Proceedings of the Fifth International Conference on Nanochannels, Microchannels and Minichannels
Proceedings of ASME ICNMM2007
ICNMM2007
5th International Conference on Nanochannels, Microchannels
and Puebla,
Minichannels
June 18-20, 2007,
Mexico
June 18-20, 2007, Puebla, Mexico
ICNMM2007-30029
ICNMM2007-30029
A REVIEW OF MODELS FOR WATER DROPLET DETACHMENT FROM
THE GAS DIFFUSION LAYER-GAS FLOW CHANNEL INTERFACE IN PEMFCs
Charles H. Schillberg
[email protected]
Satish G. Kandlikar
[email protected]
Thermal Analysis and Microfluidics Laboratory
Rochester Institute of Technology
ABSTRACT
Understanding the fundamental mechanisms of water
transport in proton exchange membrane fuel cells (PEMFCs) is
necessary for effective management of product water. Among
the transport mechanisms affecting the performance of
PEMFCs are droplet formation, growth, and detachment at the
gas diffusion layer (GDL)-gas flow channel interface. The
presence of water droplets on the GDL blocks the access of
gases to the reaction sites, increases channel pressure drop, and
creates inconsistencies in the gas velocity fields down the
channel length, all resulting in performance deterioration. In
order to gain an insight into controlling droplets, an in-depth
review of PEMFC water droplet detachment models published
in the literature is presented here. Summaries of supporting
data, modeling techniques, and conclusions are also presented.
1. INTRODUCTION
Water management remains a major concern in
effective implementation in PEMFCs. It is essential to control
the water present in a fuel cell because the essential chemical
reactions and physical transport processes are tied inextricably
together by water. Effective water management will require
careful design: at one extreme, a specific minimum amount of
water is required for maintaining high proton conductivity in
the proton exchange membrane (PEM); at the other extreme, an
excess of water leads to flooding of the fuel cell. During
flooding the reaction sites, GDL pores, and gas flow channels
can all be blocked by water. All of the aforementioned
scenarios will lead to significant performance degradation if not
dealt with properly. Among the many water transport
mechanisms in PEMFCs are electro-osmotic drag, capillary
wicking, back diffusion, and multi-component two-phase flow.
It has been observed that water droplets are the prominent
source of water entering the gas flow channels through the
GDL [1], [2]. The water droplets can simply shear off the GDL
with the gas flow, causing a minimal obstruction. This will
happen most easily with droplets that are small compared to the
channel size. However, the droplet could interact with the
channel walls or another area of the GDL upon detachment.
Droplets can adhere to these surfaces and obstruct the flow.
Over time adhered droplets can coalesce and build up into films
and slugs. Films present a higher resistance to shearing flows,
and over time they may build up into slugs. Films, slugs, and
their flows in relevance to PEMFC channels are discussed
extensively by Zhang, et al. [2]. Precise droplet control is
needed for design of more effective water management
schemes.
A brief review of definitions and terms is presented
here. A water droplet resting on a GDL is shown in Fig. 1.
H
θ
GDL
Figure 1 – Droplet Height, Contact Angle
In the present work, droplet height (H) is the distance from the
GDL to the top of the droplet. Droplet diameter (D) is used
when a droplet is considered to be circular (not shown in Fig.
1). The contact angle (θ) is the angle that the air/water interface
makes with the surface of contact (the GDL). The contact line
is the line where the three phases meet (at the air/water/GDL
interface). When a droplet has no transverse forces acting on it,
the contact angle is constant around the entire contact length.
This constant angle is called the static contact angle, θS. Under
shearing forces, as in fuel cell gas channels, the contact angle
can vary along the line of contact, which is illustrated in Fig. 2.
1
Copyright © 2007 by ASME
NOMENCLATURE
FLOW
H
θA
θR
GDL
Figure 2 – Droplet Experiencing a Shear Flow,
Dynamic Contact Angles
The downstream (θA) and upstream (θR) angles are defined as
the dynamic contact angles (advancing and receding,
respectively). The contact angle varies along the contact line
between the dynamic contact angles θA and θR. This variability
in the contact angle around the droplet is a way to measure
deformation and stability. Contact angle hysteresis, ∆θ = θA θR, is visually a measure of how much a droplet has deformed,
as well as being a measure of adhesion to the surface (GDL).
The forces acting on a droplet in a flow channel are
now discussed. The droplet in Fig. 2 is effectively experiencing
two forces: a drag force and a surface adhesion force. The drag
force is caused by the shear flow. It is a sum of a shear stress
force and a pressure force. The surface adhesion force, which
acts to hold the droplet in place, is the component of the surface
tension force which acts against the flow. When the drag force
exceeds the surface adhesion force the droplet departs from the
surface. This condition is generally employed in the analytical
models developed in the literature. The gravitational force is
typically neglected when considering individual droplets
because it is quite small when compared to the other forces on
the droplet. One way to see this is through the bond number, a
ratio of gravitational to surface tension forces:
Bo =
∆ρ g Lc
σ
(1)
where the difference between fluid densities is ∆ρ, gravitational
acceleration is g, a characteristic length is Lc, and the surface
tension is σ. Droplet size is taken as the characteristic length.
When the bond number is below 0.06, Zhang, et al. [2] suggests
that gravity can be neglected. Most droplets in flow channels
meet this condition, and so gravity can be neglected in their
consideration. However, gravity should not be neglected when
droplets coalesce into larger droplets, films, or slugs.
The objective of the current work is to study and bring
together recent attempts at modeling droplet departure in the
literature. Analytical force balance models are critically
reviewed in Section 2 and Table 2, and their supporting data
ranges are presented in Table 1. In addition, two numerical
models and two novel models are summarized and discussed
briefly in Section 3. Some conclusions about droplet dynamics
are listed along with some suggestions for future modeling
attempts in Section 4.
H
D
r
θS
θA
θR
∆θ
α
θ
Droplet Height
Droplet Diameter
Droplet Radius, r = D/2
Static Contact Angle
Advancing Contact Angle
Receding Contact Angle
Contact Angle Hysteresis, ∆θ = θA-θR
Azimuthal Direction (Fig. 6)
Contact Angle, θ = f (α)
2B
DH
∆p
τ
U
Re
Channel Height
Channel Hydraulic Diameter
Pressure Drop
Shear Stress
Gas Channel Mean Velocity
Reynolds Number, Re = U LC/νgas
Fdrag
Cdrag
F∆p
Fτ
Fσ
Drag Force
Coefficient of Drag
Pressure Drop Force
Shear Force
Surface Tension Force
σ
Surface Tension (Per Unit Length)
µ
Viscosity
ρ
Density
ν
Dynamic viscosity
_________________________________________________
Any other nomenclature is explained in adjacent text.
Units vary among authors.
2. DROPLET DEPARTURE MODELS
Droplet dynamics have been studied extensively in the
past in various applications. However, these past studies may
not describe the entire picture of water droplet dynamics in
PEMFCs. The large range of conditions experienced by the
droplets makes their dynamics quite complex. The surface
variation along and within a gas diffusion layer is enough to
severely complicate the problem [3], not to mention the other
possible variables (humidity, temperature, gas/water flow rates,
channel geometry, channel surface coatings, and number of
surfaces of contact). The models presented show reasonable
agreement with the data with which they are compared, but
they are not validated with wider ranged data sets from other
sources. Understanding these models along with their
capabilities and limitations will facilitate the development of
more robust models.
Seven models are discussed in the present work. Four
of these are simple analytical models which are reviewed in
detail in Section 2. In addition, three other models which shed
light on PEMFC droplet dynamics are briefly discussed in
Section 3. The experimental parameters used in the
development of the original models are summarized in Table 1.
A summary of modeling techniques used for the analytical
model derivations is presented in Table 2.
2
Copyright © 2007 by ASME
TABLE 1 – SUMMARY OF SUPPORTING EXPERIMENTAL WORK REPORTED IN LITERATURE
AUTHOR
Chen, et al.
Section 2.1
Kumbur, et al.
Section 2.2
CHANNEL
DROPLET
SIZE
(mm)
Simulated
with a 2 mm
gap between
plates
4 mm wide,
5 mm tall
25100,
0.2 - 2
B
H/chord
ratio (see
Sec 2.2):
0.7 - 1.6
Zhang, et al.
Section 2.3
(Also used by
He, et al.,
Re,
LC
1 mm wide,
.5 mm tall
0.1 - 0.5
1001200,
CATHODE
GAS
CONDITION
Teflon Treated, θS = 140o
Untreated, θS = 120o
Humidified
WATER
DELIVERY RATE
(µL/min)
T
O
( C)
1.4
25
0.3
(simulated)
Treated
5, 10, 20 wt% PTFE
Un-humidified
DH
20100,
Teflon Treated
20 wt% PTFE
w/micro-porous layer
θS = 150o
1
60
23
(simulated)
Humidified
80
0.8
-
Avg. Pore Size
~ 10-30µm
2.1 Chen, et al. [4]
The objective of Chen, et al.’s work is to develop two simple
analytical models for predicting droplet detachment from the
GDL. One model considers a spherical droplet and the other
considers a cylindrical droplet. The spherical droplet is
illustrated in Fig. 3 and the associated model is discussed in
detail below.
H=
H r (1 − cos(θ A ))
=
2B
2B
(3)
Figure 4 illustrates the channel schematic employed by
Chen, et al.
·p
·p’
·p’
o
o
Spherical Model Details
L
·p
L
2b
2B
r
H
H
-r cos(θA)
CURRENT
DENSITY
2
(A/cm )
Avg. Pore Size ~ 10µm
Avg. Pore Size
~ 50-150 µm
D
Section 2.4)
GAS DIFFUSION
LAYER
CHARACTERISTICS
θA
r
θA
θR
L
Figure 3 – Schematic of Spherical Droplet Geometry
Figure 4 – Schematic View of Droplet Control Volume
The droplet is modeled as a truncated sphere, shown in
Fig. 3. The height is defined assuming the contact angles of the
droplet are constant (θ = θA) around the contact line.
The channel height is 2B, and its length L. The
distance from the droplet height to the channel top wall is 2b.
The pressures immediately downstream and upstream of the
droplet are p′L and p′o, respectively. A force balance along the
direction of the flow yields the force from pressure difference
imposed across the drop, a shear force, and a drag force:
H = r (1 − cos(θ A ))
(2)
This definition of height corresponds well to their observed
images of droplets at detachment. Droplet height is normalized
with the channel height:
F∆p + Fτ + Fdrag = 0
3
(4)
Copyright © 2007 by ASME
TABLE 2 – SUMMARY OF ANALYTICAL MODELS REPORTED IN LITERATURE
AUTHOR
Chen, et al.
Section 2.1
FLOW
ASSUMPTIONS
Incompressible
Laminar
Fully developed
DRAG FORCE
SHEAR FORCE
PRESSURE FORCE
Flow between droplet
and channel top wall
is flow through
parallel plates [5],
shear at droplet
height
∆p in flow between
droplet and channel
top wall.
(Flow through
narrow slit [5])
Newtonian
Imposed on diameter
squared
Kumbur, et al.
Section 2.2
Incompressible
Laminar
Fully developed
Newtonian
SURFACE ADHESION FORCE
Flow between droplet
and channel is flow
in rectangular
enclosure [6], shear
at top wall
Imposed on diameter
squared
CONTACT ANGLES
Circumference of a
circle defined by a
sphere and a plane
intersecting at the
GDL static contact
angle
Each half of the contact
line is multiplied by a
component of surface
tension defined by each
dynamic contact angle
Circle of arbitrary
diameter to be
varied with droplet
image
measurements
Differential surface
tension force integrated
around contact line with
linearly varying contact
angles
Diameter of circle
defined by a sphere
and a plane
intersecting at the
GDL static contact
angle
Each half of the contact
circle diameter is
multiplied by a
component of surface
tension defined by each
dynamic contact angle
[8], [9]
Imposed on
product of channel
height & diameter
∆p derivation not
given
Imposed on
product of channel
height & diameter
Zhang, et al.
Section 2.3
Not
Explicitly
Stated
CONTACT LINE
Drag force correlation
adapted from Clift [7],
Not Explicitly Stated
Dynamic contact angles
assumed to be
symmetric
Circle of average
GDL pore diameter
He, et al.
Section 2.4
Not
Explicitly
Stated
Derived from 2-D model,
Not Explicitly Stated
4
Differential surface
tension force integrated
around contact line with
linearly varying contact
angles
Dynamic contact angles
assumed to be
symmetric, and to
adhere to relation in
[10]
Copyright © 2007 by ASME
The drag force found by finding the shear and pressure
forces. The flow is assumed to be laminar (Re: 25-100), fully
developed (steady state), Newtonian and incompressible (air at
25oC). The shear stress is found by assuming the flow between
the top of the droplet and the top channel wall acts as the flow
between two parallel plates. A relation from [5] is used to find
the shear stress at the top of the droplet. The shear stress is
imposed on a square with side D for the shear force:
Fτ = τ Droplet D 2 =
b ∆p ' 2
D
D
(5)
Conservation of mass is considered by equating the volumetric
gas flow rates far from the droplet and above the droplet. The
flow along the sides of the droplet is neglected:
2 BU = 2bU '
(6)
The flow between the top of the droplet and the top channel
wall is approximated to be that through a narrow slit [5]. This
assumption, along with Eq. 6, allows the immediate pressure
drop (∆p’) to be found in terms of known flow properties. To
find the force due to pressure drop, ∆p’ is imposed on the
product of the channel height and the droplet diameter:
F∆p = ∆p' (2 BD) =
− D ∆p
(2 BD)
D + ( L − D)(b / B) 3
(7)
Using Eqs. 5 and 7 the drag force is found in terms of know and
desired quantities:
− Fdrag =
− (∆p) B( D / L) 2 L(1 + H )
(1 − H ) 3 + ( D / L)[1 − (1 − H ) 3 ]
(8)
Eq. 10 is given by Chen, et al. as Eq. 16 in [4]. Equating Eqs. 9
and 10 defines the instant at droplet detachment. The final form
used by Chen, et al. is Eq. 18 in [4].
A potential problem with Chen, et al.’s model is the
use of the simple rectangular approximations of the imposed
pressure difference area (2BD), the shear stress area (D2), and
the area where gas flows above the droplet (2bD). In addition,
the surface tension force does not account for any variability of
contact angle along the contact line, and the contact length
approximation is limited in that it is dependent on only one
dynamic contact angle. These issues are discussed in Section 4.
Cylindrical Model
The same ideas and approximations used for the
spherical droplet are used again with a cylindrical droplet. The
cylinder’s axis is perpendicular to the flow and has a length of
unity. Only slight differences arise in the derivations. The final
form of the force balance used by Chen, et al. can be found in
[4] as Eq. 23.
Chen et al.’s Validation
Ex-situ droplet imaging/measurement was conducted
in a simulated flow channel. The water flow rate simulated
operation at a current density of 1.4 A/cm2. The channel height
(gap between plates) was 2 mm, and the smallest droplets
observed were on the order of 200 µm. Humidified air flowed
through the channel, and images were captured perpendicular to
the flow so that droplet height and dynamic contact angles
could be measured. These measurements are compared with the
spherical model’s predictions on plots of H versus ∆θ. These
plots are termed instability windows.
The overall pressure drop is then related to flow properties far
from the droplet. This is used to find the final form of the drag
force:
− Fdrag =
48µ gasUB
(1 − cos(θ A ))
2
(1 + H ) H
×
(1 − cos(θ A ))(1 − H ) 3 + (4 B / L ) H [1 − (1 − H ) 3 ]
(9)
The surface adhesion force is found by considering the
line of contact between the droplet and GDL. It is approximated
as the circle where the sphere meets the GDL in Fig. 3 (not
explicitly illustrated). The circle has a circumference of
πDsin(θA). The surface tension force is given by:
Fσ = σ cos(θ R )
π D sin(θ A )
2
− σ cos(θ A )(
π D sin(θ A )
2
)
(10)
Each term in Eq. 10 is a component of the surface tension force
along the direction of the flow defined by a different dynamic
contact angle. The contact length for each component
(πDsin(θA)/2) is one half of the entire contact length. The
surface tension force is rewritten to include the dimensionless
droplet height and a second parameter which allows for direct
input of contact angle hysteresis. The rewritten expression of
Figure 5 – Example Instability Window, Chen, et al. [4]
A droplet represented by a point below the line is stable, and a
droplet represented by a point above the line is unstable.
Droplets above the line will detach in the flow. As can be seen
here, and as seen on all the instability windows in [4], the
model is in reasonable agreement with the experimental data
trends.
Two-dimensional finite element simulations were
conducted of air flowing over two dimensional water droplets.
The results from these simulations are compared with
predictions from the cylindrical model on instability windows.
The model achieves reasonable agreement with the simulations,
5
Copyright © 2007 by ASME
with the best results at a vanishing Reynolds number. This
result seems to imply that inertial terms are needed in the
analytical model.
·p
2.2 Kumbur, et al. [11]
The objective of Kumbur, et al.’s work is to elucidate
conditions leading to droplet removal in PEMFCs. The focus is
on development and analysis of a model capable of predicting
droplet detachment as well as the effects that engineering
parameters have on contact angle hysteresis. Specific attention
is put toward GDL hydrophobicity treatment and cell operating
conditions.
Model Details
dFST
θ = θ(α)
H
α
2
2b
Chen et al.’s Results
A simple design parameter is derived from the
analytical models. Setting the dimensionless droplet height less
than unity will theoretically prevent droplets from interacting
with the channel top wall. This generates a useful design
parameter:
L µ gasU π
(11)
>
2B σ
12
The left hand side of Eq. 11 is the channel length to height
aspect ratio multiplied by the channel capillary number. This
allows for design of channel dimensions and operating
conditions. The same procedure is followed with the cylindrical
model. The result is the same as Eq. 11, except that the right
hand side is 1/6. It is more conservative to use π/12 for design
considerations.
Through parametric studies on instability windows,
several parameters are found to decrease departure diameter. It
is seen that as the mean air velocity in the channel or the
channel length increases, the detachment diameter decreases.
The model predicts that an increased static contact angle will
result in a lower departure diameter, though the experimental
data on this appears inconclusive.
·p
1
2B
H
θR
To find the drag force on the droplet, a control volume
enclosing the droplet is considered (Fig. 7). A force balance
along the direction of the flow yields the imposed pressure
difference force across the droplet, a shear force on the control
volume, and the drag force imposed on the droplet:
F∆ p + Fτ + Fdrag = 0
The geometry modeled by Kumbur, et al. is a spherelike droplet truncated by the GDL. The height of the droplet is
found by considering a sphere resting on the GDL at the
advancing contact angle, just as with Chen, et al. (Fig. 3). The
contact line (Fig. 6) is a circle of arbitrary diameter (c), which
is not dependent on the contact angles. This allows for a
broader definition of surface adhesion force which is not tied to
a geometry defined by contact angles.
(12)
The flow is assumed to be laminar (Re: 100-1200), fully
developed (steady state), Newtonian and incompressible (air at
60oC). From this, an approximation for the immediate pressure
drop across the droplet is found. Using a rectangular
approximation which only considers flow above the droplet, a
very simple conservation of mass equation is developed
between the flow far from the droplet and the flow above the
droplet (Eq. 6). This allows the immediate pressure drop to be
found in terms of known or desired quantities (Eq. 7 in
Kumbur, et al. [11]). The pressure force is found by
multiplying the immediate pressure drop (∆p) by the product of
the channel height and the droplet diameter:
F∆ p = ∆p (2 BD ) =
24 µB 2UH 2
( B − H ) 3 (1 − cos(θ A )) 2
(13)
To find the shear stress, the flow between the droplet and the
top channel wall is approximated as that in a rectangular
enclosure [6]. The shear force on the droplet is approximated as
the shear stress at the top wall of the channel (τtop wall)
multiplied by the droplet diameter squared:
Fτ = τ top wall D 2 =
Figure 6 – Left: Droplet Geometry Used by Kumbur, et al.
Right: Differential Surface Tension Force
θA
Figure 7 – Schematic of Droplet Control Volume
GDL
c
r
3µBU
D2
(B − H )2
(14)
Substituting Eqs. 13 and 14 into Eq. 12 allows for the solution
of the drag force in terms of channel geometry, droplet size,
flow conditions, and the advancing contact angle:
− Fdrag =
24µB 2UH 2
3µBU
D2
+
3
2
2
( B − H ) (1 − cos(θ A ))
( B − H / 2)
(15)
The surface adhesion force (Fσ) is found by integrating
a component of the differential surface tension force (dFST)
along the flow direction (Fig. 6, Right). The contact angles are
assumed to vary linearly along the azimuthal direction (α) from
θA at α = 0 to θR at α = π:
6
Copyright © 2007 by ASME
θ = θ (α ) = θ A −
∆θ
π
α
(16)
The full integral is presented by Kumbur, et al. as Eq. 5 in [11].
The surface adhesion force is rewritten to be in terms of the
surface tension per unit length (σ), the arbitrary contact circle
diameter (c), the contact angle hysteresis (∆θ), and the
advancing contact angle (θA):
Fσ = cσ
π  sin(∆θ − θ A ) − sin(θ A )
2 
∆θ − π
sin( ∆θ − θ A ) − sin(θ A ) 
+

∆θ + π
(17)
Equating Eq. 15 with Eq. 17 defines the instant at droplet
detachment.
Just as with Chen, et al., the rectangular areas used in
this derivation for the pressure force (2BD), the shear force
(D2), and the flow above the droplet (2bD) might lead to
inaccuracies. Further discussion on this topic is in Section 4.
Kumbur, et al.’s Validation
An experimental test channel was fabricated for exsitu droplet observation and measurement. The channel was
optically accessible through the top channel wall. Prisms were
installed along the channel to allow for a simultaneous side
view of the droplet. Three gas diffusion layers were tested,
each with a different PTFE content. The dynamic contact
angles, chord length/contact circle diameter (c), and height (H)
were measured with the single top/side view image. The data
generated experimentally agrees reasonably well with the
model predictions, which are reviewed in the next section.
Kumbur et al.’s Results
The experimentally measured contact angle hysteresis
data vary significantly. The ∆θ data are made into a more
useful form by fitting them with a multidimensional linear
regression. For each GDL (with different PTFE content),
contact angle hytsteresis is found as a function of three
parameters: Reynolds number, chord length normalized with
DH, and droplet height normalized with DH. These regression
fits (Eqs. 16 in [11]) are compared with predictions of the
analytical model. The effect of airflow rate on contact angle
hysteresis is studied, and it is seen that for a given droplet size
and shape (H and c held constant), the model predicts ∆θ as
linearly increasing with airflow rate. A higher ∆θ produces a
higher surface adhesion force (See Eq. 17), as long as ∆θ is less
than 90o. Thus, droplets which are more strongly adhered
require a faster flow for detachment. The drag force and the
surface adhesion force are dependent on the droplet height (H)
and chord length (c), respectively. Thus, the droplet aspect ratio
(H/c) is investigated. It is seen that taller droplets will detach at
lower Reynolds numbers, and above an aspect ratio of 1.25
droplets will detach at very low Reynolds numbers. A larger
aspect ratio causes a larger drag force, which aids in
detachment. It is also observed that at a constant droplet aspect
ratio, GDL samples with higher PTFE content (higher θS)
require lower Reynolds numbers for droplet detachment. In
addition, the model predicts that with all things held constant, a
smaller channel height is beneficial to droplet detachment.
2.3 Zhang, et al. [2]
The objective of Zhang, et al.’s work is to
quantitatively study water removal in PEMFCs, specifically
water along the GDL/channel interface. In order to accomplish
this, a transparent fuel cell is observed under different
conditions, and a simple model for droplet detachment is
developed.
Model Details
The contact length for the surface adhesion force is the
diameter of the circle created by the intersection of a sphere
and a plane meeting at the static contact angle. This is the circle
of intersection between the droplet and GDL in Fig. 3 (upon
substitution of θS for θA). The circle has a diameter of Dsin(θS).
The net surface tension force is given by:
D sin(θ S )
D sin(θ S )
− σ cos(θ A )
2
2
= 2 Dσ sin 2 (θ S ) sin( ∆θ )
Fσ = σ cos(θ R )
(18)
Each term of Eq. 18 is a component of surface tension along the
direction of the flow defined by a different dynamic contact
angle multiplied by one half of the contact length. This relation
was originally developed in [8] and [9]. Also, the dynamic
contact angles are assumed to be symmetric from the static
contact angle. A drag force correlation from [7] is used:
1
C drag ρ gas U 2 AP
2
(19)
1
D2 

θ S − sin(2θ S ) 
4 
2

(20)
Fdrag =
where:
AP =
and,
C drag =
4.62
(1 + 5.2 Re − 0.63 )
Re 0.37
(21)
and,
Re =
UD
ν gas
(22)
The projected area term in Eq. 20 is the area of the truncated
circle in Fig. 3 (when θS is substituted for θA). The drag force
(Fdrag ) is derived for a full sphere experiencing shear flow with
a uniform velocity profile. To account for this, Zhang, et al.
applies a correction factor (K1) to the drag force.
1
'
Fdrag
= K 1 C drag ρ gasU 2 AP
2
(23)
To define the system at detachment, Eq. 18 and 23 are equated:
D = K 2 C drag Re 2
(24)
The new correction factor (K2) is a conglomeration of the first
correction factor, all fluid properties, all wetting terms, and all
geometry terms. Upon substitution of Eq. 21 into Eq. 24 a
relation for droplet diameter is found in terms of a new
correction factor (K), mean air velocity and Reynolds number:
7
Copyright © 2007 by ASME
log(D) = −2.59 log(U ) + K − 1.59 log(1 + 5.2 Re −0.63 )
(25)
The use of correction factors is this model’s largest weak point.
To make use of this model, the value of K must be determined
for each combination of GDL, temperature, and channel, thus
making its usefulness limited.
Zhang, et al.’s Validation
The researchers build a fuel cell with visual access
through the channel top wall. Images of droplets before
detachment are taken and diameters are measured.
As with Kumbur, et al.’s model, the surface adhesion
force is found by integrating a component of the differential
surface tension force along the flow direction (Fig. 9):
π
Fσ = 2 ∫ σ
0
dc
cos(θ ) cos(α ) dα
2
The contact line is circle of arbitrary diameter dc. Along this
circle’s azimuthal direction, the contact angles vary linearly
from receding to advancing contact angle. An expression for
the contact angle variation is incorrectly presented as Eq. 41 in
He et al.’s original work [12]. The corrected form is presented
here:
α (θ A − θ R )
(27)
θ = θ (α ) = θ +
R
Figure 8 – Detachment Diameter vs. Air Velocity,
Zhang, et al. [2]
Zhang, et al. plot the experimental measurements against the
model’s predictions in Fig. 8. They determine a value for the
correction factor K to best fit the experimental data. With the
correction factor, a reasonable agreement is attained between
the model and the experiment.
Zhang et al.’s Results
Both the model and experimental data show the same
trend of decreasing detachment diameter with increasing air
velocity. At high flow rates, small droplets are quickly swept
away; and at lower flow rates (closer to those used in fuel
cells), detachment diameter comes close to channel dimensions,
and water tends to build up. As water builds up, different types
of two-phase flows develop (corner flow, annular flow, and
slug flow). In addition, the diameter of a droplet spreading on a
hydrophilic channel wall is related to time. A simple
polynomial is fitted to the data of diameter versus time.
2.4 He, et al. [12]
The objective of He, et al.’s work is to develop a two
dimensional, two-phase, two-fluid computational model of a
fuel cell to better understand the water transport mechanisms.
To this end, a simple analytical model is also developed which
predicts the size of a single water droplet at detachment.
Model Details
dFST
θ = θ(α)
α
dc/2
Figure 9 – Differential Surface Tension Force
(26)
π
The model by He, et al. is peculiar in that it does not include a
contact line defined by a prescribed geometry, but by the GDL
the droplets are resting on. The pores the droplets rest on are
assumed to define the effective contact line (dc = average GDL
pore diameter). This is assumed accurate until the droplet
coalesces with another, a case not considered by He, et al. The
dynamic contact angles are assumed to have a specific
relationship to each other as per work on droplets resting on an
inclined PTFE surface [10]. The contact angles are assumed to
be symmetric with each other about the static contact angle.
The static contact angle is defined by experimental studies in
[13]. The drag force is derived from a two-dimensional model:
Fdrag = ∫
π
π −θ M
2

u2
1
D
 sin 2 (α ) dα
C drag ρ gas 
2
2  sin(π − θ M ) 
(28)
where the parameters u, Cdrag, and θM are defined alongside the
drag force in He, et al.
The drag force (Eq. 28) and surface tension force (Eq.
26) are equated, defining the droplet at detachment.
He, et al.’s Validation
To test the validity of the model, its predictions are
plotted alongside data by Zhang, et al. [2]. The data falls
reasonably close to the model plot, and this is said to validate
the model.
He, et al.’s Results
The effect of GDL hydrophobicity is investigated, and
it is found that as the static contact angle increases, the
departure diameter decreases. The effect of air-water surface
tension is likewise investigated, and it is seen that as the surface
tension value decreases, the detachment diameter also
decreases.
3. OTHER DROPLET MODELS
Below is a brief summary of models which give
insight into droplet dynamics of relevance to PEM fuel cells.
3.1 Theodorakakos, et al. [1]
The purpose of Theodorakakos, et al.’s work is to
simulate detailed droplet behavior of relevance to PEMFCs.
This is done with a sophisticated numerical simulation. For
more on their techniques, see [1].
8
Copyright © 2007 by ASME
Each simulation begins with a spherical droplet,
intersecting a plane such that the contact angle is an
experimentally measured static contact angle. The simulated
channel is identical to the channel used to measure the contact
angles. The air velocity in the channel is slowly increased until
detachment, as was done in the experiments. The droplets are
seen to experience large deformations before detachment. The
numerical simulation agrees quite well with the digital
photographs (Fig. 10).
the lines plotted in Fig. 12. The unstable region is the above the
lines.
A
B
Figure 12 – Stability Plots, Theodorakakos, et al. [1]
Figure 10 – A: Digital Images of Droplets
B: Numerical Simulation of Droplets in A
Theodorakakos, et al. [1]
A detailed analysis of the simulation results shows the water-air
interface performing a vibration shortly before detachment.
This can significantly impact the contact line of the droplet and
reduce its surface adhesion force. It is also seen that the larger
of two droplets (with equal dynamic contact angles and at the
same average channel velocity) will deform less before
detachment than the smaller one. Although the flow imparts
more force onto the larger droplet due to its size, the forces
holding the droplet in place and together are much larger for the
same reason. The shape of the droplet at detachment is
observed to change from droplet to droplet. To visualize this,
droplet images are resized to the same height and compared in
Fig. 11. As can be seen, there is significant variation among the
droplet/GDL contact regions.
Figure 11 – Numerically Simulated Droplet (Gray Area)
Droplet Image Outlines (Lines)
Theodorakakos, et al. [1]
The static contact angle, contact angle hysteresis, and overall
droplet stability are seen to be related. The higher the static
contact angle, the less adhered the droplet is to the surface.
Thus, less deformation (a smaller ∆θ) is required for
detachment, and droplets detach more easily. For example, the
carbon cloth GDL sample had the highest static contact angle,
lowest contact angle hysteresis, and smallest stability region on
a plot of diameter versus flow rate. The stable region is below
The line which divides the plot into stable and unstable regions
is called the separation line. This line is studied parametrically.
Droplet position in the channel was studied. Droplets at
different positions along the width of the GDL were found to
behave similarly. Droplets contacting the other walls of the
channel as well as the GDL were studied. The droplets were
found to jump toward the more hydrophilic channel walls, and
away from the GDL, yielding a smaller stability region.
Capillary flow into the droplets, the mechanism that feeds
water from the GDL into the growing droplets, was found to
have negligible effects. An increase in temperature, which
decreases the surface tension of air and water, was found to
have a destabilizing effect on droplets. Selection of the GDL
material was found to have the largest effect on the separation
line.
3.2 Golpaygan and Ashgriz [14]
The purpose of Golpaygan and Ashgriz’s work is to
study the effect of liquid droplet properties, as well as
surrounding gas properties, on the mobility of droplets in the
channels of PEMFCs. A two-dimensional droplet is considered
in this transient numerical study. The droplet is pinned to the
GDL surface, and the contact line is not allowed to move.
When a steady state is reached, the deformation of the droplet
is used to characterize its mobility.
The effects of varying the capillary number (Ca =
µgasU/σ) are explored. Upon studying the effect of varying
surface tension values, it is seen that the droplet holds its shape
better with a larger surface tension. At a lower surface tension,
the droplet deforms more readily, and is easier to detach. The
capillary number is then studied by varying cathode gas
viscosity values. A more viscous gas creates more deformation
of the droplet before steady state is reached. The time to reach a
steady state is longer, so deformation is a slower process. At
and above a capillary number of about .15, droplet
deformation/mobility drastically increases, indicating droplet
detachment. The effect of varying surface tension is compared
with the effect of varying droplet density. With an increase in
either of these, droplets tend to deform less. However, the
effect of varying surface tension has a much larger effect than
9
Copyright © 2007 by ASME
varying droplet density. It is seen that decreasing the droplet
viscosity decreases the forces that counteract the deforming
inertia forces of the flow. Droplet viscosity is not considered in
the capillary number, and thus the capillary number is not the
only significant parameter in high inertia flows. The Reynolds
number defined in by Golpaygan and Ashgriz [14] describes
deformation more completely in this flow regime.
3.3 Palan, et al. [15], [16]
The objective of Palan, et al.’s work is to theoretically
study different types of vibration which could facilitate PEMFC
water management, and the feasibility of these methods in fuel
cell stacks. The goal of the first paper [15] is to study
atomization of water droplets on channel walls and the MEA.
The goal of the second paper [16] is to study excitation
methods which would help move droplets along with the flow,
instead of atomizing them.
Theory
In their first work, the authors use two theories to
define upper and lower bounds on acceleration levels which
atomize droplets. The first is a classical derivation which
equates the energy needed to form the droplet (viscous energy
and surface tension energy) to the vibration energy. The second
theory is called Vibration Induced Droplet Atomization (or
VIDA). The method is an experimental technique, described in
[17], [18], [19], [20], and can handle a wide range of droplet
volumes while consuming very little power.
In their second work, the authors propose three types
of waves which could potentially push droplets down a
channel. The amount of energy required by each wave to create
forces equal to droplet surface adhesion forces is studied. The
surface tension is found by considering a sphere intersecting a
plane, and is given in [21], [22]. The first type of wave is a
Flexural Plate Wave. This is the wave experienced when a plate
is excited by being struck, or shaken. Work describing these
waves, given in [19], [23], [24], [25], [26], [27] is used to find a
minimum vibrational amplitude which will induce droplet
motion. The second type is Modulated Ultrasound. This is the
wave which moves through a body of gas. It can be generated
by small speakers at the entrance and exit of a channel, each
operating at a slightly different frequency. Using formulations
by Whitworth, et al. [28], an expression for the force generated
by the wave is derived, and is used to find the necessary
vibration amplitude and acceleration. The last type of wave
proposed is a Surface Acoustic Wave [29], [30]. These are high
frequency waves which are localized at the surface of a
material. Again, the vibration parameters for droplet motion are
found.
Palan, et al.’s Results
To study energy required for droplet atomization on
channel walls, vibration amplitude and acceleration are plotted
versus droplet radius, and at different frequencies. It is seen that
larger droplets require less acceleration and amplitude to
atomize. As the frequency increases, the required amplitude
decreases significantly. This is due to the high acceleration
levels. However, extremely high acceleration levels are
required to atomize droplets which are on the scale of those in
most PEMFCs. The feasibility of implementation into a stack is
studied. ANSYS is used to simulate the vibration of a
simplified bipolar plate. When the plate is excited at its natural
frequency (with a 10 lb force), it generates extremely high
acceleration and stress values. The acceleration is capable of
atomizing many sizes of droplets, but the stress generated
would destroy the plate. Further, a small stack is simulated by
assuming the MEAs act as springs. Given the same stimulation
as the single plate, the stack experiences acceleration levels
drastically lower than the single plate, showing that structural
excitation is probably not feasible for PEMFC application.
Atomization of droplets on the MEA is studied with acoustic
excitation. Two types of acoustic excitation are chosen: A small
loud speaker pointed at an endplate (a plane wave incident on
the MEA) or a line of point sound sources along the length of
the channel. ANSYS is used to find the natural frequencies of
the channels, and COMET is used to find structural
displacements caused by the acoustic sources. Sufficient
excitation is attained with both sources of sound, though their
parasitic drains may be high.
The first type of wave proposed for droplet migration,
the Flexural Plate Wave, is analyzed. It is found that small
displacements and very high frequencies are needed. These will
generate high levels of stress, which would destroy the plates.
Modulated Ultrasound is analyzed using speakers that operate
at a frequency around 3 MHz with very small amplitudes. The
last wave proposed, the Surface Acoustic Wave, is analyzed
and found to require waves with smaller amplitudes than
Flexural Plate Waves, but larger than Modulated Ultrasound
waves. The parasitic losses required for each system comes into
question, and they are compared by how much water they can
move. The three methods are theoretically employed into a
representative fuel cell stack where they are required to move
all the product water from the system. The water is assumed to
be entirely made of 4 mm diameter droplets which all stick and
need to be removed by the vibratory method. Surface Acoustic
Waves are found to require the least energy because they use a
low frequency, even though the amplitudes are large.
4. CONCLUSIONS
Several conclusions about droplet dynamics seem to be
prevalent through all the models reviewed. Listed below, in
order of number of authors cited, are the most common results.
i) Increasing the hydrophobicity of the GDL/GFC interface
(i.e.: the static contact angle) tends to decrease droplet height at
departure. This reduces channel interactions which can result in
flooding. (Sections 2.1, 2.2, 2.4, 3.1, 3.2)
ii) Use of the design parameter developed by Chen, et al. [4]
may help in preventing droplets from clogging the channels.
L µ gasU π
[9]
>
2B σ
12
The validity of Eq. 9 is indirectly strengthened by other
authors’ conclusions. For example, Kumbur, et al. [11] came to
the conclusion that a decrease in channel height yields an ease
in droplet detachment. Golpaygan, et al. [14] came to the
conclusion that detachment would be eased with an increase in
the capillary number (Ca= µgasU/σ). (Sections 2.1, 2.2, 2.4, 3.2)
10
Copyright © 2007 by ASME
Figure 14 – Shear Stress Imposed Areas
(A: Typical, B: Suggested),
Flow Direction is Along Page
iii) Increasing the gas channel mean velocity will result in a
decrease in droplet height at departure. (Sections 2.1-2.3, 3.1)
iv) Taller droplets tend to have larger contact angle hystereses.
(Sections 2.1, 2.2)
v) GDL roughness can be neglected when considering droplets.
(Sections 2.2, 3.1)
Some suggestions are presented here which may be used to
improve the analytical models. First, given the commonly
observed droplet shape in Fig. 2, a contact line based on simply
a truncated sphere’s intersection with the GDL at a single
contact angle is not appropriate. In many instances, this
simplification can lead to an inaccurate portrayal of the contact
length and distort the surface adhesion force. A new
geometrical interpretation of the contact line is needed to
accurately portray the contact line. The rectangular
approximations for the areas that the local pressure drop and
the shear stress are imposed on also come into question. It is
believed that the use of these simple rectangular areas overapproximates the pressure and shear forces because the areas
are larger than the droplet itself. The typical imposed pressure
area, which is represented as a rectangle with a height equal to
the channel and a width equal to the droplet diameter is shown
shaded in Fig. 13-A.
A
B
To adhere more closely to the simple spherical model, the shear
force area could be replaced with πD2/4, which is illustrated in
Fig. 14-B as the gray circle. The approximations for mass
conservation between the channel far from the droplet and
around the droplet come into question, as well. The assumption
is made that there is no flow around the sides of the droplet,
and only the rectangular area above the droplet is used when
considering conservation of mass (shaded area in Fig. 15-A).
A
Figure 15 –Areas for Mass Conservation Equations
(A: Typical, B: Suggested),
Flow Direction is Into Page
An area which considers flow around the sides of the droplet is
shown shaded in Fig. 15-B. This is achieved by subtracting the
projected area (Section 2.3, Eq. 20) from the cross sectional
area of the channel.
The area considerations above are now evaluated with
an error analysis. A hypothetical square channel of side length
unity, a GDL with a static contact angle of 130o, and a range of
dimensionless droplet heights (.2, .5, .8) are considered
representative. The results are summarized in Table 3.
Imposed Area
Figure 13 – Pressure Force Imposed Areas
(A: Typical, B: Suggested),
Flow Direction is Into Page
H
To adhere more closely to a spherical model, the pressure force
area would be replaced with the imposed area from Section 2.3,
Eq. 20, which is indicated in Fig. 13-B. Typically, the area
imposed on by shear stress (gray square in Fig. 14-A) is a
square whose side length is the droplet diameter squared.
A
B
Percent Error
.2
.5
.8
∆p
500
140
50
τ
30
30
30
20
30
40
∂
∑m = 0
∂t
Table 3 – Suggested vs. Typical Areas Error Analysis
B
The Percent Error column represents the deviation of typical
areas from spherical geometry, over a range of droplet sizes.
Each typical area exhibits significant deviation from
geometrically exact areas.
-
11
In the case of the typical pressure drop area, there are
massive deviations from the exact geometry for small
droplets. As the droplet grows in size, the typical
approximation becomes better. It would be reasonable
Copyright © 2007 by ASME
to implement the suggested area as the error when
considering small droplets with the simple method is
large.
-
The typical shear stress area differs by more than
twenty five percent for all droplet sizes. Changing this
area is also feasible because its form is almost as
simple as the original.
-
For conservation of mass area consideration the error
is relatively small over most of the droplet sizes. This
amount of error might not justify the severe
complexity of including the geometrically exact
suggestion.
Future work will investigate these and more aspects of droplet
modeling in detail.
ACKNOWLEDGMENTS
This work was supported by the Thermal Analysis and
Microfluidics Laboratory of the Mechanical Engineering
Department at the Rochester Institute of Technology.
REFERENCES
1. Theodorakakos, A., Ous, T. and Gavaises, M. 2006, Dynamics of water
droplets detached from porous surfaces of relevance to PEM fuel cells. Journal
of Colloid and Interface Science, Vol. 300, pp. 673-687.
2. Zhang, F.Y., Yang, X.G. and Wang, C.Y. 2006, Liquid Water Removal
from a Polymer Electrolyte Fuel Cell. Journal of the Electrochemical Society,
Vol. 153(2), pp. A225-A232.
3. Litster, S., Sinton, D. and Djilali, N. 2006, Ex situ visualization of liquid
water transport in PEM fuel cell gas diffusion layers. Journal of Power
Sources, Vol. 154, pp. 95-105.
4. Chen, K.S., Hickner, M.A. and Noble, D.R. 2005, Simplified models for
predicting the onset of liquid water droplet instability at the gas diffusion
layer/gas flow channel interface. International Journal of Energy Research,
Vol. 29, pp. 1113-1132.
5. Bird, R.B., Stewart, W.E. and Lightfoot, E.N. Transport Phenomena. New
York : Wiley, 2002.
6. White, F.M. Fluid Mechanics, 4th ed. s.l. : McGraw-Hill, 1999. pp. 357362.
7. Clift, R., Grace, J. R. and Weber, M. E. Bubbles, Drops, and Particles.
New York : Academic Press, 1978.
8. Merte, H, Jr. and Yamali, C. 1983, Profile and Departure Size of
Condensation Drops on Vertical Surfaces. Waerme- Stoffuebertrag., Vol. 17,
pp. 171-180.
9. Shen, W. J., Kim, J. and Kim, C.-J. 2002, Controlling the Adhesion Force
for Electrostatic Actuation of Microscale Mercury Drop by Physical Surface
Modification. IEEE Conference, MEMS ’02, Las Vegas. p. 52.
10. Extrand, C.W. and Kumagai, Y. 1997, An experimental study of contact
angle hysteresis. Journal of Colloid and Interface Science, Vol. 191, pp. 378383.
11. Kumbur, E.C., Sharp, K.V. and Mench, M.M. 2006, Liquid droplet
behavior and instability in a polymer electrolyte fuel cell flow channel. Journal
of Power Sources, Vol. 161, pp. 333-345.
12. He, G.; Ming, P.; Zhao, Z.; Abdula, A.; Xiao, Y. 2007, A two-fluid model
for two-phase flow in PEMFCs. Journal of Power Sources, Vol. 163, pp. 864873.
13. Lim, C. and Wang, C.Y. 2004, Effects of hydrophobic polymer content in
GDL on power performance of a PEM fuel cell. Electrochimica Acta, Vol. 49,
pp. 4149-4156.
14. Golpaygan, A. and Ashgriz, N. 2005, Effects of oxidant fluid properties on
the mobility of water droplets in the channels of PEM fuel cells. International
Journal of Energy Research, Vol. 29, pp. 1027-1040.
15. Palan, V. and Shepard Jr., W.S. 2006, Enhanced water removal in a fuel
cell stack by droplet atomization using structural and acoustic excitation.
Journal of Power Sources, Vol. 159, pp. 1061-1070.
16. Palan, V., Shepard Jr., W.S. and Williams. 2006, Removal of excess
product water in a PEM fuel cell stack by vibrational and acoustical methods.
Journal of Power Sources, Vol. 161, pp. 1116-1125.
17. Heffington, S.N., Black, W.Z. and Glezer, A. San Diego : s.n., 2002.
Eighth Intersociety Conference on Thermal and Thermo-mechanical
phenomena in Electronic Systems. pp. 408-412.
18. James, A.J.; Vukasinovic, B.; Smith, M.K.; Glezer, A. 2003, Vibrationinduced drop atomization and bursting. Journal of Fluid Mechanics, Vol. 476,
pp. 1-28.
19. King, L.V. 1934, R. Soc. Lond. Sect. A: Math. Phys. Sci., Vol. CXLVII ,
pp. 212-241.
20. Smith, M.K. and Glezer, A. Vibration induced atomizers. US Patent
Number 6,247,525 2000.
21. Israelachvili, J. Intermolecular and Surface Forces. New York : Academic
Press, 1991.
22. Torkkeli, A., Saarilahti, J. and Haara, A. 2001, Electrostatic
transportation of water droplets on superhydrophobic surfaces. 14th IEEE
International Conference on Micro Electro Mechanical Systems (MEMS 2001),
Interlaken. pp. 475-478.
23. Alzuaga, S.; Manceau, J.F.; Ballandras, S.; Bastien, F. 2003,
Displacement of droplets on a surface using ultrasonic vibration. World
Congress on Ultrasonics, Paris. pp. 951-954.
24. Biwersi, S., Manceau, J.F. and Bastien, F. 2000, Displacement of droplets
and deformation of thin liquid layers using flexural vibrations of structures.
Influence of acoustic radiation pressure. Journal of the Acoustical Society of
America, Vol. 107, pp. 661-664.
25. Kinsler, L.E.; Frey, A.R.; Coppens, A.B.; Sanders, J.V. 2000, Energy of
Vibration. Fundamentals of Acoustics, 4th ed. New York : John Wiley & Sons,
Inc., p. 5.
26. Lyklema, J. Introduction Fundamentals of Interface and Colloid Science.
New York : Academic Press, 1993. p. 294. Vol. 1.
27. Scortesse, J., Manceau, J.F. and Bastien, F. 2002, Interaction between a
liquid layer and vibrating plates: Application to the displacement of liquid
droplets. Journal of Sound and Vibration, Vol. 254, pp. 927-938.
28. Whitworth, G., Grundy, M.A. and Coakley, W.T. 1991, Transport and
harvesting of suspended particles using modulated ultrasound. Ultrasonics,
Vol. 29, pp. 439-444.
29. Ash, E.A. and Paige, E.G.S. Rayleigh-Wave Theory and Application. New
York : Springer-Verlag, 1985.
30. Viktorov, I.A. Rayleigh and Lamb Waves: Physical Theory and
Applications. New York : Plenum Press, 1967.
12
Copyright © 2007 by ASME