Proceedings of the ASME 2022 International Mechanical Engineering Congress and Exposition
IMECE2022
October 30-November 3, 2022, Columbus, Ohio
IMECE2022- 96126
A METHOD TO ACCOUNT FOR THE EFFECTS OF THERMAL OSMOSIS IN PEM
FUEL CELLS
N. Ingarra
Doctoral student
C.J. Kobus, Ph.D.
Associate Professor
J. Maisonneuve, Ph.D.
Associate Professor
Department of Mechanical Engineering
Oakland University
Rochester, MI 48309
ABSTRACT
The objective of this research is to detail a method to account for
(TO) as a driver of net water flow in PEM fuel cell experiments
to properly determine its magnitude and direction compared to
electro-osmotic drag (EOD) and back diffusion (BD). Presently
there are two primary drivers of water transport in proton
exchange membrane (PEM) fuel cells: electro-osmotic drag
(EOD) and back diffusion (BD). These modes of water transport
are explicitly visible in the Nernst-Planck equation. Thermal
Osmosis (TO) on the other hand is a water transport mechanism
that has often been dismissed as negligible, most times without
justification. This paper details a method that does not rely on
empirical correlations of which there are many and which do not
necessarily agree with one another. To isolate the effect of each
driver, a method to decompose the net water flow into
components is developed here. This will lead to more accurate
conclusions regarding TO-driven water flow leading to better
understanding of its implications and applications. In turn, an
accurate understanding of TO will lead to better design and
operation of PEM fuel cells.
The ionic conductivity of the membrane can be expressed in
terms of the species charge, ionic mobility and concentration as
shown in Equation 2.
𝜎 = |𝑧|𝐹𝑐𝑢
Where σ is the ionic conductivity, z is the charge of the species,
u is the ion mobility, F is Faraday’s constant and c is the
concentration of species.
If Ohm’s law is used the conductivity of the membrane can be
related to the voltage gradient and current as shown in Equation
3.
𝑗 = 𝜎𝛻∅
(1)
Where Jw is the water flux, z is the charge of the species, u is the
ion mobility, F is Faraday’s constant and c is the concentration
of species, ∅ is the voltage and D is the diffusion coefficient.
The Nernst-Planck equation states that the net water flow can
only change by concentration gradient along with voltage
gradient. The first term with the voltage gradient can be
converted to electro osmotic drag (EOD). The second term is the
Fickian Diffusion.
(3)
Where J is the current, σ is the membrane conductance and φ is
the voltage.
The Electro osmotic drag is a function of current density, the
water transport from EOD is shown in Equation 4
INTRODUCTION
In PEM fuel cells the net water flow across the membrane is
modeled through the Nernst-Planck equation as shown:
𝐽𝑤 = −𝑧𝑢𝐹𝑐𝛻∅ − 𝐷𝛻𝑐
(2)
𝐽𝐸𝑂𝐷 = 𝑛𝑑
𝑗
𝐹
(4)
Where nd is the electro-osmotic drag coefficient and it represents
quantity of water molecules per hydrogen ions, j is the current
density and F is Faraday’s constant. The electric osmotic drag
coefficient is a function of the hydration state of the membrane
as well as the membrane temperature.
If the ionic conductivity relationship of Equation 2 and electro
osmotic drag in Equation 4 are combined and substituted in
Equation 1, the standard thermodynamics water flow through the
membrane can be expressed in Equation 5
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𝜎
𝐽𝑤 = −𝑛𝑑 𝛻∅ − 𝐷𝛻𝑐
𝐹
(5)
One keynote about net transport of water there is no thermal
osmosis term but according to non-equilibrium thermodynamics
thermal osmosis exists.
To understand thermal osmosis and water transport across the
membrane of the PEM fuel cell, water flow will be modeled
through non-equilibrium thermodynamics (NET). In NET, there
are three fundamental driving forces: thermal, chemical
potential, and electro-potential gradients. In addition to mass
being moved by these forces, heat and charge can also be moved
by these thermodynamic forces. The movement of heat, mass,
and charge are shown in Equations 6-8.
Jq=Lqq(∇∙(1/T) +Lqµ/T(∇∙μ) +LqΦ/T(∇∙φ)
(6)
Jw=Lqµ(∇∙(1/T) +Lµµ/T(∇∙μ) +LµΦ/T(∇∙φ)
(7)
j=LqΦ(∇∙(1/T) +LµΦ/T(∇∙μ) +LΦΦ/T(∇∙φ)
(8)
Where Jq is the heat flux, Jw is the water flux and j is the current.
The temperature is expressed in terms of T while µ is the
chemical potential and lastly the voltage is expressed as φ, where
Lij represents constants.
In the NET, the chemical potential (µ) must be decomposed into
measurable variables. Equation 9 shows how the chemical
potential is affected by the electric field, where as Equation 10
shows the chemical potential without the electric field.
μ=μo(T, P)+RTln(c)+Fzψ
(9)
μ=μo(T, P)+RTln(c)
(10)
The chemical potential is expressed as µ. The concentration of
the species is c, z is the charge of the species, and Ψ is the
voltage. The electric field adds an additional to the chemical
potential.
In NET, there are two heat and mass transfer effects that are not
accounted for in standard thermodynamics, these effects are the
Soret and Dufour Effect. The Soret effect mass transport from a
temperature gradient. The Dufour Effect is the energy transport
from the concentration gradient
The Soret and Dufour Effect occur simultaneously; one of them
cannot occur without the other. The Dufour effect depends on the
substance, on the phase of the material, and the Lewis (Le)
number. The Lewis number is the mass diffusion coefficient (D)
divided by thermal diffusivity (α).
In cases where liquid is separated by porous media, the Lewis
number is on the order of 10-2. When porous media separates
gasses, the Lewis number 102 therefore the Dufour effect in
gasses cannot be neglected but can be in liquids.
Experimentation performed on liquids would not have the Soret
and Dufour effect whereas experiments done with gasses will
account for the Soret and Dufour Effect.
The effects of thermal osmosis should be added to the existing
PEM fuel cell model and the results should be compared to
existing models. To fully understand thermal osmosis, the
mechanism behind it must be understood
NOMENCLATURE
Jw net water flow across the membrane kg/(m2s)
z charge of species
u ion mobility m2/(s V)
c concentration of species mol/m3
, φ,electro-potential V
 electro-potential V
D Fickian diffusion coefficient m2/s
 ionic conductivity S/m
j
current density A/cm2
nd electro-osmotic drag coefficient
F Faraday’s constant 96485 C/mol
Jq heat flux, W/m2
 hydration state of the membrane
Lqq Fourier law of conduction thermoconductivity W/m K
Lq Coefficient of thermal &chemical potential K kg/(s m)
Lq Coefficient of thermal & chemical potential K kg/(s m)
Lq Coefficient of thermal &voltage potential A K/m
Lq Coefficient of thermal &voltage potential A K/m
L Fickian diffusion coefficient m2/s
L thickness of membrane m
T Temperature, K
x Spatial coordinate, m
Coupling Fuel Cell Water Flow with Non-Equilibrium
Thermodynamics
Electro osmotic drag and back diffusion can be explained though
non-equilibrium thermodynamics the first mechanism of water
movement in the PEM fuel cell, electro osmotic drag (EOD), is
based upon the membrane hydration state and current density as
shown in Equation 4.
𝐽𝐸𝑂𝐷 = 𝑛𝑑
𝑗
𝐹
(4)
As in standard thermodynamics there is Fickian diffusion as
shown in Equation 5.
𝐽𝐵𝐷 = −𝐷
𝜕𝑐
𝜕𝑥
(5)
The Fickian diffusion follows Fick’s law of diffusion where the
diffusion is a function of the diffusion coefficient and the
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concentration gradient. The current density and membrane
conductance and voltage gradient can be related through Ohm’s
Law as shown in Equation 11
𝛻∅ =
𝐽
(11)
𝜎
Where J is the current, σ is the membrane conductance and φ is
the voltage. This relationship correlates EOD to the voltage
gradient term as shown in Equation 12.
𝐿𝜇𝜑▽𝜑
𝑇
𝑛𝑑 𝜎▽𝜑
=
(12)
𝐹
With an additional simplification, the unknown NET coefficient
can be expressed in terms of Nernst Planck terms as shown in
Equation 13.
𝐿𝜇𝜑
𝑇
=
𝑛𝑑 𝜎
(13)
𝐹
Therefore, EOD can also be explained through NET. The next
term that needs to be explained through NET is back diffusion.
The next mode is water transport through diffusion. Siemer[1]
developed a way to convert the chemical potential term of
Equation 14 NET to Fickian diffusion as shown in Equation 5.
𝐿𝜇𝜇 𝜕𝜇
𝑇 𝜕𝑥
=
𝐿𝜇𝜇 𝜕𝜇 𝜕𝑐
𝑇 𝜕𝑐 𝜕𝑥
=𝐷
𝜕𝑐
𝜕𝑥
(14)
As a result of this linkage, back diffusion can also be explained
through NET. The NET equations have a thermal osmosis term,
whereas current fuel cell models do not have thermal osmosis
incorporated into them. Based on the Nernst-Planck equation,
the effects of thermal osmosis are neglected.
Presently it is unknown if the effects of thermal osmosis are
small or large. Researchers inside and outside the fuel cell
industry have tried to model and explain thermal osmosis. The
thermal osmosis portion of the net water flow is shown in
Equation 15.
𝐽𝑤 = −
𝐿𝑞𝜇 𝜕𝑇
𝑇 2 𝜕𝑥
(15)
The next step will be to add thermal osmosis to the membrane
water flow to see how this new mode of water transport affects
the heat and water management of the PEM fuel cell.
Adding thermal osmosis to the Nernst-Planck equation will
make the Nernst-Planck equation the same as the NET equations.
INCONSISTENCIES IN THERMAL OSMOSIS
Researchers have attempted to understand thermal osmosis, they
have either done it through modeling and/or experimentation.
Kim and Mench[2], Villalienga[3], Tasaka[4], Zaffou[5],
Bradaen[6] Fu[7], Thomas[8] and Perry[9] have done thermal
osmosis experimentations. Each researcher had gotten different
results.
Villaluenga[3], Fu[7]and Tasaka[4] had gotten thermal osmosis
water flow to go in the direction of cold to hot and Zaffou[5],
Braden[6] and Perry[9] had gotten thermal osmosis flow to go in
the direction of hot to cold. The thermal osmosis coefficient
ranges from. 7.5E-6 kg/(m2s °C) through 4.1E-4 kg/(m2s °C).
Bradaen[6]had done transient thermal osmosis experiments and
Perry had performed experiments under sub-freezing conditions.
There were several inconsistencies with each experiment.
Villaluenga[3]had done experiments with methanol (CH3OH)
and water, Kim and Mench[2], Tasaka[4] had done their
experiments with liquid water. Zaffou[5]and Thomas[8] had
done their experiment with 100% humidified reactant.
Empirical thermal osmosis models were developed and some of
these models were used in PEM fuel cell models.
Fu[7], Dickenson[10]has concluded that thermal osmosis could
be neglected. Goshtasbi[11],had mentioned that there is
uncertainty with thermal osmosis data
Fu had done experimentation to look at thermal osmosis, based
on Fu experimental data they had believed that thermal osmosis
can be neglected but there were issues with the experimental data
which could prove that thermal osmosis is not neglectable but a
substantial mover of fluid.
RESULTS
Fu had done an experiment on a non-operation fuel cell where
the hot side and cold side were at a fixed temperature difference
and variable Relative Humidity (RH) across the membrane but
with the cold side RH fixed at 100%. The hot side humidity
varied from 20% to 80%. To separate the Fickian diffusion, Fu
assumed Motupully’s [12]diffusion model. By assuming a
diffusion model, it makes the thermal osmosis diffusion
dependent on the Fickian diffusion.
Fu’s data was analyzed and decomposed. The data was
decomposed through the following method
Thermal osmosis has been examined inside and outside the fuel
cell industry.
Studies outside the fuel cell industry focus on the heat and mass
transfer end of thermal osmosis, and studies inside the fuel cell
industry focus on heat, mass, and charge transfer.
𝐽 = 𝑎0 + 𝑎1
𝐽=
3
𝜌
𝐸𝑊
𝐷𝜆
𝑑𝜆
𝑑𝑥
𝑑𝜆
𝑑𝑥
𝑑𝜆
+ 𝑎2 ( )2
𝑑𝑥
(16)
(17)
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The next step involves taking the derivative of both function in
𝑑𝜆
respect to ( )
𝑑𝑥
𝜌
𝐸𝑊
𝑑𝜆
𝐷𝜆 = 𝑎1 + 2𝑎2 ( )
𝑑𝑥
(18)
The only two modes of water transport through the experiment
are Fickian diffusion and thermal osmosis. Since the relative
humidity is varied and the thermal osmosis would be constant,
then the portion of the net water flow that is in the ao constant is
the thermal osmosis water flow. In addition, the Fickian
diffusion coefficient is also determined.
coefficient is 2.34E-5 kg/(m2 s °C) or(6.1163E-9 kg/(m s °C)
𝑑𝑇
pending if △ 𝑇 or is used.
𝑑𝑥
To quantify if thermal osmosis can be neglected, the thermal
osmosis flow was compared to the back diffusion flow in three
of the cases. As shown in the figure below the thermal osmosis
is almost equal to the back diffusion. Based on this case study,
thermal osmosis should not be neglected, it should be examined
further.
The thermal osmosis experimental data was processed different,
the net water flow was originally plotted against relative
humidity, the first step in the post processing involved plotting
the net water flow against the concentration gradient (dλ/dx))
Figure 2- Fu Net Water Compared to BD and TO
Experimentation in thermal osmosis needs to be explored further
to see how it can be applied to PEM fuel cells and other
applications where there is a temperature difference.
Figure 1- Fu Net Water Flow Experimental Results
As a result of solving for the diffusion coefficient the obtained
experimental value is 7.36E-10m2/s which is in expected range
of Fickian Diffusion models. If only BD was present the constant
in the curve fitting (ao) would be zero, but that is not the case.
Since there is a temperature difference across the membrane,
thermal osmosis and back diffusion are the only two driving
forces present in Fu’s experiment. When the data is fit in terms
of the concentration gradient (dλ/dx), a constant appears in the
curve fitting. Since thermal osmosis is the other driving force, it
would be constant in the curve fitted equation.
The sign in front of the constant will give the direction of the
thermal osmosis water flow. Since the sign in front of the
constant is negative it is going against the Fickian diffusion.
This means that the thermal osmosis water flow is going in the
direction of hot to cold not cold to hot.
The next step of the analysis involves computing the thermal
osmosis coefficient, the thermal osmosis water flow is divided
by the temperature gradient. The estimated thermal osmosis
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