.1% saline with glass beads at 20° incline
Mesh
Diameter (cm)
Time (s)
60
180
600
3600
Inf.
a
140–325
100–140
0.0075
0.0127
Wicking distance (cm)
5.9
7.0
9.0
10.5
15.3
16.3
23.8
25.9
a
30.0
70–100
0.0180
50–70
0.0254
40–50
0.0360
30–40
0.0500
20–30
0.0720
7.8
12.0
18.5
23.3
23.7
7.3
10.2
14.0
17.8
17.8
8.5
11.0
14.3
16.3
17.9
6.5
7.0
8.5
9.7
9.7
4.5
5.5
5.8
5.8
5.8
Limit exceeded size of equipment.
end is just at the surface of a large excess of the
wicking fluid but not wetted. To begin an experimental measurement, a low wave is propagated
through the bath to wet the samples lying in the
various channels simultaneously and wicking ensues. The distance the wetted front travels is
recorded periodically. The tests were run in an
air-conditioned laboratory.
3. Results
The bead data in Table 1, similar to, though of
much lower precision than, the tube data of
LeGrand and Rense [4], can be plotted in an
analogous way. In Fig. 2 we plot the maximum
distance of fluid rise (circles) versus, somewhat
arbitrarily, bead diameter. Correcting the capillary
pressure equation (Eq. (3)) for the test angle
(ƒ =20° off horizontal: distance traveled, L =
Lvertical/sin ƒ= 2.92× Lvertical), and substituting
bead diameter for tube radius, gives the line Leq
which roughly fits the circles. This suggests that,
with respect to capillary pressure, that the ‘characteristic radius’ of the capillaries in packed beds of
beads is close to the bead diameter.
The other points (squares) are the distances
reached at the indicated times for each bead size.
The 1-h distance calculated from the Lucas equation, using the same characteristic radius as used
for the limit line Leq (i.e. bead diameter), is the
upper positive slope line. This misses the measured
1-h values by a considerable amount. In order to
even approximately fit these points, we would
have had to substitute a characteristic radius 100
times smaller into the Lucas equation (lower 1-h
line). In other words, the radius of an equivalent
capillary tube giving the same drag (not capillary
pressure) as the 140– 325 mesh beads would be
about 0.7mm, rather than the 75 mm of the midpoint diameter. Apparently, the radius which is
most important for viscous drag is much smaller
than that which determines the capillary pressure.
Chatterjee [13] proposed handling this situation
by combining both rvisc and rcap into an ‘effective
radius,’ reff (roughly= (rvisc)2/rcap). His early time
data for 6ertical (ƒ= 90°, sin ƒ= 1, L= h) wicking in paper suggest an effective radius for the
drag-limited regime from about 0.3 mm for the
slowest to 3.0 mm for the fastest wickers. He does
not give enough data to determine the ultimate
capillary rise and thus characteristic radius for an
equivalent capillary.
Thus the interesting result from the bead wicking data was that, while indeed behavior similar to
that predicted by the Lucas–Washburn equation
occurred, it appeared one needed two different
characteristic dimensions rather than one (e.g. an
‘effective radius’). What follows is a more quantitative investigation of the problem of capillary
flow in irregular channels. We develop an explanation of the results and a model from a more
fundamental perspective.
4. Analysis and discussion
4.1. General case
One could imitate the surfaces of porous media
with complex boundary conditions, simulate surface tension driven flow in them with a fluid
T.L. Staples, D.G. Shaffer / Colloids and Surfaces A: Physicochem. Eng. Aspects 204 (2002) 239–250
243
Fig. 2. Wicking of 0.9% NaCl in beds of glass beads (20° off horizontal).
dynamics program, and then search for generalizations. This is a highly computationally intensive approach, albeit with a substantial literature.
Instead, we sought a simple incremental generalization of the Lucas– Washburn analysis. As a
first step in solving the ultimate problem for any
geometry, we considered capillary flow in a circular, but not necessarily straight, tube. As derived
in Appendix A, capillary flow in a vertical circular
capillary with the radius varying axially should be
described by the following equation:
t=
&
L
0
&
L
dz
4
[R(z)]
0
dL
2k cos q − R(L)zgL
8p[R(L)]3
(4)
In this expression, as before, L is the wetted length
of the capillary at time t, the radius of which is
given by the function in z, R(z) and all other
variables retain their meaning.
A critical part of solving this problem is calculating the viscous drag, which requires integrating
1/R 4. For any stipulated function, R(z), one can
numerically integrate the right hand side of Eq.
(4) and thereby simulate the capillary rise of a
Newtonian fluid in a tube of arbitrary dimensions.
As was the case for Eq. (2), the fluid parameters
used are viscosity, surface tension, and density.
Also required is a contact angle between the fluid
and the capillary wall. This is perhaps the most
difficult property to measure in the system. In
many cases, the utility of the analysis described
here may be to extract that value from uptake
rates. Other values used in the calculation are the
geometrical descriptors of the capillary.
4.2. Sinusoidal capillaries
One explicitly integrable function, which nevertheless describes many practical structures, is a
linear combination of sines and cosines, combined
to make an arbitrarily complex surface. In what
follows, we will describe our conclusions from
modeling using a sinusoidal description of the
tube shape, as illustrated in Fig. 3.
244
T.L. Staples, D.G. Shaffer / Colloids and Surfaces A: Physicochem. Eng. Aspects 204 (2002) 239–250
This figure shows an undulating tube with narrow throats separating large cavities. Such a
figure can be described as a rotation of a sine
wave about a line outside its range. Specifically,
the following function describes the radius:
R(z) =
Dvis (Dcap − Dvis)
2y
z
+
1 +cos
4
u
2
(5)
where Dvis is the throat diameter, Dcap the cell
diameter, and u the wavelength of undulations in
the vertical z-direction. Dcap is always greater
than Dvis; when equal, the solution collapses to
the Lucas–Washburn equation for a straight tube
(Eq. (2)). Open cell foams are an example class of
materials with large cells separated by smaller
openings termed pores. Although we will adopt
this ‘cell/pore’ terminology in our discussion here,
we note that it is not universally adopted in the
literature.
Eq. (5) is particularly useful for modeling systems in which the cells are spheroidal or ellipsoidal, such as foams. As illustrated in Fig. 3, one
full cycle (2y) of the cosine encloses a cell; hence,
for a spheroidal cell, u is approximately equal to
Dcap. For example, a cell with a diameter of 100
mm (0.01 cm) and a midthroat-to-midthroat distance (wavelength) of 100 mm would be
spheroidal. A shorter wavelength would describe
a cell that was more squat (oblate ellipsoidal), and
a larger value would yield a more slender cell
(prolate ellipsoidal). Diameters rather than radii
were used in this presentation because it is the
Fig. 3. Schematic of sinusoidal capillary.
diameter that is typically measured by microscopic and related techniques, from which our
information about structure morphology and dimensions generally comes.
Using SimuSolv™1 software to simulate the
distance versus time curves for various values of
the parameters gave expected results for the case
in which Dvis and Dcap are equal, exactly matching
results calculated by Eq. (2), setting R=Dcap/2
(= Dvis/2). With Dcap \ Dvis, there were wiggles in
the curve, but clearly there was also a slower
approach to the equilibrium height with smaller
Dvis. Some typical plots are shown in Fig. 4.
An initially surprising observation from the
simulation was the absence of any dependence of
the overall ‘envelope curve’ of L(t) on the wavelength, when u was kept small. Of course the fine
structure (wiggles) do depend on the u. One might
think that a smaller wavelength, i.e. more
‘throats’ in the flow path, would lead to greater
drag and slower flow. Because this is not the case,
we can infer that the geometrical drag term in Eq.
(4),
&
L
0
dz
=
[R(z)]4
&
L
0
dz
[A+B cos(2yz/u)]4
(6)
is independent of u, if the limits of integration are
in phase, i.e. L= 2ynu, where n is an integer. This
condition is automatically fulfilled, in a practical
sense, if u L. In other words, for cells small
relative to the overall distance of capillary flow,
which is the typical situation in a porous medium,
we cannot realistically distinguish a wicking distance of, for example, 10 cm from one of 10.01
cm, the difference of one 100 mm cell. Hence, cell
dimensions in the flow direction do not matter
very much. Payatakes et al. [10,11], as noted
earlier, found that u does matter in defining the
onset of turbulence, a result which is not in
conflict with our analysis of much slower (laminar) flow rates.
We had speculated that the data on glass beads
could be fit with a two-radius differential equation. With this hypothesis in mind we attempted
to fit the envelope curves from the simulation to
an equation of the form
1
Trademark of The Dow Chemical Company.
T.L. Staples, D.G. Shaffer / Colloids and Surfaces A: Physicochem. Eng. Aspects 204 (2002) 239–250
ln 1 −
Fig. 4. Simulations of L vs. t.
L
L
D zg
+
= −
t
Leq
Leq
32pLeq
2
vis
(7)
wherein Leq is calculated from Eq. (8), using Dcap.
Leq =
4k cos q
Dcapzg
245
(8)
The denominator on the right of Eq. (7) has 32
instead of 8 as in Eq. (2), and the numerator of Eq.
(8) has a 4 instead of a 2 as in Eq. (3) because of
the change from radii to diameters.
We have now reduced the problem for any
specified system of fluid and wall material (i.e. given
k, q, z, and p) to a relation among the four
remaining parameters: displaying L versus t for any
Dvis and Dcap. One approach to generalizing the
results is to plot the distance traveled at a given time
over a wide range of diameter combinations as a
surface. Consider the plane defined by Dvis and
Dcap, and further consider the values to be displayed logarithmically to accommodate the wide
range of dimensions seen in porous media. If
perpendicular to this plane is plotted liquid height
(L), also logarithmically, then for any time t there
is a surface which defines the distance traveled L,
for every sinusoidal tube with any specified combination of Dvis and Dcap values. Of course there is
a similar surface for every other value of t as well.
The surfaces for longer times enclose the surfaces
for shorter times, stacking up and building outward
like successive layers of an onion. Now Eq. (7) can
be written as
L
L
C 2D 2capzg
ln 1−
+
=−
t
(9)
Leq
Leq
32pLeq
for which C (51) is the ratio of Dvis to Dcap.
D
C= vis
(10)
Dcap
Any value of C would represent a diagonal on the
Dvis − Dcap plane, and a slice along the diagonal
where C= 1, for example, would yield a plot such
as Fig. 1.
Eq. (9) can be implicitly differentiated (holding
L constant) to yield dt/dDcap, which can be set to
zero to yield the following:
T.L. Staples, D.G. Shaffer / Colloids and Surfaces A: Physicochem. Eng. Aspects 204 (2002) 239–250
246
(L/Leq)2
+3[ln(1− L/Leq) +L/Leq]= 0
1− L/Leq
(11)
The value for L obtained from solving Eq. (11) is
Lmax. Eq. (11) further says that the maximum
wicking distance for a given time occurs at the
same fraction of equilibrium height (L/Leq) regardless of the pore diameter, Dvis. By solving Eq.
(11), one can determine the value of this fraction
to be 0.645.
To finish the analysis, one must still find the
‘right’ Leq-value, i.e. the optimum diameter. To
illustrate the process, consider Fig. 5. It is essentially a reproduction of Fig. 1 with fewer lines and
no data. Any fixed-C plane will look the same.
Of the four variables (Dcap, Dvis, L and t), the
Leq-line is a function of only Dcap, and thus this
line is the same regardless of the value of C. It has
a slope of − 1 on a logarithmic plot. The other
straight line is an asymptotic constant-t line; it is
described by the equation,
L=C
'
Dcapk cos q 1/2
t
4p
(12)
which is analogous to the early time approximation of Eq. (2) mentioned previously. It has slope
of + 1/2 on a logarithmic plot. Where these two
lines cross is the Leq value for the optimum diameter given the fixed time t, i.e. L*eq. The maximum height Lmax for that time is then 0.645 L*eq.
The overall fastest time will be when C= 1, a
straight tube; all other equal time lines will be
scaled by log C.
4.3. Extensions of the analysis
Our original goal was to create a model that
would allow calculation of wicking distance versus time for any set of the input parameters
previously discussed. As derived in Appendix A,
Eq. (4) is presented here as the complete description for vertical circular tubes. Finally, Eq. (9)
represents a major simplification of the general
model, eliminating the largely irrelevant microscopic fluctuations but fitting with our earlier
observations for glass beads. What are the limitations of our analysis? If we can identify them,
these limitations will suggest opportunities for
Fig. 5. Any L–Dcap-plane (C constant).
T.L. Staples, D.G. Shaffer / Colloids and Surfaces A: Physicochem. Eng. Aspects 204 (2002) 239–250
247
Fig. 6. Series arrangement of multiple diameters.
extending the analysis further. We will now discuss
specific examples.
4.3.1. Multiple sinusoidal constrictions
A sinusoidal tube shape obviously oversimplifies
reality. In a real substance there will be a distribution of broad and narrow passages rather than
single Dvis and Dcap values. These distributions
may be quite narrow in the case of packed uniform
spheres, but systems such as foams or nonwovens
would be expected to have broad distributions of
both pore and cell sizes. As noted earlier, Eq. (4)
does not require a periodic structure. Any stipulated set of successive diameters could be used and
the resultant L versus t curve calculated by pointwise integration. However, to meaningfully describe a real flow channel would require
measurements of the diameter every few microns,
if that is the scale of the fluctuations. This is clearly
prohibitive for routine use, but can the generality
of our simple two-parameter cosine expression be
improved by using just a ‘few’ more input values?
When considering the effects of nonuniformity,
it can be useful to look at a series arrangement of
irregularly sized tubes. Consider a tube with aperiodic fluctuations over some range, as illustrated in
Fig. 6. It seems reasonable that there may be a
‘superperiod,’ the wavelength of which is indicated
as \ in the figure, that fairly represents the collection of microscopic variations in cell and pore size
occurring in a real structure. If this was not the
case, one could divide the whole sample into
regions, e.g. skin and core, wherein this was the
case.
Our problem reduces then to finding how fluid
wicks through an aperiodic tube of modest length,
\, where \ is large enough to be representative of
the full range of variability but small enough to be
described by a manageable number of parameters.
A convenient approach is to describe the tube as
a sum of various half-period segments of cosine
functions, arranged such that Dvis and Dcap of
adjacent elements are equal, as indicated in the
Fig. 6. This could then be built into a simulation
program rather handily. Presumably one could
generalize the results from such simulations in
248
T.L. Staples, D.G. Shaffer / Colloids and Surfaces A: Physicochem. Eng. Aspects 204 (2002) 239–250
such a way as to reduce the number of parameters
required.
4.3.2. Parallel flow paths
A separate but related question is how to handle the inevitable distribution of dimensions that
occurs in parallel flow paths. If the ‘series problem’ above is adequately treated, there should be
a grand average L versus t curve that represents
the ‘typical’ flow path. The parallel arrangement
of an assembly of such tubes is moot. In other
words, all the averaging has been done in treating
the series arrangement. The question of short
circuits and dead end cavities may bear some
consideration.
4.3.3. Cur6ature of surfaces
A different issue concerns the curvature of the
surfaces, in both the x – y plane and the z-axis.
For example, is there loss in generality from selecting a circular cross-section, which is much
easier to treat mathematically than less symmetrical descriptions. An even more basic issue is the
‘sense of curvature’ — is the surface of the flow
channel ‘closed’ or ‘open?’ An example of an open
curvature would be the cusp-shaped passages that
result from packed beds of spherical beads. An
example of the importance of this question in a
real system concerns experimental results showing
a difference in wicking between reticulated foams
and blown foams [14]. The former have with very
open curvature while open cell blown foams have
residual walls and thus a more closed curvature.
Nonwovens would be expected to resemble reticulated foams in this respect.
Fig. 7. Detail for contact angle.
Fig. 7 illustrates another question to consider
when evaluating the effect of geometrical variables on fluid and surface parameters. Because of
curvature, the force vectors resulting from the
wettability of the surface, as measured by the
contact angle, do not all point ‘up,’ as we have
tacitly assumed in our model. Because little good
data exists on contact angles anyway, we can let
the value float and treat it rather like a fit parameter, i.e. some ‘average’ value will best describe the
data.
5. Summary
With a relatively simple relation for wicking
distance versus time, such as the Lucas–Washburn equation, modeling capillary flow in a
macroscopic article is manageable problem. One
can divide the area of interest (i.e. inside the
boundary conditions) into smaller regions and use
this relation to calculate flow from section-to-section with time. The Lucas–Washburn equation,
however, only applies to straight capillaries,
which would not be very accurate descriptors of
complex porous media. The work here shows that
flow in much more complicated channels can be
described with an equation only slightly more
complicated than the Lucas– Washburn equation.
A flow model requiring only one additional
parameter, the characteristic diameter controlling
viscous drag Dvis, could be used. We would expect
such a model to give significantly better agreement with experiment than would be the case for
a model with only one characteristic flow dimension determining both capillary suction and viscous drag, R in the Lucas–Washburn equation.
This was indeed found to be the case for data on
wicking through beds of glass beads.
More specific details from this work are the
following:
1. An overall analysis of the problem for capillary flow in circular tubes of irregularly varying radius has been presented. The detailed
derivation is given in Appendix A.
2. The case for sinusoidal tubes was shown to
give simple flow curves when uL, i.e. when
the scale of fluctuations was small.
T.L. Staples, D.G. Shaffer / Colloids and Surfaces A: Physicochem. Eng. Aspects 204 (2002) 239–250
249
To obtain the total pressure drop DP over L, we
can integrate. Note that dV/dt can be pulled
outside the integral because, at any time t, it is
constant over all z for an incompressible fluid. It
is not generally constant over time, however.
DP =
&
P
dP =
0
8p(dV/dt)
y
&
L
0
dz
[R(z)]4
(A2)
For a vertical capillary, the driving force DP is the
sum of the capillary suction pressure developed at
the meniscus (z= L) and the weight of the
column of fluid, viz.
DP =
Fig. 8. Generalized capillary.
2k cos q
− zgL
R(L)
(A3)
Combining these two elements yields
3. Curves simulated when u is small can be quite
closely approximated by a two diameter
equation.
4. Extensions of this model were outlined.
2k cos q
8p(dV/dt)
− zgL =
R(L)
y
&
L
0
dz
[R(z)]4
(A4)
In a circular tube, the volume flow rate is related
to the speed of the front as follows:
dV
dL
= y[R(L)]2
dt
dt
Acknowledgements
The authors would like to appreciate Fred
Buchholz, Larry Wilson, David Allan and Andy
Graham for reading drafts of this work and making many useful comments. Professor Robert
Prud’homme steered us to other sources and deserves our thanks. We would also like to acknowledge The Dow Chemical Company for allowing
us to submit this work.
This is also true at all other z less than L. Making
this substitution yields
2k cos q
8py[R(L)]2(dL/dt)
− zgL =
R(L)
y
&
L
0
dz
[R(z)]4
(A6)
Canceling y and rearranging gives
dL
=
dt
Appendix A. Derivation of flow equation for
irregularly shaped capillaries
(A5)
2k cos q
−zgL
R(L)
L
dz
8p[R(L)]2
4
[R(z)]
0
&
&
(A7)
which can be further rearranged to
Consider a vertical circular capillary, the radius
of which is given by some stipulated function
R(z), as illustrated in Fig. 8. A liquid is rising in
this tube and the meniscus sits at height L.
For an incompressible fluid, Poiseuille’s Law
says that the volume flow rate past any point z
(z 5 L) at time t is given by:
dV y[R(z)] dP
=
dt
8p
dz
8p[R(L)]2
dt=
0
(A1)
dz
[R(z)]4
2k cos q
− zgL
R(L)
and
&
dL
(A8)
L
dz
4
[R(z)]
0
dt=
dL
2k cos q− R(L)zgL
8p[R(L)]3
4
L
(A9)
T.L. Staples, D.G. Shaffer / Colloids and Surfaces A: Physicochem. Eng. Aspects 204 (2002) 239–250
250
Integration from 0 to t on the left and 0 to L on
the right gives the following:
t=
&
L
0
&
L
dz
4
[R(z)]
0
dL
2k cos q − R(L)zgL
8p[R(L)]3
References
[1] R. Lucas, Kolloid-Z. 23 (1918) 15.
[2] E.W. Washburn, Phys. Rev. 17 (1921) 273.
(A10)
[3] P.K. Chatterjee (Ed.), Absorbency, Elsevier, New York,
1985.
[4] E.J. LeGrand, W.A. Rense, J. Appl. Phys. 16 (1945) 843.
[5] J.A. Deiber, W.R. Schowalter, AIChE J. 25 (1979) 638.
[6] P. Fedkiw, J. Newman, AIChE J. 23 (1977) 255.
[7] D.A. Heagland, R.K. Prud’homme, AIChE J. 31 (1985) 236.
[8] R.A. Greenkorn, Flow Phenomena in Porous Media,
Marcel Dekker, New York, 1983.
[9] A.D. Mahale, PhD Thesis, Princeton University, 1994.
[10] A.C. Payatakes, C. Tien, R.M. Turian, AIChE J. 19 (1973)
58.
[11] A.C. Payatakes, C. Tien, R.M. Turian, AIChE J. 19 (1973)
67.
[12] European Patent 0 532 002 B1, May 14, 1997.
[13] P.K. Chatterjee, Svensk Papperstidning 74 (1971) 503.
[14] US Patent 6,071,580, June 6, 2000.