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Eruptive water transport in PEMFC: A single-dropcapillary model

G. Flipo a, C. Josset a, G. Giacoppo b, G. Squadrito b, B. Auvity a,*,J. Bellettre a

a Polytech Nantes, Laboratoire de Thermocin�etique de Nantes (CNRS UMR 6607), Nantes, Franceb Istituto di Tecnologie Avanzate per l'Energia (CNR-ITAE), Messina, Italy

a r t i c l e i n f o

Article history:

Received 14 March 2015

Received in revised form

5 June 2015

Accepted 16 June 2015

Available online xxx

Keywords:

PEM fuel cell

Eruptive water transport

Capillary model

GDL

* Corresponding author.E-mail address: bruno.auvity@univ-nante

http://dx.doi.org/10.1016/j.ijhydene.2015.06.00360-3199/Copyright © 2015, Hydrogen Ener

Please cite this article in press as: Flipo G, etof Hydrogen Energy (2015), http://dx.doi.org

a b s t r a c t

In this article, the liquid water eruptive transport occurring during the water breakthrough

from the Gas Diffusion Layer (GDL) of a PEMFC into the gas channel is investigated. A

dedicated experimental set-up is used enabling the simultaneous measurement of pres-

sure inside the water system and visualization of droplet formation. A single-drop capillary

model is proposed to explain the eruptive nature of droplet formation. The model is built

on four physical equations involving Laplace's law, a water system compressibility law, the

pressure losses occurring during droplet formation, and a spherical droplet geometrical

function. Two numerical resolution methods are implemented and compared. For this

model, three parameters have to be identified from the experimental data. The identifi-

cation procedure is explained in the paper and the comparison with experimental results

shows the ability of the present model to reproduce the eruptive nature of liquid water

breakthrough from the GDL.

Copyright © 2015, Hydrogen Energy Publications, LLC. Published by Elsevier Ltd. All rights

reserved.

Introduction

A fuel cell coupled with renewable hydrogen production is a

clean power generation system. However, to date, the exces-

sive cost of both hydrogen and fuel cell systems has prevented

their large-scale commercialization. Besides, for PEM fuel

cells, water management is still a crucial issue to optimize the

performance of the cell and ensure performance stability over

time. On one hand, liquid water is needed for the transport of

protons through the membrane and, on the other hand, the

water produced by the electrochemical reaction at the cata-

lytic sitesmust be removed to allow access to the reaction gas.

s.fr (B. Auvity).82gy Publications, LLC. Publ

al., Eruptive water transp/10.1016/j.ijhydene.2015.0

Water management is thus a fine balance between drying and

flooding modes, both leading to a significant decrease in per-

formance [1].

Water management studies focusing on transport phe-

nomena in the gas diffusion layer (GDL) have highlighted the

existence of preferential pathways [2,3] for liquid water

through the GDL, which lead to a finite number of water

breakthrough locations. Preferential pathways are caused by a

non-homogeneous pore size distribution [4,5] and/or wetta-

bility heterogeneity within the GDL [6]. Different invasion

patterns are observed according to the wettability of the GDL

[7], the hydrophobic character of which has been proved to be

beneficial [8,9].

ished by Elsevier Ltd. All rights reserved.

ort in PEMFC: A single-drop capillary model, International Journal6.082

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n en e r g y x x x ( 2 0 1 5 ) 1e92

One particular feature of the liquid water transport is the

non-continuity of the flow. This is true both inside the GDL

and during the breakthrough into the gas channel. Manke

et al. [10] have shown that, despite a constant current in the

fuel cell and thus a constant water production, the quantity

of water that breaks through the GDL and arrives in the gas

channel is not constant. This phenomenon, named “eruptive

transport” [10], has been reported in both ex situ experi-

mental cell conditions [3,11,12] and in situ investigations [10].

Different methods have been used to highlight this feature of

the flow: pressure temporal evolution [3,11], image process-

ing [12] and X-ray radiography [10]. In fields other than

PEMFC, this fluid displacement front in porous media is

called “Haines Jumps” [13,14]. It results from a pore-scale

interfacial jump and associated pressure fluctuation. The

pressure increase is caused by water pushed into the system

without interface movements, and will hereafter be called

“system compressibility”. In their study, Gurau et al. [15]

present a physical and mathematical model solved with a

CFDmethod. This numerical approach is used to describe the

multiphase phenomena at the cathode GDL/channel inter-

face and explain the mechanisms that trigger the eruptive

water ejection. The local saturation inside the porous GDL

changes over time, depending on the location and stage of

droplet formation.

To obtain a better physical insight into this unstable

transport phenomenon, previous studies have often consid-

ered media with a less complex structure than industrial or

geological porous media and have focused on one capillary

[16e18]. In the present study, we propose a one-drop capillary

model in order to explain the water eruptive transport phe-

nomenon. A previous experimental study [19] was carried out

on the influence of the gas distribution channel wettability on

liquid droplet removal. This showed that the gas channel has

a significant effect on the water flow pattern. Because of films

and residual droplets, the experimental repeatability is diffi-

cult [20,21] and there is day-to-day fluctuation. In order to

model the eruptive water transport, it was decided to remove

the channel and to study the emergence of liquid water

droplets from a single capillary pore opening freely into air.

This configuration, with liquid water injected in one nee-

dle, is relevant because of the finite number of breakthrough

locations in a GDL. There is no proof that water is produced in

liquid form at catalytic sites but it is known that liquid water

breaks through the GDL to reach the gas distribution channel

Fig. 1 e Experim

Please cite this article in press as: Flipo G, et al., Eruptive water transpof Hydrogen Energy (2015), http://dx.doi.org/10.1016/j.ijhydene.2015.0

[22e25]. Some research studies have considered water trans-

port only in the liquid phase [25]. Other recent studies [26,27]

have shown that condensation can be caused by a thermal

gradient across the porous layer.

The motivation to build a model for eruptive water trans-

port is also driven by the need to improve current pore

network modeling. The invasion percolation mechanism is

satisfactorily taken into account in the porous media. There-

fore, the pore network simulations reproduce well the

increasing saturation of the GDL when the liquid water pres-

sure increases at the interface with the catalyst layer [7,28,29].

However, to reproduce the withdrawal of liquid water from

the GDL at the interface with the gas channel, models of

eruptive ejection dynamics are needed. Suchmodels will then

be implemented as boundary conditions in pore network

simulations.

In the present article, we present our experimental results

on emerging water droplet formation obtained in ex situ con-

ditions and propose a capillary model.

Droplet formation and detachment

Experimental set-up

A diagram of the experimental set-up is shown in Fig. 1. It

was composed of a needle connected to a syringe pump

(Legato 110, KD Scientific). The needle was mounted

perpendicular to the ground. Experiments with needles of

various internal diameters (200 mm, 330 mm, 410 mm ± 20 mm)

were carried out. For a better visualization of the water

droplet, a fluorescent dye was added to distilled water and an

ultraviolet lamp was used. Gas bubbles were carefully

removed from the system. Semi-rigid pipes, which can

expand with pressure, were used.

The water pressure in the system was measured using a

pressure transducer (differential pressure sensor,

EndressþHauser, Deltabar S) and the liquid droplet formation

was visualized using a CCD camera (Prosilica GE, Allied Vision

Technologies). The control of the syringe pump and the data

acquisition of the experimental set-up were developed in a

Labview 11.0 (National Instrument) environment. The camera

and the pressure sensor were connected to the data acquisi-

tion card (USB Series 43e63) to obtain images synchronized

ental set-up.

ort in PEMFC: A single-drop capillary model, International Journal6.082

Fig. 3 e Water pressure signal and droplet radius vs. time.

i n t e rn a t i o n a l j o u rn a l o f h y d r o g e n en e r g y x x x ( 2 0 1 5 ) 1e9 3

with pressure signals. The pixel resolution of the camera was

200 � 360 with a frame rate of 10 Hz.

The droplet detachment is controlled by Tate's law as fol-

lows. When the weight of the droplet becomes more than the

capillary forces, the droplet falls.

Vdroprg ¼ gkRdrop (1)

where Vdrop is the droplet volume, r is the water density, g is

gravity, g is the surface tension, k is a constant, and Rdrop is the

droplet radius of curvature.

In a fuel cell, thewater flow rate,V·

H2O, is proportional to the

current density:

V·

H2O ¼ ð1þ 2aÞjSact

2F�MH2O

rH2O

(2)

where a is the water flux, j is the current density, MH2O is the

molar mass of water, F ¼ 96485 C/mol is the Faraday constant,

rH2O is the water density and Sact is the active area. To mimic

the water flow rate encountered in a PEMFC application, it is

supposed that the water produced by an active area of

Sact ¼ 0.6 cm2 (2 mm � 30 mm) exits through only one pore of

the GDL. In this study, we have chosen to take into account

only thewater produced during the reaction, therefore a¼ 0 in

equation (2). For a current density between 0.4 and 3 A/cm2,

the water flow rate V·

H2O is between 1.2 and 10.5 mL/min. This

water flow rate range is investigated in the present paper.

Experimental results

Droplet images were recorded and post-processed with an

image binarization technique (Fig. 1) to determine the droplet

radius. Fig. 1 shows one image of a droplet with the corre-

sponding circles approximating the droplet shape. From this

circle, the droplet radius is determined.

The droplet growth can be decomposed into three stages

described below and illustrated by a diagram in Fig. 2. The

phenomenon is periodical and very stable over time. Fig. 3

Fig. 2 e Stages of the droplet formation.

Please cite this article in press as: Flipo G, et al., Eruptive water transpof Hydrogen Energy (2015), http://dx.doi.org/10.1016/j.ijhydene.2015.0

presents the temporal evolution of water pressure in the

system along with that of the post-processed droplet radius

for a water flow rate of 8 mL/min.

Stage 1: The droplet radius of curvature corresponding to

the liquid meniscus radius of curvature is greater than the

capillary radius (Rdrop1>Rt). As water is pushed into the system

and no droplet appears, pressure increases. The droplet radius

diminishes to reach Rt at the end of stage 1. This phase cor-

responds to the capillary energy storage in the GDL repre-

sented by the well-known pressure/saturation curve. In our

case, this is a mechanical elasticity energy storage or a pres-

sure energy storage. This phenomenon will be called “system

compressibility” in the present paper. In stage 1, the liquid

meniscus is partially obstructed by the needle's solid walls

thus the droplet radius of curvature cannot be determined

experimentally. It is measured only when it becomes greater

than the capillary radius.

Stage 2: This corresponds to the exact time when the

droplet radius is equal to the inner radius of the needle,

Rdrop2 ¼ Rt. At this time, the maximum pressure is reached.

Stage 3: The water pressure decreases and the droplet

radius of curvature increases, which is classically expressed

by Laplace's law, see equation (3), Section 3.1. When the

droplet becomes too heavy compared to the capillary forces, it

detaches.

Although the experimental set-up is composed of one

needle, the pressure signals have the same dynamics as those

observed by Liu et al. [30] in their experiments on water

transport through the GDL. The droplet radius obtained from

image post-processing gives the droplet volume while its

temporal evolution enables the water flow rate in the droplet

to be determined. Fig. 4 shows the volume flow rate injected

into the system at a constant flow rate, V·

syst, and the instan-

taneous volume flow rate in the droplet during stage 3, V·

drop,

for three injection conditions, 6 mL/min, 8 mL/min and 10 mL/

min. It can be clearly observed that although liquid water is

injected constantly into the system, the water flow rate in the

droplet is not constant and suddenly increases. This shows

that the droplet is forming in an eruptive way. The water flow

rate inside the droplet is driven by the pressure difference

between the system and the droplet. During stage 3, the

ort in PEMFC: A single-drop capillary model, International Journal6.082

Fig. 5 e Water flow rate in the droplet for three water flow

rates injected into the system.

Fig. 4 e Injected flow rates/flow rate inside the droplet vs.

time.

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n en e r g y x x x ( 2 0 1 5 ) 1e94

pressure inside the droplet is lower than the pressure inside

the needle.

Fig. 5 shows the temporal evolution of the water flow rate

in the droplet for three injection flow rates with the time set to

0 at the beginning of stage 3. Regardless of the flow rate

injected into the system, the water flow rate injected into the

droplet is the same. The droplet growth dynamics, stage 3, are

fairly independent of the system compressibility phase, stage

1. The droplet flow rate decreases at the end of the droplet

formation because the pressure between the system and the

droplet stabilizes.

Capillary model

As previously mentioned, Laplace's law, the system

compressibility and the droplet geometry are important pa-

rameters in the droplet formation and detachment. The

pressure losses occurring during stage 3 also have to be taken

into account. In the following, the variables are functions of

time. R·and R

··are the temporal derivative and the second

temporal derivative, respectively, of the radius of curvature.

Fig. 6 is a diagram of the two elements considered in the

model: the water system, composed of the syringe, con-

necting tubes, the pressure sensor and the needle, and the

droplet.

Equations

The pressure difference between wetting and non-wetting

fluid is given by Laplace's law, which links capillary pressure

with the radius of curvature of the droplet:

Pdrop � Patm ¼ 2gR0P

·

drop ¼ � 2g

R2drop

R·

drop (3)

where Pdrop is the pressure inside the droplet, Patm is the at-

mospheric pressure surrounding the droplet, g is the surface

tension and Rdrop is the droplet radius of curvature.

Please cite this article in press as: Flipo G, et al., Eruptive water transpof Hydrogen Energy (2015), http://dx.doi.org/10.1016/j.ijhydene.2015.0

A compressibility law [31] is introduced:

DP ¼ ADV0P·

syst ¼ A�V·

syst � V·

drop

�(4a)

where Psyst is the pressure in the system, V·

syst is the water flow

rate injected into the system and V·

drop is the water flow rate in

the droplet. The compressibility coefficient A is system-

dependent. It has to be determined from experimental data

obtained during stage 1 of the droplet formation, as explained

in Section 3.3.1. As in stage 1, the droplet flow rate can be

neglected and the following equation used to determine the

coefficient A:

A ¼ P·

syst

V·

syst

(4b)

In stage 3, the flow from the system to the droplet is

generated by the pressure difference. In fact, even if V·

syst is

small, V·

drop can be greater because of water accumulated in

the system (see Fig. 4). When the droplet is forming, pressure

losses are present and they depend on the flow rate entering

the droplet. Neglecting pressure losses in themodel could lead

to nonphysical results. Pressure losses due to droplet forma-

tion are given by the following relation:

Psyst ¼ Pdrop þ V·

dropK (5)

where Psyst is the pressure in the system, Pdrop is the pressure

inside the droplet and V·

drop is the water flow rate inside the

droplet. The pressure loss coefficient K depends on the water

flow rate inside the droplet.

During the droplet formation, the droplet shape is first a

spherical cap in stage 1, Rdrop1, and then the inverse of a

spherical cap in stage 3, Rdrop3. The changes in the droplet

volumes are calculated using the geometrical function of Rdrop

, called fðRdropÞ and defined as follows:

V·

drop ¼ f· �Rdrop

�R·

drop (6)

ort in PEMFC: A single-drop capillary model, International Journal6.082

Fig. 6 e Diagram of the system.

where f�Rdrop

� ¼8>><>>:

p

3

�Rdrop �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2drop � R2

t

q �2�2Rdrop þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2drop � R2

t

q �stage 1

43pR3

drop �p

3

�Rdrop �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2drop � R2

t

q �2�2Rdrop þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiR2drop � R2

t

q �stage 3

i n t e rn a t i o n a l j o u rn a l o f h y d r o g e n en e r g y x x x ( 2 0 1 5 ) 1e9 5

Equations (3), (4) and (6) are used in equation (5) to obtain

the final equation of the model:

Kf·

ðRÞR··þ�Kf

··

ðRÞR·þ�Aþ K

· �f·

ðRÞ � 2gR2

�R·�AV

·

syst ¼ 0 (7)

Numerical method

The choice of the initial condition Rdropð0Þ needed to integrate

this equation is explained in Section 3.3.1.

In stage 1, when the droplet is a spherical cap shape,

pressure losses are negligible. Equation (7) can be simplified as

follows and is easily solved:

R·¼ AV

·

syst

Af·

ðRÞ � 2gR2

(8)

When the droplet radius is equal to the capillary radius

Rdrop ¼ Rt, equation (7) is not resolvable because of

discontinuity.

In stage 3, equation (7) is a non-linear equation and there is

no general method to solve it. In the present study, two

methods are used to solve this equation: a fixed point method

and, due to simplifications, the solution of Riccati's equation.

Fixed point methodThe classic fixed point method is used as it is iterative, robust

and well adapted for non-linear equations, see Ref. [32]. The

time step is fixed at 0.01 s. This value is fine enough to ensure

Please cite this article in press as: Flipo G, et al., Eruptive water transpof Hydrogen Energy (2015), http://dx.doi.org/10.1016/j.ijhydene.2015.0

satisfactory convergence of the resolution. The chosen values

for the residues are ε1 and ε2 defined as:

jRðt� 1Þ � RðtÞj< ε1 and���R· ðt� 1Þ � R

·ðtÞ

���< ε2 with ε1 ¼ ε2

¼ 5:10�5½m�:

Riccati's equationR and R

·are taken as the value obtained at the previous time

step and f·ðRÞ and f

··ðRÞ become constant for each step. Y ¼ R

·is

taken to obtain Riccati's equation as follows:

Y·¼ C1Y þ C2Y

2 þ C3 (9)

where C1 ¼ 2g

R2Kf·ðRÞ

� AþK·

K , C2 ¼ �f··ðRÞ

f·ðRÞ

and C3 ¼ AV·1

Kf·ðRÞ.

One of the particular solutions of equation (7) is:

Yp ¼�Cþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC21 � 4C2C3

q2C2

(10)

The homogeneous solution of Riccati's differential equa-

tion is given by:

Y ¼ 1

e�ðC1þ2C2YpÞ � C2C1þ2C2Yp

þ Yp (11)

R·is given by Riccati's solution and R is obtained thanks to

Euler's approximation.

The numerical results obtained with both the fixed point

method and Riccati's method has an eruptive trend. Indeed,

water is pushed linearly into the system but the droplet is

ort in PEMFC: A single-drop capillary model, International Journal6.082

Table 1 e Compressibility coefficient for three needleradii.

Needleradius [mm]

Rdrop(0) [mm] A Precision (%)

0.1 0.11 11 3.1

0.16 0.18 14.5 3.1

0.2 0.24 13.7 2.9

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n en e r g y x x x ( 2 0 1 5 ) 1e96

formed suddenly. Both numerical methods yield the same

result. In the following, Riccati's method is used.

Parameter identification

Initial droplet radiusThe initial radius of curvature Rdropð0Þ is important because it

governs the time during which the system is compressed, i.e.

the duration of stage 1. This parameter is unique and very

difficult to measure experimentally. The initial radius of cur-

vature is thus determined numerically by the integration of

equation (8) from the initial radius of curvature Rdropð0Þ to the

capillary radius Rt. During stage 1, the droplet growth is

negligible, which implies that f·ðRÞ is almost constant. Thus,

f·ðRÞ may be considered null, as well as f

··ðRÞ. Equation (8) be-

comes Rdrop ¼ �AV·

systR2

2g .

Table 1 shows the value of Rdropð0Þ for the three different

needle radii used in this study. As expected, Rdropð0Þ is higher

than Rt due to the needle wall surface wettability, see the di-

agram in Fig. 2.

Coefficient of compressibilityAs explained in Section 2.2, in stage 1, no droplet appears so

the water flow rate V·

drop in the droplet is negligible. Thus, the

water pressure change is a linear function of water flow rate,

which is time-dependent, equation (4a). The compressibility

coefficient A is identified by a linear regression, see Fig. 7. The

parameter A is fixed for a given experimental set-up, i.e.

piping systemþ needle. In fact, for three water flow rates V·

syst

between 6 and 10 mL/min, the compressibility coefficient is

Fig. 7 e Determination of the coefficient of compressibility

A.

Please cite this article in press as: Flipo G, et al., Eruptive water transpof Hydrogen Energy (2015), http://dx.doi.org/10.1016/j.ijhydene.2015.0

shown to be constant with a precision of ±3%. This is within

the experimental error. The compressibility coefficient is thus

an intrinsic characteristic of our experimental set-up. There-

fore, if the needle is changed, the system compressibility may

also change, see Table 1.

Coefficient of pressure lossesThe coefficient of pressure losses K needs to be adjusted for

our experimental configuration. The pressure losses are

mainly due to friction inside the needle. The Reynolds num-

ber, calculated from the flow rate in the droplet V·

drop and the

capillary radius, lies between 43 and 135. By analogy with a

straight tube in a laminar and established regime, K is initially

supposed constant. Preliminary calculations conductedwith a

constant value for K provide an order of magnitude for K

values. To obtain Fig. 8, the radius of the needle is 0.1 mm and

the water flow rate is 8 mL/min. The droplet formation time

and the dynamics of the formation are driven by the value of

K. A high value of K is needed at the beginning of the droplet

formation to follow the droplet growth dynamics. A low value

of K is needed at the end of the droplet formation to respect

the growth time. It is thus concluded that a constant value for

K is not satisfactory. However, as the flow is not established in

time and space inside the needle, there are no theoretical

solutions for the pressure loss coefficient K.

Therefore, a phenomenological law is chosen for the co-

efficient K as follows:

K ¼ c

V· n

drop

þ d (12)

where K is a function of the water flow rate inside the droplet

V·

drop and is composed by three parameters (c, d, n), which

enable the best value of pressure losses to be found. K has a

high value Kini for a low water flow rate and a low value for a

high water flow rate, i.e. at the end of the droplet formation.

This configuration of K (12) ensures that it will not be less than

zero because, for physical reasons, the lowest value of K

cannot be negative. To determine the parameters (c, d, n) that

Fig. 8 e Simulated growth of the droplet radius for two

values of K taken as constant. Comparison with the

experimental results.

ort in PEMFC: A single-drop capillary model, International Journal6.082

Fig. 9 e Contour lines of the error between experimental

and numerical results. (a) on a (d, n) map with

c ¼ 1100000000, (b) on a (n, c) map with d ¼ 100, (c) on a (d,

c) map with n ¼ 4.

i n t e rn a t i o n a l j o u rn a l o f h y d r o g e n en e r g y x x x ( 2 0 1 5 ) 1e9 7

Please cite this article in press as: Flipo G, et al., Eruptive water transpof Hydrogen Energy (2015), http://dx.doi.org/10.1016/j.ijhydene.2015.0

correspond to the best fit with the experimental results, the

minimum error between the experimental and the numerical

radius is calculated as follows (Fig. 8):

Error ¼ 1

nb

Xstage 3

��Rexp � Rnum

�� (13)

where nb is the number of numerical steps during stage 3.

With this definition, if the time for the numerical droplet

growth is not the same as the experimental one, the error

increases.

Fig. 9 shows the error for a test with a capillary radius of

0.1 mm and a water flow rate of 8 mL/min. Three contour lines

of error are plotted for different values of couples (d, n), (n, c)

and (d, c) and a value kept constant. It can be seen that the

error is minimal for c ¼ 1100000000, d ¼ 100 and n¼ 4. In Fig. 9

(a), the contour lines remain constant regardless of the value

of d while in Fig. 9 (b) the contour lines remain constant

regardless of the value of c. The values of c and d are not very

sensitive to the choice of the value of K. As the sensitivity of

the numerical model is weak, several pairs of values may be

taken with a close error.

Tests with different water flow rates and a similar capillary

radius, Rt ¼ 0.1 mm, are presented in Fig. 10. The same law for

K, c ¼ 1100000000, d ¼ 100 and n ¼ 4, is used for these three

tests with different water flow rates injected into the system.

It can be observed that the model reproduces the experiment

well. The duration of stage 1 is shortest in the test inwhich the

water flow rate is greatest. However, the duration of stage 3 is

fairly constant, see Fig. 5. As the injected water flow rate in-

creases from 6 ml/min to 10 ml/min, the duration of stage 1

decreases from 56 s to 22 s while for stage 3, the duration

change is hardly measurable, as seen in Fig. 5.

In Fig. 11, the water flow rate is kept constant (8 mL/min)

and different capillary radii are tested. The law for K is

different for each capillary radius. The (c, d, n) triplet yielding

the minimum error is retained for each case. For a capillary

radius of 0.1 mm, 0.16 mm, or 0.2 mm, the triplet values are

(1100000000, 100, 4), (9100000000, 100, 5), or (100000000, 100, 5),

respectively. The value obtained for d is the same for the three

tests, which can be explained by the low sensitivity. As before,

Fig. 10 e Experiment vs. model. Different water flow rates

for the same capillary radius.

ort in PEMFC: A single-drop capillary model, International Journal6.082

Fig. 11 e Droplet growth obtained for identical volume flow

rates injected into the system with three different capillary

radii: model vs. experiments.

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n en e r g y x x x ( 2 0 1 5 ) 1e98

experimental and numerical results fit well. The droplet

growth is slowest for the largest needle radius. As the radius of

the capillary increases, the time needed to reach it, i.e. stage 2,

becomes longer. Indeed, from the initial radius of curvature

Rdropð0Þ to the capillary radius Rt, the volume needed to form

the droplet is greater for a larger needle radius. Thus, for the

samewater flow rate, the time to reach stage 2 is longer with a

larger capillary radius.

Conclusion and prospects

A simple, stable and predictable experimental apparatus has

been developed to study eruptive water droplet evolution.

Looking at previous experimental observations [14] and at the

existing literature, the apparatus enables the acquisition of

synchronized images with water pressure inside the system.

The eruptive water transport has been experimentally

observed in a configuration with one droplet that detaches by

weight. The physical phenomena taken into account in the

numerical model are modeled using Laplace's law, a system

compressibility law, pressure losses and droplet geometrical

functions. Two numerical methods have been implemented

to solve the resulting non-linear equation and they were

shown to yield identical results. The experimental and nu-

merical results fit well, confirming both the resolution process

and the physical meaning of the developed model.

The presented model may thus be considered for imple-

mentation in a pore network simulation as boundary condi-

tions. As a first step toward this goal, an extension of the

present model in a dual pore network is envisaged. Actually,

as experimentally evidenced by Manke et al. [10] in in situ

conditions and Bazylak et al. [33] in ex situ conditions, liquid

water may exit alternately from one pore and a neighboring

one. The extension of the present model would then repro-

duce the oscillations of the water outlet from a GDL.

Please cite this article in press as: Flipo G, et al., Eruptive water transpof Hydrogen Energy (2015), http://dx.doi.org/10.1016/j.ijhydene.2015.0

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