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 e n e r g y 4 6 ( 2 0 2 1 ) 3 3 2 1 8 e3 3 2 4 0
Available online at www.sciencedirect.com
ScienceDirect
journal homepage: www.elsevier.com/locate/he
Optimal operating conditions of PEM fuel cells in
commercial aircraft
€der a,*, F. Becker a, J. Kallo b, C. Gentner a
M. Schro
a
German Aerospace Center (DLR), Institute of Engineering Thermodynamics, Hein-Saß-Weg 22, 21129, Hamburg,
Germany
b
German Aerospace Center (DLR), Institute of Engineering Thermodynamics, Pfaffenwaldring 38-40, 70569,
Stuttgart, Germany
highlights
Novel
graphical abstract
computationally
efficient
model for PEM fuel cell water
management.
System-level
performance
is
linked to water management effects at cell-level.
Stack operating conditions (pressure, stoichiometric ratio, humidity) are optimized.
Optimal
operating
conditions
depend on flight phase and system
sizing.
The trade-off between system efficiency
and
system
mass
is
explored.
article info
abstract
Article history:
This work investigates the integration of polymer electrolyte membrane fuel cells (PEMFC)
Received 19 April 2021
into recently proposed hydrogen aircraft concepts. Based on a numerical optimization of the
Received in revised form
stack's operating conditions, the interrelated aspects of efficiency and system mass are
8 July 2021
explored. A novel 1D two-phase PEMFC stack model is developed that captures water
Accepted 14 July 2021
management effects in detail and yet features sufficiently low computational cost to be used
Available online 19 August 2021
for system-level optimization. The stack model is validated for a wide range of temperatures, current densities and oxygen concentrations. In combination with auxiliary compo-
Keywords:
nent models, it can relate the effects of the investigated parameters on cell-level water
Fuel cells
management to system-level effects. This allows for an improved understanding of the
Commercial aircraft
underlying design tradeoffs, particularly regarding pressurized operation and stack over-
Water management
sizing. The results show that the conditions that maximize the overall system efficiency for
Optimization
a given flight phase deviate significantly from those that merely maximize stack efficiency.
Auxiliary power
© 2021 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.
System efficiency
* Corresponding author.
€ der).
E-mail address: matthias.schroeder@dlr.de (M. Schro
https://doi.org/10.1016/j.ijhydene.2021.07.099
0360-3199/© 2021 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.
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 e n e r g y 4 6 ( 2 0 2 1 ) 3 3 2 1 8 e3 3 2 4 0
Nomenclature
Symbol
A
ccm
cdi
cf
cp ;
Ci;j
D
Erev
F
h
i
J
k
krel
Mi
Ma
mi
mi;sp
m_ i;j
ndrag
nk
ni;j
n_i;j
N
p
P
Q1 , Q2
R
Rsp
RHi
q
Q_
s
b
DS
active cell area ðm2 Þ
factor in conformal mapping approach ð Þ
coefficients in pressure drop correlation, i ¼ 1…2,
Eq. (49) ( )
scaling factor for average saturation, Eq. (69), ð Þ
cv specific heat capacities ðJ kg1 K1 Þ
molar concentration of species i in control volume
j ðmol m3 Þ
diffusion coefficient ðm2 s1 Þ
reversible cell voltage ðVÞ
Faraday's constant ðAs mol1 Þ
enthalpy ðJ kg1 Þ
current density ðA m2 , A cm2 Þ
dimensionless function in Leverett approach ð Þ
GDL intrinsic permeability ðm2 Þ
relative permeability ð Þ
molar mass of species i ðkg mol1 Þ
Mach number ð Þ
mass of component i (kg)
specific component mass, units depend on
context as defined in Table 7
mass flow of species i into/out of control volume j
ðmol s1 Þ
coefficient for electro-osmotic drag ð Þ
exponent for relative permeability ð Þ
amount of species i in control volume j ðmolÞ
molar flow of species i into/out of control volume j
ðmol s1 Þ
number of components ð Þ
pressure or partial pressure ðPa; barÞ
electric effective power ðW; kWÞ
objective functions for optimizations
ideal gas constant ðJ mol1 K1 Þ
specific gas constant ðJ kg1 K1 Þ
relative humidity in control volume ið Þ
switch parameter in flooding sub-model, Eq. (68),
ðÞ
heat flow ðWÞ
liquid water saturation in GDL ð Þ
molar standard-state entropy of reaction
ðJ mol1 K1 Þ
Introduction
In order to reduce the impact of aircraft emissions on human
health and the climate, the aerospace sector is investigating
low-emission power sources that can assist or replace conventional kerosene-based options [1,2]. Low temperature
polymer electrolyte membrane fuel cells (PEMFC) are considered as one of several promising technologies [3] that could be
used to achieve long-term emission reduction targets [4].
t
T
U
vD
vm
V
V_
x, y, z
Xi;j
33219
time ðsÞ
temperature ðK; CÞ
voltage ðVÞ
Darcy velocity ðm s1 Þ
molar volume ðm3 mol1 Þ
volume ðm3 Þ
volume flow ðm3 s1 Þ
spatial variables ðmÞ
mole fraction of species iin control volume jð Þ
Greek letters
cathode transfer coefficient ð Þ
ac
thickness of layer k ðmÞ
dk
ε
GDL porosity ð Þ
hc , hm , hU voltage losses ðVÞ
h
efficiency ð ; %Þ
Q
contact angle ð Þ
k
ionic conductivity ðU1 cm1 Þ
lmem ; lCCL water content of ionomer ð Þ
lO2 ; lH2 stoichiometric ratios for O2 and H2 ð Þ
m
dynamic viscosity ðPa sÞ
r
density ðkg m3 Þ
s
surface tension ðN m1 Þ
Subscripts
A
anode
C
cathode
cap
capillary
comp
compressor-related quantity
eff
effective
fl
fluid
g
gaseous
in
inlet, inward
l
liquid
out
outlet, outward
p
pore
ref
reference quantity, reference conditions
rel
relative
cell
cell-related quantity
stack
stack-related quantity
sys
system-level quantity
tot
total, combined quantities, all species in mixture
Recently, a large aircraft manufacturer announced three new
aircraft concepts where hydrogen is burnt in gas turbines for
propulsion while fuel cells provide electric energy for auxiliary
loads and/or hybrid electric propulsion [1]. Based on this
background, this work investigates the use of PEMFCs to
provide electric auxiliary power onboard future commercial
aircraft.
The use of PEMFCs for propulsion of small passenger
aircraft [5,6] and small electric loads in commercial aircraft [7]
has been successfully demonstrated. However, several
33220
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 e n e r g y 4 6 ( 2 0 2 1 ) 3 3 2 1 8 e3 3 2 4 0
technological challenges must be overcome when using
PEMFCs to meet the electric auxiliary load demand of commercial aircraft (several hundred kW). Specifically, one must
achieve a high system efficiency and a low system mass
(stacks and balance-of-plant components). These two interrelated aspects are affecting the aircraft's overall design in
various direct and indirect ways, as summarized in Table 1.
Achieving good performance for both these aspects is critical
for a PEMFC-system to be feasible and competitive to conventional technology options.
As a part of the overall assessment of the feasibility of such
a PEMFC-based concept, this work analyzes which systemlevel efficiencies can be achieved, discusses the underlying
tradeoff between mass and efficiency, and shows how the
efficiency can be maximized by choosing the optimal stack
operating conditions for each flight phase. The operating
conditions of PEMFCs include the current density, stack temperature, stack pressure, oxygen stoichiometric ratio and the
amount of humidification. At cell-level and for ground-based
applications, their effects are well known [8e10]. However,
for aircraft applications the widely varying environmental
conditions throughout a flight mission result in significant
differences.
Previous work on PEMFCs as auxiliary power sources in
commercial aircraft had a strong focus on overall system
design aspects. Guida et al. [11] developed a methodology for
Table 1 e Summary of the major direct and indirect
effects of the PEMFC-system's efficiency and mass on the
overall aircraft design. The PEMFC-system is defined here
as the stacks and their balance-of-plant components
(compressors, cooling system, etc.), a detailed definition
of the system components and the system-level
efficiency is provided in section System description.
PEMFC-system
parameter
Effect
Overall
aircraft
design
parameter
System efficiency
Affects required
cooling system mass
and size per required
net electric power
output
System efficiency
Direct effect on
required fuel mass
System efficiency
Affects required
hydrogen tank size
and mass (based on
required fuel mass)
Affects the required
reactant air flow per
given power output
Direct contribution
to aircraft mass
Indirect contribution
to aircraft mass
(increased mass
leads to higher
required propulsive
power)
aircraft mass
volumetric
constraints
aerodynamic drag
from cooling air
inlets
fuel consumption
for design mission
aircraft mass
aircraft mass
volumetric
constraints
System efficiency
System mass
System mass
aerodynamic drag
from reactant air
inlets
aircraft mass
fuel consumption
for design mission
optimizing the specific energy of such systems. The feasibility
of several system integration concepts was assessed by Pratt
et al. [12]. Several aviation-related studies focused on systemlevel effects of stack operating conditions in more detail.
Campanari et al. [13] investigated different configurations for
supplying compressed air to the stacks and determined
favorable operating pressures. Lüdders et al. [14] discussed the
multi-functional integration of PEMFCs and optimized several
system-level design parameters simultaneously. Possible
benefits of operating the stacks at lower current densities
were discussed by Kadyk et al. [15]. Another set of studies
focused on practical aspects for the integration of PEMFCs into
€ ter et al. [16] investigated the effect of low
future aircraft. Schro
ambient pressures and pressure drops. The impact of operating parameters and system architecture on the stack's water
management was analyzed by Werner et al. [17].
These studies [11e17] all have in common that the overall
system analysis is conducted based on relatively simple and
computationally efficient PEMFC models or interpolated
experimental data. Owing to their wider perspective on the
overall system, these studies either do not explicitly consider
the effects of the stack's operating conditions on its water
management [11e16] or do so only for a limited range of
operating conditions [17]. A similar pattern can be observed in
studies on the operating conditions of PEMFCs in groundbased applications: studies that focus only on a stack or single cell [8,10,18] capture water management and particularly
flooding effects quantitatively and at high detail, while
system-level studies typically consider flooding only qualitatively1 [19e21], empirically [22] or not at all [23].
The water management of PEMFCs aims at achieving
optimal humidity conditions within the cells so that neither
membrane dehydration nor flooding with excess liquid water
occurs [9,24]. Both phenomena have been shown to be relevant for typical operating conditions [25,26]. They significantly
affect the feasible operating range and achievable efficiency
and are therefore highly relevant to the overall system design.
Consequently, an improved understanding of the effects of
cell-level water management on system-level performance and
the system-level optimal operating conditions of PEMFCs is
needed.C.
The approach that is presented here addresses this challenge with the following improvements:
1. System-level effects of all main stack operating conditions
are considered simultaneously instead of one at a time:
Pressure, oxygen stoichiometric ratio and humidity are
optimized for different current densities and a prescribed
stack temperature.
2. The optimization is conducted with a phenomenological
stack model that is comprehensively validated for water
management effects (including flooding), while still
maintaining the overall-system perspective
In order to conduct this optimization, one requires a PEMFC
model that can capture the combined effects of these operating conditions on the cell's water management and yet
1
Their model equations include liquid water effects but the
model is not validated for flooding conditions.
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 e n e r g y 4 6 ( 2 0 2 1 ) 3 3 2 1 8 e3 3 2 4 0
33221
Table 2 e Comparison of exemplary PEMFC modeling approaches. The computational cost is approximated relative to the
other items in the table based on the models’ governing equations. PDE: partial differential equation, ODE: ordinary
differential equation, DAE: differential algebraic equation.
Ref.
Futter et al. [8]
Vetter et al. [27]
Goshtasbi et al. [18]
This work
Abdin et al. [9]
Liu et al. [28]
Type
transient, 2D, nonisothermal
Steady-state, 1D,
non-isothermal
transient, pseudo2D, non-isothermal
steady-state, 1D,
isothermal
steady-state, 1D,
isothermal
steady-state, 1D,
isothermal
Flooding
Membrane
dehydration
Governing equations
✓
✓
PDEs
moderate to high
✓
✓
PDEs
moderate
✓
✓
PDEs, DAEs
moderate to low (faster than real-time)
✓
✓
ODEs
low
e
✓
ODEs
very low
e
✓
ODEs
very low
features sufficiently low computational cost to be useable for
system-level optimization.
Several exemplary PEMFC modeling approaches are
compared in Table 2. Examples for high fidelity, phenomenological (physics based) models are described in Refs.
[8,18,27]. They capture the effect of varying operating conditions with high accuracy and would, in principle, be well
suited to answer the investigated research question. However,
even real-time capable approaches among them are too
computationally expensive for system-level optimization.
Less computationally demanding phenomenological
models [9,28] are often based on a detailed description of the
processes in the cells, but typically do not consider liquid
water transport in the diffusion media and thus do not capture flooding effects. A semi-empirical PEMFC model that
considers flooding phenomena and yet achieves low computational cost was developed by McKay et al. [24]. However, this
approach has the drawback that it relies on large amounts of
experimental data. Due to the large number of relevant
operating parameters, it can be prohibitively time consuming
to test all their combined effects. Phenomenological models
are therefore better suited to answer the investigated research
question because they inherently require less experimental
data on a particular stack to be parameterized.
Based on these considerations and to the authors’ knowledge, there is at present no phenomenological PEMFC model
that allows for quantitatively assessing water management at
sufficiently low computational cost to conduct the aforementioned optimization of stack operating conditions. Here,
we develop a 1D two-phase PEMFC model that captures both
flooding and membrane dehydration effects in a wide range of
operating conditions, and yet can be used to simulate several
hundred steady-state operating points per second and per
processor core.2 The model is validated with experimental
data for the effects of current density, stack temperature and
oxygen content of the supplied air that were measured with a
commercial stack.
2
For comparison, the open source reference 1D model by
Vetter et al. [27] simulates about 5 operating points per second on
the same computer (Intel i9 processor, 32 GB RAM, Matlab version
R2019a).
Computa-tional cost
By combining the stack model with auxiliary component
models, we relate the effect of the investigated parameters
on cell-level water management to system-level effects. This
enables more realistic simulation results for the optimal
operating conditions and achievable system efficiencies of
PEMFCs in aircraft applications. The outputs of the developed methodology can be easily integrated into high-level
system-sizing frameworks such as those proposed in Refs.
[11,12,14].
System description
In conventional commercial aircraft, electric power is typically provided via two main pathways:
- Generators driven by main engine shaft, typical output:
4 250 kVA for large single-aisle aircraft [29].
- Auxiliary power unit (APU), typical output: 2 225 kVA for
large single-aisle aircraft [29].
The APU is operated on ground or in case of an emergency,
the main engines supply electric power during nominal flight.
A small ram air turbine is typically included as another inflight emergency power source. Additionally, a ground
power connection is used when the aircraft is parked. A
detailed description of electric loads onboard more-electric
aircraft was given by Campanari et al. [13]. Because the
largest consumer (electric environmental control system) is
used continuously throughout the flight, it can be assumed
that the electric auxiliary load demand remains fairly constant throughout the different flight phases.
This work investigates an exemplary scenario where the
engine-driven generators and APU are replaced by PEMFCs
(several generators might still be included for redundancy). To
arrive at generalized results, the investigations are conducted
based on size-independent quantities and without making
assumptions for a particular aircraft size and electric load
demand. The underlying assumption is that the achievable
component efficiencies and optimal operating points do not
significantly depend on the exact number of stacks and cells
per stack in systems of this scale.
33222
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 e n e r g y 4 6 ( 2 0 2 1 ) 3 3 2 1 8 e3 3 2 4 0
System architecture
The investigated system architecture is depicted in Fig. 1. To
enable operation at high altitudes, the stacks are supplied
with compressed air. Under cruise conditions, an air
inlet allows for converting some of the ambient air's kinetic
energy to an increased pressure at the compressor inlet. These
components are comparable to those used in electric environmental control systems of today's aircraft [29]. The compressed air is cooled and humidified before it is supplied to the
stacks.
At low ambient pressures one could recover part of the
compressor's power consumption by expanding the stacks'
pressurized cathode exhaust in a turbine. This would require
additional components for the dehumidification of the
exhaust in order to prevent frost formation in the turbine [13].
Pressure drops along the flow path reduce the amount of
recoverable power. Because it is unclear at present whether
the achievable efficiency improvement of such an approach
can justify the additional mass and complexity, this study
focusses on an architecture without energy recovery.
The stacks are cooled with a conventional liquid coolant
that transfers their waste heat to the ambient air. When the
aircraft speed is not sufficient to force air through the channel,
a fan is used. In the underlying aircraft concept, the hydrogen
will likely be stored in liquid form [1]. Here, it is considered to
be present at the required pressure and stack temperature at
the system boundary. The underlying assumption is that the
pre-conditioning of the hydrogen can be achieved with passive components (heat exchangers, pressure reduction components) and hence does not significantly affect the system's
efficiency. The stacks are operated with an anode recirculation loop as described in section Mass balances. Fig. 1 shows
an exemplary system configuration. The required number of
stacks Nstacks and cells per stack Ncell for a given effective
power output under design conditions will be a simulation
result. The model assumes that all stacks are operated under
the same conditions. The number of compressors Ncomp is a
free design parameter; the model assumes that all compressors are each operated under identical conditions if there is
more than one.
The stacks and the air supply system are simulated in
detail because these components affect the system's efficiency most significantly. The cooling system simply acts as a
heat sink in this investigation. A generalization of the analysis
to consider the cooling system in detail is left for future work,
since the design of cooling systems for commercial aircraft is
a complex design problem on its own [30].
Efficiency definition
The system's net direct-current (DC) power output is
Psys;eff ¼ Pstack;tot PBoP
(1)
where Pstack;tot ¼ Nstacks Pstack is the combined stack power. The
compressor's power consumption is considered with PBoP ¼
Fig. 1 e Investigated system architecture, exemplary configuration with Nstacks ¼ 2 and Ncomp ¼ 1. Bypass tubing as well as
flow and pressure regulation components for control purposes are not depicted; the stacks' anode recirculation loop is
shown in detail in Fig. 2.
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 e n e r g y 4 6 ( 2 0 2 1 ) 3 3 2 1 8 e3 3 2 4 0
33223
Fig. 2 e Schematic fuel cell stack and modeling domain (not to scale, coolant circuit not depicted). ACL: anode catalyst layer,
CCL: cathode catalyst layer, MPL: micro-porous layer (a) top view of stack (b) section of a single cell.
Ncomp Pcomp . The power consumption of the other balance-ofplant components (for example fuel and coolant pumps) are
neglected because they are small compared to Pstack;tot . Moreover, we focus on operating states where Pfan;RA ¼ 0. Situations
where the cooling air fan is necessary (e.g. when waiting on
the runway) are not as relevant to the overall design, because
the system is sized for cruise conditions where the compressor's power loss is more significant and the fan is not needed.
The system's overall efficiency is
hsys;HHV ¼
Psys;eff
m_ H2 ;sys DhHHV;H2
(2)
where DhHHV;H2 is the higher heating value (HHV) of hydrogen
and m_ H2 ;sys ¼ m_ H2 ;A;in Ncell Nstacks is the supplied hydrogen flow.
Mathematical model development
PEMFC stack model
The model is implemented in Matlab code and simulates a
stack's steady-state behavior by solving the governing equations for one hypothetical average cell. The modeling domain
is depicted in Fig. 2 and will be discussed further below. A
straight channel, counter-flow configuration is shown exemplarily; the model makes no assumptions regarding these two
aspects. The model's main assumptions are.
- 1D discretization
- isothermal
- catalyst layers are considered infinitely thin, except when
calculating their ohmic resistance
- the micro-porous layers (MPL) are considered infinitely
thin
- the vapor and liquid phase are assumed to be in equilibrium at all times when solving the governing differential
equations for their steady-state solution
- ideal gases, ideal gas mixtures, incompressible liquid
phase
High-fidelity models often resolve the temperature differences that occur between different locations in the stack
[8,18,27]. In alignment with previous computationally efficient
models [9,28] these temperature differences of several Kelvin
are neglected here and an average temperature Tstack ¼ Tcell is
used.
The average cell voltage in the stack is calculated based
on [31].
33224
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 e n e r g y 4 6 ( 2 0 2 1 ) 3 3 2 1 8 e3 3 2 4 0
Table 3 e Stack model parameters that are fitted in a nonlinear optimization problem.
Sym-bol
i0,ref
ilim,ref
ac
cT
cH
nk
cD1
dGDL;eff
Description
Reference exchange current
density (A m-2)
Reference limit current density (A
m-2)
Cathode transfer coefficient ()
Factor for spatial variation of
temperature in channel ()
Factor for spatial variation of
relative humidity in channel ð Þ
Exponent for relative
permeability ð Þ
Combined GDL-specific
parameters ðm1 Þ
Effective GDL thickness ðmÞ
Lower bound l
Upper bound u
(6)
Eq.
10
450
assumed
(8)
104
106
assumed
(5), (7)
(47)
0.2
0.55
0.5
0:95
based on [31]
based on [56]
(48)
0:5
1:5
assumed
(60)
2
5
based on [50]
(62)
1,1010
1,106
assumed
(65)
1:1,104
2:5,103
dGDL based on [42,53],
ccm based on [55]
Ucell ¼ Uocv hc hU hm
(3)
where the open circuit voltage (OCV), cathode side activation
losses hc , ohmic losses hU and cathode side mass transport
losses hm depend on a number of effects as described below.
The anode activation overpotential is neglected [32,33]. The
stack's power is
Pstack ¼ i A Ucell Ncell
(4)
Cell voltage
(5)
The exchange current density i0 is calculated from a fitted
value i0;ref for the reference conditions listed in Table 4.
CO ;C
i0 ¼ i0;ref , 2
CO2 ;ref
(6)
Cathode side mass transport losses are modelled with [31].
1 RTcell
ilim
ln
hm ¼ 1 þ
ac
4F
ilim i
(7)
where ilim fðCO2 ;C ,DO2 ;eff Þ. By using Eq. (7), the model assumes a
linear distribution of oxygen across the GDL and calculates the
corresponding additional activation losses and changes in
open circuit voltage via the limit current density approach
that is described in Ref. [31]. In alignment with Refs. [9,31], the
model assumes that a water vapor concentration gradient
across the GDL has no direct effect on the cell voltage. The
limit current density ilim depends on the operating conditions
and is determined based on Eq. (8) with a fitted reference value
and the diffusion coefficients described in section Mass
transport in cathode GDL.
CO ;C DO ;eff
ilim ¼ ilim;ref , 2 , 2
CO2 ;ref DO2 ;ref
!
dmem
dCCL
i
þ
hU ¼
kmem fCCL 1:5 kCCL
(9)
for an assumed ionomer fraction fCCL ¼ 0:15 in the CCL [36].
1
1
k ¼ ð0:005139l 0:00326Þ,exp 1268
303 Tcell
The model determines activation losses hc based on [31].
with i0 f CO2 ;C
The cathode transfer coefficient ac is fitted to experimental
data as well.
The major part of the cell's ohmic losses arises from proton
transport in the membrane and CCL [32,34]. These contributions are considered with [32,35].
The conductivity of Nafion in ðU,cmÞ1 is calculated from
Ref. [37].
where i is the current density and A the active cell area.
RTcell
i
ln
hc ¼
;
i0
ac 2F
Ref. For bounds
(8)
(10)
which is used for both kmem ðlmem;eff Þ and kCCL ðlCCL;eff Þ. Here, l is
the water content of the respective component as defined in
section Membrane humidity. Since the model is not discretized along the channel length, lumped average conductivities are determined based on effective values lmem;eff and
lCCL;eff (see section Fluid temperature and humidity).
The OCV is calculated with the Nernst equation and a fitted
value DUcr ¼ const for the voltage drop due to hydrogen
crossover [38].
Uocv ¼ Erev þ
0:5 RTcell
pH2 pO2
ln
DUcr
2F
p0 p0
(11)
The partial pressures pH2 ¼ pA XH2 ;A and pO2 ¼ pC XO2 ;C;mid;gp
are determined in the mass balance sub-model. The reversible
cell voltage Erev is [31].
Erev ¼ Erev;0 þ
b
DS
ðTcell T0 Þ
2F
(12)
with parameters from Table 4. The voltage efficiency hstack;V
and overall stack efficiency based on DhHHV;H2 are
hstack;V ¼
Ucell
Erev
(13)
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 e n e r g y 4 6 ( 2 0 2 1 ) 3 3 2 1 8 e3 3 2 4 0
determined from Eq. (17), which holds true for steady state
operation.
Table 4 e Stack model parameters that are not fitted.
Symbol
Description
CSO3 ;mem
sulfonic charge
concentration in ionomer
Erev;0
standard-state reversible
cell voltage [31]
EW
ionomer equivalent weight
[42]
F
Faraday's constant
fCCL
ionomer fraction in CCL [36]
higher heating value of
DhHHV;H2
hydrogen
ndrag
coefficient for EOD [42,57]
qsw
switch condition in flooding
sub-model
R
ideal gas constant
sres
residual liquid water
saturation [50]
b
DS
molar standard-state
entropy of reaction
ε
GDL bulk porosity [36,48]
hydrogen stoichiometric
lH 2
ratio
ionomer density [33]
rdry;Naf
Standard and reference conditions
pressure at standard
p0
conditions
T0
temperature at standard
conditions
RHref
relative humidity at
reference conditions
reference oxygen mole
XO2 ;ref
fraction
CO2 ;ref
reference oxygen
concentration
DO2 ;ref
reference diffusion
coefficient
hstack;HHV ¼
Value
Unit
rdry;Naf =EW
mol m3
1:23
V
0¼
(17)
The average mole fraction of water XH2 O;C in Vref;C is
kg mol
96485
0:15
1:418,108
As mol1
J kg1
1
1:8
8:314
0:08
XH2 O;C ¼
nH2 O;C
ntot;C
(18)
and includes both liquid and gaseous water. The volume Vref;C
and the amount of substance in it are split into a gaseous part
and a liquid part denoted with the indices “g” and “l”.
Vref;C ¼ Vref;C;g þ Vref;C;l
(19)
J mol1 K1
nH2 O;C ¼ nH2 O;C;l þ nH2 O;C;g
(20)
163
J mol1 K1
ntot;C ¼ ntot;C;l þ ntot;C;g
(21)
0:6
1:05
is
1980
kg m3
1,105
Pa
298:15
K
where Tfl;C is the cathode fluid temperature (see section Fluid
temperature and humidity). The relation of Vref;C;g and Vref;C is
0
Vref;C;g ¼ Vref;C 0:209
where rH2 O;l is the density of liquid water evaluated at Tcell [39].
p0 XO2 ;ref
RT0
mol m3
Eqs. (20)e(23) form a system of algebraic equations with two
independent variables nH2 O;C;l and Vref;C;g . The analytical solu-
Eq. (52)
m2 s1
tion for Vref;C;g is
If the gas phase is saturated, the amount of water in Vref;C;g
nH2 O;C;g ¼
(14)
Mass balances
The mass balance for a single average cell in the stack is
formulated with the reference fluid volumes Vref;C and Vref;A as
depicted in Fig. 2. Vref;C includes only the fluid volume in
cathode channels and GDL pores, while Vref;A includes the
fluid volume in the entire anode recirculation loop. Each volume is split equally among the cells.
The amount of water nH2 O;C in Vref;C is given by
dnH2 O;C
iAlO2 XH2 O;C;in
¼
þ n_H2 O;CCL XH2 O;C n_tot;C;out
dt
4F XO2 ;C;in
(15)
where the first term is the water supplied with the incoming
air flow. Here, lO2 is the ratio of supplied and consumed molar
flows of oxygen. The molar flow of water from the CCL is
iA
2F
iAlO2
iA
n_tot;C;out
þ n_H2 O;CCL 4F
4FXO2 ;C;in
1
1:1
Pstack
n_H2 ;in;A MH2 Ncell DhHHV;H2
n_H2 O;CCL ¼ n_H2 O;mem;C þ
33225
(16)
where n_H2 O;mem;C is the molar flow to or from the membrane
and the second term is the produced water. The total molar
flow of fluid n_tot;C;out that exits the reference volume is
psat Tfl;C ,Vref;C;g
R,Tfl;C
MH2 O
,nH2 O;C;l
rH2 O;l
8
>
>
> if RHC ¼ 1 :
>
>
>
>
Vref;C ,rH2 O;l
rH O;l 1
psat Tfl;C
>
<
nH2 O;C 2
R,Tfl;C
MH2 O
MH2 O
Vref;C;g ¼
>
>
> if RH < 1 :
>
C
>
>
>
>
:
Vref;C
(22)
(23)
(24)
The case distinction for the trivial single-phase case is
made based on the relative humidity RHC that is defined
below. The analytical solution for nH2 O;C;l is
8
if RHC ¼ 1 :
>
>
>
>
r
>
>
< Vref;C Vref;C;g H2 O;l
MH2 O
nH2 O;C;l ¼
>
>
>
if
RH
<
1
:
C
>
>
>
:
0
(25)
The total amount of substance can then be calculated to
ntot;C;l ¼
Vref;C;l ,rH2 O;l
MH2 O
(26)
pC ,Vref;C;g
R,Tfl;C
(27)
ntot;C;g ¼
where pC is the known pressure of the cathode side fluid. The
lumped relative humidity of the cathode fluid is
33226
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 e n e r g y 4 6 ( 2 0 2 1 ) 3 3 2 1 8 e3 3 2 4 0
nH O;C;g
p
C RHC ¼ 2
ntot;C;g psat Tfl;C
(28)
With the assumption of a uniform current density across the
cell area, the oxygen mole fraction decreases linearly along
the channel length. The average value XO2,C,mid,gp is
XO ;C; in þ XO2 ;C;out;gp
XO2 ;C;mid;gp ¼ 2
2
(29)
where XO2 ;C;out;gp is the mole fraction in the gas phase at the
outlet. The mole fraction for the two-phase mixture (considering both liquid and gaseous water) is determined from
iA lO2 1
1
,
XO2 ;C;out ¼
4F
n_tot;C;out
(30)
(31)
The average value CO2 ;C used in Eqs. (6) and (8) is related to
XO2 ;C;mid;gp via
CO2 ;C ¼
XO2 ;C;mid;gp pC
RTcell
(32)
The anode side mass balance is set up in a similar manner.
As shown in Fig. 2, the investigated stack features an anode
recirculation loop. The H2 purge valve is opened at a given
frequency of about 1 s1 while the inlet valve is controlled to
supply just enough hydrogen to maintain a constant anode
side pressure. The model considers these discrete purge
events as a quasi-steady-state continuous supply and removal
of fluid from Vref;A with a small excess hydrogen stoichiometric ratio lH2 ¼ 1:05. Because the investigated system does
not use an external humidification of the supplied hydrogen,
only conditions without liquid water on the anode side are
considered here. Extending the model to consider open-anode
configurations, external hydrogen humidification and liquid
water on the anode side is straightforward because the
equations for the cathode side water balance can be re-used
for the anode side with minor adjustments.
The amount of water nH2 O;A in Vref;A is given by
dnH2 O;A
¼ n_H2 O;mem;A XH2 O;A ,n_tot;A;out
dt
(33)
iAðlH2 1Þ
þ n_H2 O;mem;A
2F
n_H2 O;mem;C ¼ n_H2 O;mem;A
(34)
(35)
The term
n_H2 ;A;in ¼
iAlH2
2F
nH2 O;A
ntot;A
(38)
where pA and Tfl;A are the pressure and temperature of the
anode side fluid (see sections Fluid temperature and humidity
and Fluid pressure). By calculating the relative humidity
RHA ¼
XH2 O;A ,pA
psat Tfl;A
(39)
the model's code checks that RHA < 1 holds true in the investigated operating conditions.
The model considers diffusion and electro-osmotic drag
(EOD) as transport mechanisms for water in the membrane.
For entirely liquid-equilibrated membranes the contribution
of diffusion would need to be replaced by a pressure-driven
flow [40]. As shown in section Stack model validation, both
vapor-equilibrated and partially liquid-equilibrated conditions (sact > 0) occur in the simulations. Weber and Newman
[41] developed a model that considers both mechanisms
simultaneously for partially liquid-equilibrated membranes.
However, due to its complexity that approach is not well
suited for computationally efficient models. A less complex
approach for combining both transport modes would be to
simply add their contributions, but has an unclear physical
basis [40]. The pressure-driven transport mode under
partially liquid-equilibrated conditions is therefore neglected here, which is a reasonable simplification for systemlevel investigations. The model therefore likely underestimates the anode-side membrane humidity under
flooding conditions.
The molar flow of water through the membrane is calculated from
n_H2 O;mem;C ¼
ndrag ,iA
þ n_H2 O;diff;C
F
Diffusion in the membrane is governed by
(41)
The water concentration CH2 O;mem is related to the water
content lmem ¼ CH2 O;mem =CSO3 ;mem of Nafion with its sulfonic
charge concentration [31]. A correlation DH2 O;mem ðl; TÞ from
Liso et al. [43] is used for the diffusion coefficient of water in
Nafion. The membrane water contents at the interfaces to the
adjacent gas phases are approximated with [44].
l ¼ llt ðRHÞ þ
(37)
(40)
where the first term describes the EOD and the second term
diffusion. The drag coefficient is assumed as ndrag ¼ 1 [42].
(36)
is the supplied flow of hydrogen. The molar fraction XH2 O;A of
water in Vref;A is
XH2 O;A ¼
Vref;A ,pA
RTfl;A
vCH2 O;mem
v2 CH2 O;mem
¼ DH2 O;mem ,
vt
vx2
with
n_tot;A;out ¼
ntot;A ¼
Membrane humidity
The corresponding mole fraction in the gas phase is
ntot;C
XO2 ;C;out;gp ¼ XO2 ;C;out
ntot;C;g
Moreover, it holds XH2 ;A ¼ 1 XH2 O;A . Due to the recirculation, XH2 ;A is assumed to be constant in Vref;A . If there is no
liquid water in Vref;A , the total amount of substance ntot;A is
lht ðRHÞ llt ðRHÞ
ðTcell 303Þ
50
(42)
where llt ðRHÞ and lht ðRHÞ are sorption equilibrium relations of
Nafion at different temperatures from Refs. [37,45] with aw ∶ ¼
RH.
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 e n e r g y 4 6 ( 2 0 2 1 ) 3 3 2 1 8 e3 3 2 4 0
llt ¼ 0:043 þ 17:810aw 39:850a2w þ 36:000a3w at 303 K
(43)
lht ¼ 0:300 þ 10:800aw 16:000a2w þ 14:100a3w at 353 K
(44)
Because only vapor-equilibrated transport modes are
considered here, the maximal water content is kept at lsat ðTÞ ¼
lmem ðRH ¼ 1; TÞ based on Eq. (42). The water contents at the
membrane/gas phase interfaces are used as boundary
conditions.
,lðRHA ; TÞ
CH2 O;mem ðx ¼ 0Þ ¼ C
CH2 O;mem ðx ¼ dmem Þ ¼ CSO3 ;mem ,lðRHC ; TÞ
SO
3 ;mem
(45)
Interfacial resistances to water transport at the membrane's surfaces are neglected. The computing time is significantly reduced by approximating DH2 O;mem ðxÞ with an average
value DH2 O;mem across the 1D domain. In this case, the steadystate solution to Eq. (41) is a linear increase of lmem ðxÞ. The
resulting molar flow is given by Fick's law.
CSO3 ;mem
½lðRHA ; TÞ lðRHC ; TÞ
n_H2 O;diff;C ¼ A,DH2 O;mem
dmem
(46)
Eqs. (15)e(46) form a system of two first-order ODEs that is
solved for its steady-state solution using Matlab's solver
ode15s().
33227
effective properties of the anode side fluid are therefore
assumed as Tfl;A ¼ Tcell and RHA;eff ¼ RHA .
Fluid pressure
The cathode side pressure drop DpC ¼ pC;in pC;out is modelled
with a polynomial
2
DpC ¼ cd1 V_ C;tot þ cd2 V_ C;in;tot
(49)
where V_ C;in;tot is the volume flow entering the cathode side of
the cells. V_ C;in;tot is calculated from the known supply air
composition based on section Mass balances. The polynomial's coefficients are fitted based on the measured pressure drop for different mass flows from the experiments
described in section Underlying experimental data. The
average pressure pC along the channel length is approximated
with
pC ¼ pC;in DpC
2
(50)
where pC;in is the known pressure at the cathode inlet. In the
investigated stack, the anode side pressure pA is controlled to
be 0:06,105 Pa above pC;in . The pressure drop on the anode side
is neglected due to the small hydrogen flow.
Mass transport in cathode GDL
Fluid temperature and humidity
When the cathode fluid is supplied at a temperature below
Tcell , it is heated up and humidified as it flows through the
channels. Therefore, its relative humidity changes considerably along the channel length (y-axis in Fig. 2 (a)), which
cannot be captured directly by the 1D model. Below, two fitted
parameters cT and cH are introduced, which allow for a
simplified linearized consideration of these effects.
The value RHC in Eq. (28) can be considered an average
value across the channel length and is determined based on
an effective cathode fluid gas phase temperature Tfl;C . This
temperature is calculated with a fitted coefficient cT that relates it to the fluid temperatures at cathode inlet and outlet
(assuming Tfl;C;out ¼ Tcell ).
Tfl;C ¼ Tfl;C;in þ cT Tfl;C;out Tfl;C;in
(47)
Moreover, the cell's average ohmic resistance cannot be
directly calculated from the channel-length averaged relative
humidity as the ionomer's ohmic resistance depends nonlinearly on RH. This effect is considered with another fitted
factor cH . It is used to calculate an effective humidity RHC;eff ,
based on which the effective water contents lmem;eff and lCCL;eff
are determined.
RHC;eff ¼ cH ,RHC
(48)
When operating the stack with anode recirculation and
lH2 ¼ 1:05, the amount of supplied hydrogen is small
compared to the amount of hydrogen that is already present
within the recirculation loop (in steady-state operation). The
Per Eq. (8), ilim is affected by oxygen diffusion in the cathode
GDL. The effective diffusion coefficient is calculated from
Ref. [46].
DO2 ;eff ¼ DO2 ;N2 ,ε3:6 ð1 sav Þ3
(51)
where ε is the GDL's porosity and DO2 ;N2 ðTfl;C ; pC Þ the bulk binary
diffusion coefficient from Ref. [47]. The effect of water vapor
on oxygen diffusion is neglected [42]. Since the modeling
approach only requires the ratio DO2 ;eff =DO2 ;ref , this is a
reasonable simplification. The reference diffusion coefficient
DO2 ;ref is calculated from
DO2 ;ref ¼ ε3:6 DO2 ;N2 T0 ; p0
(52)
Oxygen diffusion is affected by the GDL's liquid water
saturation sact , which is defined as
sact ¼
VGDL;p;l
VGDL;p
(53)
with the pore volume filled with liquid water VGDL;p;l and the
total pore volume VGDL;p . The model calculates sact ðxÞ in the
GDL for a 1D domain under a channel area (see Fig. 2(b)). Since
2D effects are not resolved by the model, an average saturation sav is used in Eq. (51) as discussed below.
Generally, cathode flooding can occur at the CCL surface,
MPL, GDL and channels [25,48]. However, the saturation in the
MPL is typically lower than that of the GDL [48]. In alignment
with Ref. [42], the MPL is not explicitly considered here and its
effect on oxygen transport is included in the fitted effective
GDL properties. Moreover, it can be assumed that flooding of
33228
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 e n e r g y 4 6 ( 2 0 2 1 ) 3 3 2 1 8 e3 3 2 4 0
the channels would only occur after the GDL is already severely
flooded [25]. Because the system would not be designed to
operate in states where the channels are flooded, the model
only considers flooding of the GDL. The model furthermore
neglects interfacial resistances to mass transport.
Transport of liquid water in the GDL is modelled with a
gradient of capillary pressure as the driving force. In hydrophobic media, capillary pressure is commonly defined as [46].
pcap ¼ p1 pg
(54)
where pl and pg denote the liquid and gas phase pressure. The
model uses a 1D form of Darcy's law for the liquid phase [49]. It
holds
k,krel
Vp1
vD ¼ m
(55)
where m(T) is the water viscosity, k the absolute permeability,
krel(s) the relative permeability and vD the Darcy velocity.
When assuming a uniform gas phase pressure across the GDL,
Vpl ¼ Vpcap. Moreover, pcap is commonly considered to be a
function of liquid water saturation [48,49] so that Eq. (55) can
be rewritten as
vD ¼ k,krel vpcap
ds
dx
m
vs
(56)
where s is the effective saturation. Generally, there exists a
residual saturation sres due to water being trapped in isolated
pores of the GDL [50]. The actual saturation sact used for
calculating DO2,eff is calculated with [50].
sact ¼ sres þ s , ð1 sres Þ
(57)
The model assumes sres ¼ 0.08 [50]. In conditions with
RHC<1, sres is set to zero.
The relation pcap(s) is modelled with a Leverett approach
[51].
pcap ε 12
¼ JðsÞ
s cosðQÞ k
(58)
where s(T) is the water surface tension [52], Q the GDL's contact angle and J(s) an empirical function. The model is
parameterized with a Leverett-type function J(s) that is fitted
to the secondary injection curve measured Gostick et al. [53]
for an exemplary SGL10BA GDL under compressed conditions
(the exact GDL type of the modelled stack is not known to the
authors). Assuming that the fitted dimensionless function J(s)
can also describe the behavior of other GDLs, it is used with
another set of parameters (ε, k, Q) when inserting its analytical
derivative into Eq. (59).
ε 1 dJðsÞ vpcap
2
¼ s cosðQÞ
k
ds
vs
(59)
The GDL's relative permeability is modelled with [8,42,48].
krel ¼ snk
(60)
where nk depends on the GDL type [49,50] and is fitted with
stack-level experimental data. When RHC ¼ 1, capillary flow
becomes the only transport mode [49,54]. Accordingly, the
Darcy velocity vD is calculated from
8
>
if RHC ¼ 1 :
>
>
>
>
>
>
> MH2 O ,n_H2 O;CCL
<
rH2 O ,A
vD ¼
>
>
>
>
if RHC < 1 :
>
>
>
>
:0
(61)
Eq. (56) is combined with Eqs. (59)e(61) to the nonlinear
ODE
ds
1
¼ CD1 CD2 n
dx
s k , ½JðsÞ 0
(62)
where constant properties are grouped into unknown GDLspecific properties cD1 and known properties cD2.
pffiffiffiffiffi 1
CD1 ¼ cos Q, kε
CD2 ¼
mvD
s
(63)
(64)
The combined parameter cD1 is fitted to stack-level
experimental data while cD2 is calculated based on water
properties sðTÞ and mðTÞ from Ref. [39]. A low saturation
s0 ¼ 102 at the GDL-channel interface [42,48] is assumed
when solving the initial value problem for sðxÞ with Matlab's
solver ode45().
There are two spatial effects to be accounted for. The
model considers them in a simplified manner in order to avoid
the computational cost of 2D or 3D discretization. 2D effects
due to channel ribs (x;z-plane in Fig. 2) are taken into account
based on conformal mapping [55]. That approach relates 2D
effects of a partially conducting boundary (channel and ribs)
to an increased effective layer thickness in the 1D formulation
for a fully conducting boundary. For a known cell geometry,
the ratio ccm ¼ dGDL;eff =dGDL of effective and actual GDL thickness can be calculated [55]. Eq. (62) can then be solved between
locations x1 ¼ dmem and
x2 ¼ dmem þ ccm dGDL
(65)
However, because the exact GDL thickness under compressed conditions is not known to the authors, the term
dGDL;eff is directly fitted to stack-level experimental data.
The average of sact ðxÞ across the “effective” 1D domain is
S*av
¼
1
dGDL;eff
xð2
sact ðxÞdx
(66)
x1
However, this value does not include variations of saturation along the channel length (y-direction in Fig. 2). Flooding
can be expected to occur in cell regions where the gas phase is
saturated. Typically, this region gradually expands from the
cell's outlet towards the inlet if the humidity in the cell is
increased [25]. The following simplified relation based on an
empirical model parameter q is used to mimic such a gradual
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 e n e r g y 4 6 ( 2 0 2 1 ) 3 3 2 1 8 e3 3 2 4 0
increase of saturation once the fluid in the lumped control
volume reaches RHC ¼ 1.
8
>
>
<0
sav ¼ cf ,s*av
>
>
: s*
av
q¼
for q 1
for 1 < q < qsw
for q qsw
nH2 O;C
p
C ntot;C psat Tfl;C
(67)
(68)
In contrast to RHC, which only considers water vapor in the
gas phase and is limited to one (see Eq. (28)), the definition of q
allows for values q>1. If q exceeds a limit value, s*av is used
without modification. The limit is set to qsw ¼ 1.8 by
comparing the model's results to the measurements described
in section Underlying experimental data. If q < 1, there is no
liquid phase. In between the two cases, the saturation is
scaled with a factor cf2[0,1] which is chosen as
cf ¼ 1 qsw q
qsw 1
(69)
and increases linearly as q approaches qsw.
Stack model parameter identification
Several stack model parameters are fitted to experimental
data while others are chosen based on literature values. The
parameter DUcr (Eq. (11)) and the coefficients of the pressure
drop correlation (Eq. (49)) are determined directly from the
experiments described in section Underlying experimental
data.
The 8 remaining fitted parameters are summarized in
Table 3. They are determined by minimizing the simulation
error in a nonlinear optimization problem
.
.
.
.
Q
min .
1 C Such that l i < Ci < u i for i ¼ 1…8
C
.
Q1 ð c Þ ¼
2
N X
.
. .
wj Um;j Us;j
(70)
(71)
j¼1
where the objective function Q1 includes the complete model
and solver code. The search space is constrained by the
bounds given in Table 3. Vector c contains the fitted parameters from Table 3, Um and Us of size N-by-1 are the measured
and simulated cell voltages (N ¼ 30). Weights w are used to
slightly increase the contribution of operating points with
higher current densities. To find the global minimum within
the search space, Q1 is minimized with Matlab's genetic algorithm ga() with a population size of 6400 and 2000 generations. The simulation results with the fitted parameters are
compared to the measurements in section Stack model
validation.
The stack's geometric parameters A, dCCL , dmem , Vref;C and
Vref;A are chosen/assumed based on information from the
stack manufacturer. Fluid properties psat ðTÞ, mðTÞ and rH2 O;l ðTÞ
are interpolated between data points with 1 C increments
from Ref. [39]. The remaining model parameters are summarized in Table 4.
33229
Underlying experimental data
A custom-built fuel cell test bench is used to measure the
effect of stack temperature, current density and oxygen mole
fraction at the cathode inlet under steady-state, galvanostatic operation. The experiments are conducted on two
different 40 cell Hydrogenics HD4 stacks with a rated power
output of 4 kW. This stack type is not particularly well suited
for aircraft applications because it has a lower specific power
ðkW kg1 Þ than more recent stack designs [58,59]. Because it
uses industry-standard membrane and electrode materials it
is nonetheless a reasonably representative testbed for
investigating the effect of the stack operating conditions on
size-independent quantities (power per cell area, voltage
losses, etc.).
The stack is supplied with varying mixtures of N2 and O2 at
the cathode and pure H2 at the anode. Per the stack's specifications, it is operated without external humidification. The
stack's current and voltage as well as a number of temperatures, pressures and mass flows are measured at 0:1 s time
steps. Since no temperature sensors are placed inside the
stack, the coolant outlet temperature is assumed to represent
the stack's temperature with sufficient accuracy and is used as
Tstack in the following. The experiments are carried out with a
stepwise adjustment of one operating parameter at a time.
After each step change the stack is operated for several minutes until it reaches steady-state conditions. The data points
that are presented in section Stack model validation are
average values of the respective signals over 30 s once a
steady-state is reached (hysteresis effects were not studied). A
detailed description of the stack test bench is provided by our
colleagues Montaner Rı́os et al. [60].
Auxiliary component models
In order to determine the stack's optimal operating conditions,
the effects of several auxiliary components on the overall
system efficiency need to be simulated. However, the model
only needs to consider the auxiliary components' rated operating points and does not need to provide detailed information on their internal processes. The air inlet and compressor
are therefore modelled based on thermodynamic considerations and typical component efficiencies. For the heat
exchanger and humidifier, it is sufficient to include their
pressure drop in the model and assume that they fulfill their
task as required at the system's design conditions. Under the
condition T4 zTstack no significant heat transfer occurs in the
humidifier as both the wet and dry fluid have nearly the same
temperature. The auxiliary component models are described
in Appendix A.
Results and discussion
Stack model validation
As a first step, the stack model is validated for varying stack
temperatures, oxygen contents and current densities. Table 5
33230
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 e n e r g y 4 6 ( 2 0 2 1 ) 3 3 2 1 8 e3 3 2 4 0
Table 5 e Operating conditions in the experiments used
in this work.
Parameter
Data in Fig. 3
2
i
Tstack
0:77 A cm
varied
pC;out
XO2 ;C;in
lO 2
1:01 to 1:03 bar
varied
2:5
RHC;in
0
Data in Fig. 4
varied
50…60 C (set by stack controller
based on i)
1:02 to 1:08 bar
0:209
2:2 above 0:7 A cm2 (increased by
stack controller at lower i)
0
summarizes the measured operating conditions that are used
to parameterize and validate the model. The experiments
focus on moderate current densities because these are most
relevant to aircraft applications [14,15] (see section The massefficiency trade-off).
Effect of stack temperature and oxygen content
Fig. 3(a)e(c) show the measured and simulated effect of
varying operating temperatures and oxygen concentrations.
Increased oxygen contents can be relevant for emergency
operation [61] and furthermore provide a good basis for validating oxygen concentration dependent effects. Considering
the system-level modeling approach, a good agreement of
simulation and measurements is achieved. It should be noted
that the simulation results are directly compared to the
dataset that the model's parameters were fitted with. However, since a very wide range of operating conditions can be
captured, it is unlikely that the model only matches the
experimental data well because some of the data points were
used to parameterize it.
As shown in Fig. 3 (a) and (d), both flooding (sav > 0Þ and
membrane dehydration (RHC < 1, RHA < 1) occur within the
investigated temperature range for operation with the oxygen
content of ambient air. The observed trends in cell voltage and
the optimal operating temperature of about 60 C under these
conditions are predicted well by the model.
With higher XO2 ;C;in no abrupt drop of cell voltage is observed
at higher temperatures. This can be explained with the lower
total gas flow n_tot;C;in that is required for achieving the desired
oxygen stoichiometric ratio, which results in less severe
membrane dehydration. The GDL saturation remains similar at
increased XO2 ;C;in but results in less severe mass transport losses due to the oxygen concentration effect in Eq. (8).
Fig. 3 e Influence of stack temperature and oxygen mole fraction on the average cell voltage in the stack at i ¼ 0:77 A cm2 ,
experimental conditions given in Table 5. Measurements first published by Becker et al. [62]. (a) to (c): Measured and
simulated cell voltage, (d) to (f): simulated relative humidity and saturation of GDL pores with liquid water, (g) to (i):
simulated voltage losses. Note: legends for entire row.
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 e n e r g y 4 6 ( 2 0 2 1 ) 3 3 2 1 8 e3 3 2 4 0
33231
The results in Fig. 5 show that for a given pC;in and Tstack
there is a distinct range of combinations of RHC;in and lO2 that
result in high stack efficiencies. At low temperature and
pressure, the favorable operating range is limited by membrane dehydration at low RHC;in and high lO2 and by cathode
flooding at large RHC;in . At increased operating pressures the
stack efficiency becomes less sensitive to changes of external
humidification and stoichiometry. The highest stack-level
efficiency is achieved for high temperatures, high pressures
and large lO2 , because these conditions lead to high oxygen
concentrations within the cells in combination with a good
membrane conductivity (see section Cell voltage).
Optimal operating conditions at system level
Fig. 4 e Current-voltage characteristic at the stack's rated
operating conditions, experimental conditions in Table 5.
(a) Measured and simulated cell voltage (b) simulated
saturation and relative humidity (c) simulated voltage
losses.
Effect of current density
The measured and simulated current-voltage characteristic at
the stack's rated operating temperature are shown in Fig. 4 (a).
The curve has the typical shape for PEMFCs, in which activation losses dominate at low current densities and mass
transport losses become more relevant at higher current densities. The model can accurately simulate the stack's behavior
for a wide range of current densities. At i < 0:4 A cm2 the
model becomes less accurate. This is expected with the linearization described in section Fluid temperature and humidity.
Influence of operating conditions at stack level
Next, the model is used to simulate the effects of the previously introduced parameters RHC;in , lO2 , Tstack and pC;in on the
achievable stack efficiency. It should be noted that the model
is not directly validated for the effects of lO2 , RHC;in and pC;in :
However, it can be assumed to capture the effects of these
parameters with sufficient accuracy as it links them to the
validated effects of oxygen concentration and humidity conditions in the cells.
To show how the system's auxiliary components affect the
optimal operating conditions, the effect of lO2 and pC;in on
both stack-level and system-level efficiency is explored for
typical cruise phase environmental conditions (Mach 0:74 at
11278 m, see Table 8) and a moderate current density of
0:7 A cm2 . As shown in Fig. 6, it is found that the compressor's power consumption reduces the system efficiency
significantly at high altitudes. The combination of lO2 and
pC;in that result in the highest system-level efficiency deviates considerably from that one that merely results in the
highest stack efficiency. As shown in section Influence of
operating conditions at stack level, this affects the optimal
values for the other stack operating parameters as well.
Consequently, all main stack operating parameters should
be optimized simultaneously while considering both stacklevel and system-level effects.
This nonlinear optimization is conducted by including the
overall system model in an objective function
2
Q2 RHC;in ; lO2 ; pC;in ¼ hsys;HHV
(72)
and optimizing the stack's operating parameters RHC,in2[0,
1], lO22[1.8, 3] and pC,in2[1.1 bar,2.5 bar] to maximize
hsys,HHV(RHC,in,lO2,pC,in). Because heat exchangers can be
sized smaller with higher temperature gradients to the
ambient, Tstack is kept constant at 85 C. Including Tstack in
the optimization problem would be straightforward. To
prevent convergence to a local optimum and avoid the stochastic nature of evolutionary algorithms, Q2 is first evaluated at 103 equally spaced grid points in the threedimensional search space. The ten most promising grid
points are then used as initial guesses for Matlab's gradientbased interior-point algorithm. If the gradient-based optimization converges to multiple minima, the one with the
lowest penalty value Q2 is used.
The optimization is conducted with the constraint of a
given current density ides that the stacks are operated at when
meeting the system's design load. Within the feasible range,
ides is a free design parameter. With this approach (referred to
hereafter as “sizing mode”), the total number of cells Ncell,tot ¼
NcellNstacks that is needed to achieve a given power output
Psys,eff is a result of the optimization and depends on the cell
voltage at the optimal conditions Ucell,opt ¼ f(ides,RHC,in,lO2,pC,in). It holds
33232
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 e n e r g y 4 6 ( 2 0 2 1 ) 3 3 2 1 8 e3 3 2 4 0
Fig. 5 e Stack efficiency (HHV-based, see Eq. (14)) at different combinations of oxygen stoichiometric ratio lO2 and relative
humidity at cathode inlet (based on Tfl;C;in ¼ Tstack , i ¼ 0:7 A cm2 , contours interpolated from four sets of 100-by-100 single
operating points). (a) Low temperature, low pressure (b) low temperature, high pressure (c) high temperature, low pressure
(d) high temperature, high pressure.
Psys;eff
Ncell;tot ¼
Acell;sizing ,Ucell;opt ,ides
(73)
where the denominator describes the power per cell for a
given active cell area. Because the stack model result is used
in quantities (Ucell,opt, ides) which are independent of the
stack's active cell area, Acell, sizing is a free design parameter
and does not necessarily need to match the active cell area
that is used by the stack model. The number of cells per stack
can be set below a reasonable maximum (e.g. 455 [58]) by
choosing Nstacks appropriately. The required air mass flow
and compressor power consumption are obtained based on
Ncell,tot and the normalized per-cell results from the stack
model.
The model is also suited to instead conduct the optimization with the constraint of a given Ncell,tot (non-sizing mode).
In this case, the model solves for the required current density
by iteratively minimizing Q2 for different ides until the required
power output is achieved (using iterative guesses for ides from
Matlab's algorithm fminsearch).
An exemplary result in terms of absolute numbers for a
typical in-flight power output of 605 kW [13] is shown in Table
6. The optimization algorithm can effectively find the combination of pC,in, RHC,in and lO2 that results in the highest
system-level efficiency (visualized in Fig. 6 (b) for a constant
RHC,in). For an exemplary cell area of Acell,sizing ¼ 250 cm2 and
ides ¼ 0.7 A cm-2, 18 stacks with 419 cells each are required to
achieve the effective power output of 605 kW. One could either
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 e n e r g y 4 6 ( 2 0 2 1 ) 3 3 2 1 8 e3 3 2 4 0
33233
Table 6 e Detailed results for one exemplary operating point (fluid states 1 to 6 are defined in Fig. 1, optimized operating
parameters pC,in, lO2 and RHC,in are obtained with the “sizing mode” that is described in section Optimal operating
conditions at system level, environmental conditions for cruise phase of commercial aircraft as listed in Table 8).
Symbol
Description
System-level results
hsys,HHV
hstack,HHV
hstack,V
Nstacks
Ncomp
Pstack,tot
Pcomp,tot
Psys,eff
Stack-level results
Tstack
pC,in
RHC,in
lO2
lH2
Acell,sizing
ides
Ucell
Ncell
Fluid states
p1
p2
p3
p4
p5
p6
T1
T2
T3
T4
T5
T6
RH1
Value
Type
System efficiency based on HHV, Eq. (2)
Stack efficiency based on HHV, Eq. (14)
Stack voltage efficiency, Eq. (13)
Number of stacks
Number of compressors
Combined power output of stacks
Combined power consumption of compressors
Effective (net) power output
29.50 %
41.21 %
54.35 %
18
2
845.11 kW
240.11 kW
605.00 kW
Result
Result
Result
Parameter
Parameter
Result
Result
Input
Stack temperature
Optimized stack pressure
Optimized humidity at cathode inlet
Optimized O2 stoichiometric ratio
H2 stoichiometric ratio (with anode recirculation)
Active cell area used for sizing
Current density at design conditions
Average cell voltage in stacks
Number of cells per stack (rounded to integer)
85 C
1.60 bar
0.49
1.8
1.05
250 cm2
0.7 A cm-2
0.64 V
419
Parameter
optimized
optimized
optimized
Parameter
Parameter
Parameter
Result
Result
Ambient pressure
Pressure at filter inlet
Pressure at compressor inlet
Pressure at heat exchanger inlet
Pressure at humidifier inlet
Pressure at stack inlet (¼pC,in)
Ambient temperature
Temperature at filter inlet
Temperature at compressor inlet
Temperature at heat exchanger inlet
Temperature at humidifier inlet (¼Tstack)
Temperature at stack inlet (¼Tstack)
Ambient relative humidity
0.20 bar
0.27 bar
0.26 bar
1.75 bar
1.65 bar
1.60 bar
42 C
17 C
17 C
225 C
85 C
85 C
0.5
Input
Result
Result
Result
Result
optimized
Input
Result
Result
Result
Parameter
Parameter
Input
Table 7 e Assumed mass-related parameters in the system analysis.
Parameter
Value
Description
Background/Reference
mcell,sp
3.5 kg m-2 in Fig. 8, varied in Fig. 9
Mass per cell normalized with
active cell area
mcomp,sp
5.333,10-4 kg W-1
Compressor mass per rated
power consumption
mhum,sp
82.353 kg/(kg/s)
humidifier mass per rated dry
air mass flow
mcell,sp is defined here as an average cell mass per
active cell area and includes the averaged mass
contribution of the endplates and structural stack
components:
mcell,sp ¼ mstack/(NcellAcell). According to public
manufacturer data, today's stacks achieve
approximately 2.5 to 4 kg m-2 [58]
[63] (The pressure ratio of this compressor is
somewhat below the required one. The normalized
value is nevertheless reasonably accurate for the
purpose of this study.)
[64] (typical membrane humidifier for fuel cell
systems)
decrease the required number of stacks by operating them at
higher current density, or further increase their efficiency by
choosing an even lower ides. This tradeoff is discussed in the
following section.
The mass-efficiency trade-off
To reduce the required tank size for a given load demand, it
can be beneficial to operate the stacks at lower current
33234
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 e n e r g y 4 6 ( 2 0 2 1 ) 3 3 2 1 8 e3 3 2 4 0
Table 8 e Investigated flight phases and environmental conditions, exemplary flight profile based on Ref. [30]. The ambient
relative humidity is kept constant at RH1 ¼ 0:5.
Flight
phase
Geopotential altitude (m)
Ambient Pressure (bar)
Ambient Temperature ( C)
Aircraft velocity (Mach number)
Takeoff
Climb
Cruise
Descent
Approach
0
2438
11278
6096
457
1.014
0.752
0:200
0:469
0:958
40
22
42
4
36
0.250
0:440
0:740
0:550
0:390
densities than those that lead to the maximum power output
[14,15]. This increases the required stack mass but can result
in a net improvement of the total system mass due to a lower
required tank mass at higher efficiencies [15]. Besides the
lower tank mass, there are additional benefits when oversizing the stacks. At low ides, the supplied air is used more efficiently by the stack (higher electric energy output per mol of
supplied air). This reduces the required air flow and consequently also the compressor's power consumption and
balance-of-plant component mass for a given Psys,eff (with
lO2 ¼ const).
To visualize this trade-off between improved efficiency and
increased mass, the optimizations for the 605 kW scenario from
Table 6 are repeated for different ides. The change in stack mass
is calculated with mstacks ¼ f(ides) ¼ NstacksNcellAmcell,sp where
the required Nstacks and Ncell are simulation results and mcell,sp
is the cell mass per active cell area. The change in compressor
and humidifier mass is approximated with
mcomp ¼ mcomp;sp Pcomp Ncomp
(74)
mhum ¼ mhum;sp m_ c;tot
(75)
where mcomp,sp is the compressor's assumed mass per rated
power, mhum,sp the humidifier mass per rated mass flow and
m_ C;tot ¼ f ðides Þ ¼ m_ c;in Ncell Nstacks the required air flow. Table 7
summarizes the assumed parameters. The mass of the
remaining balance-of-plant components (air inlet, tubing,
wiring, etc.) is assumed to remain constant for different ides.
An increased efficiency also reduces the waste heat per power
Fig. 6 e Effect of varying lO2 and pC,in on stack-level and
system-level efficiency (i ¼ 0.7 A cm-2, Tstack ¼ 85 C,
RHC,in ¼ 0.8, environmental conditions for cruise phase as
listed in Table 8, contours interpolated from 100-by-100
grid of single operating points). (a) Stack efficiency (b)
system efficiency.
Fig. 7 e Change in (a) efficiency and (b) component mass for
different design point current densities ides and a constant
required power output of 605 kW (environmental
conditions from Table 6, mcell,sp ¼ 3.5 kg m-2, optimization
was repeated for 50 linearly spaced values of ides in the
depicted range).
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 e n e r g y 4 6 ( 2 0 2 1 ) 3 3 2 1 8 e3 3 2 4 0
33235
Fig. 8 e Optimized stack operating conditions for different design point current densities ides, datapoints correspond to
those in Fig. 7. (a) Cathode inlet pressure, (b) relative humidity at cathode inlet, defined with T5 ¼ 85 C, (c) oxygen
stoichiometric ratio.
output. This reduction in cooling system mass is not further
investigated here (see system boundary in section System
architecture), but would further benefit lower ides.
The efficiency improvement for different ides is depicted in
Fig. 7 and is found to be nearly linear in the investigated range.
The required stack mass increases significantly at lower current densities, while the balance-of-plant component mass is
improved slightly due to the more efficient use of the supplied
air. It can be concluded that a net-improvement in system
mass by oversizing the stacks needs to be compensated for by
reductions of the hydrogen tank mass and does not occur
from benefits to balance-of-plant components alone.
The corresponding optimal stack operating conditions
are depicted in Fig. 8 and were found to vary only slightly
for different ides. The optimization results for pC,in and
RHC,in oscillate somewhat (less than ±2%) around an averaged solution. This behavior arises from the chosen optimization approach but has no significant effect on the
overall results. As described in section Optimal operating
conditions at system level, the operating conditions are
optimized by conducting multiple runs of a local optimization algorithm with several different initial guesses.
Therefore, the obtained solution can be expected to be
close to but not exactly at the global minimum. As shown
in the corresponding data in Fig. 7 (a), the oscillations in
the optimized parameters have no significant effect on the
optimization results and the resulting optimized efficiency
curve remains fairly smooth.
Because the results scale linearly with the system's effective power output (see Eq. (73)), they can be normalized to
arrive at a size-independent relation of these quantities (see
Fig. 9). For mcell,sp ¼ 3.5 kg m-2, one can “trade” an efficiency
increase from 23.60% to 30.84% by increasing the mass of the
stacks and the air supply system by 0.2 kg per kW effective
power output when designing the system. With different assumptions for mcell,sp similar trends can be observed. The
location of the optimal point on this curve depends on the
aircraft type and flight mission (duration, altitude, speed) and
can therefore not be determined at this subsystem-level. The
developed model can support the overall aircraft design process by providing the proposed relation between added mass
and efficiency.
Optimal conditions in different flight-phases
Different environmental conditions were found to significantly affect the achievable efficiency. This aspect is explored
in Fig. 10 for several representative flight phases (see Table 8).
In this section, no particular load demand per flight phase is
assumed. Instead, the optimization is conducted for a range of
typical design-point current densities [14,15] to show more
generally which efficiency and optimal stack operating conditions can be expected when designing the system to operate
under the respective conditions.
The results in Fig. 10 (a) suggest that the achievable efficiency hsys;HHV varies widely for different load requirements
and flight phases. When the system is for example designed
and oversized to operate at a low current density of 0:4 A cm2
during take-off, the resulting efficiency would be 43:12 % according to the model. On the other hand, when designing and
sizing the stacks so that they run at a higher current density of
1 A cm2 during take-off, the efficiency would only be 33:30 %.
The model furthermore suggests that the system efficiency for
a given current density is generally lower during the cruise
Fig. 9 e Relation of system efficiency and added component
mass per effective power output (cruise conditions,
Dm ¼ DmstacksþDmcompþDmhum, delta defined with respect
to operating the system at the current density that leads to
the lowest overall system mass).
33236
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 e n e r g y 4 6 ( 2 0 2 1 ) 3 3 2 1 8 e3 3 2 4 0
Fig. 10 e Optimization results for the investigated system architecture. (a) System efficiency at optimized conditions (b)
stack efficiency at optimized conditions (c) optimized pressure (absolute) (d) optimized stoichiometric ratio (e) optimized
humidity at cathode inlet.
phase. In this exemplary case, the higher altitudes cause the
efficiency to drop as low as 24:81 % when operating the stacks
at 1 A cm2 during the cruise phase. This effect is mainly
caused by the higher compressor power consumption at lower
ambient pressures, and is somewhat reduced by the higher
Mach number. As shown in Fig. 10 (b), the stack-level efficiency
remains nearly constant throughout the different flight phases
with the optimized conditions.
The optimal efficiency occurs with moderate operating
pressures between 1:59 and 1:90 bar (absolute), which is below
typical maximum rated pressures for ground-based applications [58]. Because the optimal stoichiometric ratio was found
to be at the chosen minimum limit of 1:8 for the investigated
conditions, it can be concluded that stack efficiency improvements of higher lO2 do not outweigh the additional
compressor power.
One can use the fuel-to-electricity efficiencies of
kerosene-based solutions in today's aircraft to put these results into context. Typical APUs achieve efficiencies of <
10 % [65], engine-driven generators up to 32 % [12] (fuel-toelectricity, based on HHV of kerosene). At lower altitudes, the
PEMFC system exceeds these efficiencies by far. At higher
altitudes, it can match or slightly exceed them when oversizing the stacks and operating them at lower current densities. It should be noted that recent stack designs [58,59]
achieve improved efficiencies compared to the investigated
Hydrogenics HD4 stack. The reported numbers should
therefore not be understood as the full potential of this
technology but rather as a baseline for future improvements.
The phenomenological approach makes it possible to
analyze such improvements in PEMFC technology without
major modifications to the model once further information
becomes available.
To assess the system's feasibility regarding its effect on
overall aircraft mass, one needs to include the mass of the
cooling system and additional components (wiring, air inlets,
etc.) in the analysis. Moreover, several positive and negative
aircraft-specific effects need to be considered, including
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 e n e r g y 4 6 ( 2 0 2 1 ) 3 3 2 1 8 e3 3 2 4 0
33237
Table 9 e Comparison of typical specific powers of the discussed technology options. PEMFC system mass numbers are
based on existing volume-optimized PEMFC systems for ground-based applications. These numbers will likely improve for
future weight-optimized aircraft systems.
Component
Specific power
Existing kerosene-based aircraft systems
Generators driven by the main engine shaft
Conventional APU, including its generators
1:0 kW kg1
1:6 kW kg1
[67]
[68], based on APU of the Boeing 787, which is most suitable
for a comparison with PEMFCs because it supplies, unlike
most APUs, only electrical power to the aircraft
(no pneumatic or hydraulic power)
3:0 kW kg1
0:6 kW kg1
[58,59]
[69]
< 0:6 kW kg1
The mass of the cooling system and power converter will further
reduce the system-level specific power, the exact value for
aircraft-rated systems with several hundred kW power
output is unknown at present
Existing PEMFC systems
Stacks
Stacks and balance-of-plant components,
without cooling system and power converter
System (stacks, balance-of-plant components,
cooling system and power converter)
improvements in main engine efficiency if less generators
need to be powered by the engine shaft [66], an increased
aerodynamic drag due to additional air inlets [12] and possible
synergies from a multi-functional integration [7,14]. Because
there are no aircraft-rated systems at this scale so far and data
from small experimental aircraft is likely not representative
for large commercial aircraft, no reliable approximation of the
system's overall mass can be made at this point. However,
based on the exemplary numbers in Table 9 it seems likely
that the stacks and balance-of-plant components will be
heavier than the kerosene-based options. For novel aircraft
concepts that have a significant electric load demand and use
hydrogen as the main energy carrier [1], the higher efficiency
of PEMFCs at maximum and part load could nonetheless make
them a favorable option despite the increased mass of the
energy conversion system itself.
Conclusion
This work investigated the use of PEMFCs as an electric
auxiliary power source for future hydrogen-based aircraft
concepts with a focus on the interrelated aspects of efficiency,
mass and optimal operating conditions. A steady-state, onedimensional, two-phase PEMFC model was developed and
validated with data from a commercial stack. It was shown to
capture key aspects of PEMFC water management while
achieving sufficiently low computational cost for system-level
optimization. The stack model was combined with several
auxiliary component models to assess the effect of the stack's
operating parameters on both stack-level and system-level.
The simulations show that the operating conditions that
result in the highest overall system efficiency deviate
considerably from those that merely result in the highest
stack efficiency. With this aspect in mind, the stack's operating conditions were numerically optimized to maximize the
system efficiency. The maximal achievable efficiency depends
strongly on the flight phase and system sizing aspects and is
generally achieved with moderate pressurization (1:59 to
1:90 bar absolute), low oxygen stoichiometric ratios (1:8) and
moderate cathode humidification.
Background/Reference
In terms of efficiency, PEM fuel cell systems are found to be a
promising option to supply auxiliary electric power in future
hydrogen aircraft concepts. At high altitudes (about 11 km),
sufficiently high system-level efficiencies can be achieved by
oversizing the stacks. The developed model can be used to
assess this trade-off between improved efficiency, added stack
mass and reduced balance-of-plant component mass in detail.
For future work, it can be interesting to consider the cooling system and additional balance-of-plant components in
higher detail in order to ultimately determine the PEMFCsystem's total mass and volume. This would allow for a direct
comparison with other technology options such as batteries
and generators driven by hydrogen-burning engines. Ultimately, operational and safety-related aspects such as cost
and durability as well as the net environmental impact will
need to be considered as well before arriving at a reliable
conclusion which technology option is suited best. These aspects are currently under investigation as part of a large multidisciplinary research project at the German Aerospace Center
[70].
Declaration of competing interest
The authors declare that they have no known competing
financial interests or personal relationships that could have
appeared to influence the work reported in this paper.
Acknowledgements
This work was supported by the Federal Ministry of Transport
€ ger Jülich
and Digital Infrastructure of Germany via Projekttra
and NOW GmbH (research project number 03B10701) and by
the Federal Ministry for Economic Affairs and Energy of Ger€ ger (research project number
many via DLR Projekttra
20M1909B, LuFoVI-1). The authors thank their colleagues Igor
Sokolov, Simon Coners, Stefan Bleeck and Gema Montaner
Rı́os for their support with setting up the fuel cell test bench
used in the experiments.
33238
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 e n e r g y 4 6 ( 2 0 2 1 ) 3 3 2 1 8 e3 3 2 4 0
The pressure drop in the air filter is considered with p3 ¼
p2 Dpfilt . With the parameters listed in Table A1 the corresponding power consumption is given by
Appendix A. Auxiliary component models
The air inlet and compressor are modelled based on thermodynamic considerations and typical component efficiencies as
described below. The models assume ideal gases, ideal gas
mixtures and temperature-independent heat capacities. They
make use of the specific gas constant Rsp;air and specific heat
capacities cp;air and cv;air of the ambient air (Nomenclature of
fluid states 1 to 8 according to Fig. 1, assuming that state 6 ¼
00
00
00
60 ¼ 6 , 7 ¼ 70 ¼ 7 and 8 ¼ 80 ¼ 8 ).
m_ C;tot ¼ m_ C;in Ncell Nstacks
Pcomp ¼
Dh23 m_ C;tot
hcomp;m hcomp;el hcomp;pc
DpHX
Dphm;d
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
gair Rsp;air T1 ,Ma
2cp;air
T2 ¼ T1 þ
p2;stat ¼ p1
g gair1
T2 air
T1
Dphm;w
Dpstack;C
(A.1)
for subsonic flow. The maximum achievable static pressure at
the diffuser outlet is
(A.2)
with gair ¼ cp;air =cv;air . The pressure recovery of a non-
(A.8)
Table A.1 e Parameters of the auxiliary component models
Parameter
Description
Value
Air inlet model
The model calculates the pressure increase in an adiabatic
diffuser with a given pressure recovery. The fluid velocity at
the diffuser inlet is assumed to equal the aircraft's true
airspeed (given as Mach number Ma). When neglecting the
remaining fluid velocity at the diffuser outlet, it holds [71].
(A.7)
Dpfilt
hcomp;s
hcomp;m
hcomp;el
hcomp;pc
Pressure drop in air to air heat
exchanger at its rated mass
flow [74]
Pressure drop in dry side of
membrane humidifier at its rated
mass flow [64]
Pressure drop in wet side of
membrane humidifier at its rated
mass flow [64]
Pressure drop in stack: correlation
evaluated at respective volume
flow per cell, see section Fluid
pressure
Pressure drop in air filter at its
rated mass flow [75]
Isentropic compressor efficiency
[13]
Mechanical compressor efficiency
[76]
Electric motor efficiency [77]
Power converter efficiency
1:0,104 Pa
5:0,103 Pa
1:1,104 Pa
DpC ¼ f ðV_ C;tot Þ
5:0,102 Pa
0:76
0:97
0:94
0:95
isentropic diffuser is defined as [72].
hpr ¼
p2 p1
p2;stat p1
(A.3)
The model assumes hpr ¼ 0:75 [72] for a straight lip inlet
when calculating p2 from Eqs. (A.1) to (A.3).
Pressure drops in air supply components
The pressure p4 that is required to achieve a given pressure
p60 ¼ p600 ¼ pC;in at the stack's cathode inlet is.
p4 ¼ pC;in þ DpHX þ Dphm;d (A.4); The minimum feasible
pressure pC;in is limited by
pC;in p1 þ Dpstack;C þ Dphm;w
(A.5)
The component's pressure drops at their rated mass flows
are summarized in Table A1. To estimate the stack's cathode
side pressure drop, the overall system model uses normalized
data from the investigated 4 kW short stack (see section Fluid
pressure).
Compressor model
With the usual simplifications of neglecting the heat transfer
between fluid and compressor as well as the fluid velocity
difference between inlet and outlet, the enthalpy change in a
single stage compressor is given by Refs. [71,73].
Dh34 ¼
1
hcomp;s
cp;air T3
Rcsp;air
p4 p;air
1
p3
(A.6)
references
[1] Airbus S.A.S. Press release from Sept. 21, 2020. Available:
https://www.airbus.com/newsroom/stories/these-newAirbus-concept-aircraft-have-one-thing-in-common.html.
[Accessed 3 March 2021].
[2] MTU aero engines AG. Integrating hydrogen propulsion into
aircraft. Available: https://aeroreport.de/en/innovation/
integrating-hydrogen-propulsion-into-aircraft. [Accessed 3
March 2021].
[3] ElringKlinger AG. Press release October 14, 2020. Available:
https://www.elringklinger.de/sites/default/files/releases/
press-releases/2020/201014_elringklinger_-_press_release.
pdf. [Accessed 5 March 2021].
[4] European Commission. Flightpath 2050. Europe’s Vision for
Aviation 2011. https://doi.org/10.2777/50266.
[5] Romeo G, Borello F, Correa G, Cestino E. ENFICA-FC: design of
transport aircraft powered by fuel cell & flight test of zero
emission 2-seater aircraft powered by fuel cells fueled by
hydrogen. Int J Hydrogen Energy 2013;38(1):469e79.
[6] Nishizawa A, Kallo J, Garrot O, Weiss-Ungethüm J. Fuel cell
and Li-ion battery direct hybridization system for aircraft
applications. J Power Sources 2013;222:294e300.
[7] Renouard-Vallet G, Saballus M, Schmithals G, Schirmer J,
Kallo J, Friedrich KA. Improving the environmental impact of
civil aircraft by fuel cell technology: concepts and
technological progress. Energy Environ Sci
2010;3(10):1458e68.
[8] Futter GA, Gazdzicki P, Friedrich KA, Latz A, Jahnke T.
Physical modeling of polymer-electrolyte membrane fuel
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 e n e r g y 4 6 ( 2 0 2 1 ) 3 3 2 1 8 e3 3 2 4 0
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
cells: understanding water management and impedance
spectra. J Power Sources 2018;391:148e61.
Abdin Z, Webb CJ, Gray EM. PEM fuel cell model and
simulation in MatlabeSimulink based on physical
parameters. Energy 2016;116:1131e44.
Salva JA, Iranzo A, Rosa F, Tapia E, Lopez E, Isorna F.
Optimization of a PEM fuel cell operating conditions:
obtaining the maximum performance polarization curve. Int
J Hydrogen Energy 2016;41(43):19713e23.
Guida D, Minutillo M. Design methodology for a PEM fuel cell
power system in a more electrical aircraft. Appl Energy
2017;192:446e56.
Pratt JW, Klebanoff LE, Munoz-Ramos K, Akhil AA,
Curgus DB, Schenkman BL. Proton exchange membrane fuel
cells for electrical power generation on-board commercial
airplanes. Appl Energy 2013;101:776e96.
Campanari S, Manzolini G, Beretti A, Wollrab U. Performance
assessment of turbocharged pem fuel cell systems for civil
aircraft onboard power production. J Eng Gas Turbines Power
2008;130(2).
Lüdders HP, Strummel H, Thielecke F. Model-based
development of multifunctional fuel cell systems for MoreElectric-Aircraft. CEAS Aeronautical Journal 2013;4(2):151e74.
Kadyk T, Winnefeld C, Hanke-Rauschenbach R, Krewer U.
Analysis and design of fuel cell systems for aviation.
Energies 2018;11(2):375.
€ ter J, Graf T, Frank D, Bauer C, Kallo J, Willich C.
Schro
Influence of pressure losses on compressor performance in a
pressurized fuel cell air supply system for airplane
applications. Int J Hydrogen Energy 2021;46(40):21151e9.
Werner C, Busemeyer L, Kallo J. The impact of operating
parameters and system architecture on the water
management of a multifunctional PEMFC system. Int J
Hydrogen Energy 2015;40(35):11595e603.
Goshtasbi A, et al. A mathematical model toward real-time
monitoring of automotive PEM fuel cells. J Electrochem Soc
2020;167(2).
Kim DK, et al. Efficiency improvement of a PEMFC system by
applying a turbocharger. Int J Hydrogen Energy
2014;39(35):20139e50.
Kim DK, et al. Parametric study on interaction of blower and
back pressure control valve for a 80-kW class PEM fuel cell
vehicle. Int J Hydrogen Energy 2016;41(39):17595e615.
Chen H, Liu B, Liu R, Weng Q, Zhang T, Pei P. Optimal interval
of air stoichiometry under different operating parameters
and electrical load conditions of proton exchange membrane
fuel cell. Energy Convers Manag 2020;205:112398.
Qin Y, Du Q, Fan M, Chang Y, Yin Y. Study on the operating
pressure effect on the performance of a proton exchange
membrane fuel cell power system. Energy Convers Manag
2017;142:357e65.
Hoeflinger J, Hofmann P. Air mass flow and pressure
optimisation of a PEM fuel cell range extender system. Int J
Hydrogen Energy 2020;45(53):29246e58.
McKay DA, Ott WT, Stefanopoulou AG. Modeling, parameter
identification, and validation of reactant and water
dynamics for a fuel cell stack. 2005. https://doi.org/10.1115/
IMECE2005-81484. Available:.
Hussaini IS, Wang C-Y. Visualization and quantification of
cathode channel flooding in PEM fuel cells. J Power Sources
2009;187(2):444e51.
rida W, Harrington DA.
Le Canut J-M, Latham R, Me
Impedance study of membrane dehydration and
compression in proton exchange membrane fuel cells. J
Power Sources 2009;192(2):457e66.
Vetter R, Schumacher JO. Free open reference
implementation of a two-phase PEM fuel cell model. Comput
Phys Commun 2019;234:223e34.
33239
[28] Liu H, Li P, Wang K. Optimization of PEM fuel cell flow
channel dimensionsdmathematic modeling analysis and
experimental verification. Int J Hydrogen Energy
2013;38(23):9835e46.
[29] Sinnett M. Boeing: 787 No-bleed systems: saving fuel and
enhancing operational efficiencies. Available: https://www.
boeing.com/commercial/aeromagazine/articles/qtr_4_07/
AERO_Q407_article2.pdf. [Accessed 3 March 2021].
[30] Rheaume JM, Macdonald M, Lents CE. Commercial hybrid
electric aircraft thermal management system design,
simulation, and operation improvements. In: 2019 AIAA/IEEE
electric aircraft technologies symposium (EATS); 2019. p. 1e23.
[31] O'Hayre R, Cha S-W, Colella W, Prinz FB. Fuel cell
fundamentals. 3rd ed. Hoboken: John Wiley & Sons, Inc.;
2016.
[32] Mao L, Wang C-Y. Analysis of cold start in polymer
electrolyte fuel cells. J Electrochem Soc 2007;154(2):B139.
[33] Gerteisen D, Heilmann T, Ziegler C. Modeling the
phenomena of dehydration and flooding of a polymer
electrolyte membrane fuel cell. J Power Sources
2009;187(1):165e81.
[34] Li G, Pickup PG. Ionic conductivity of PEMFC electrodes. J
Electrochem Soc 2003;150(11):C745.
[35] Ohma A, et al. Analysis of proton exchange membrane fuel
cell catalyst layers for reduction of platinum loading at
Nissan. Electrochim Acta 2011;56(28):10832e41.
[36] Jiang F, Fang W, Wang C-Y. Non-isothermal cold start of
polymer electrolyte fuel cells. Electrochim Acta
2007;53(2):610e21.
[37] Springer TE, Zawodzinski TA, Gottesfeld S. Polymer
electrolyte fuel cell model. J Electrochem Soc
1991;138(8):2334.
[38] Vilekar SA, Datta R. The effect of hydrogen crossover on
open-circuit voltage in polymer electrolyte membrane fuel
cells. J Power Sources 2010;195(8):2241e7.
[39] E. W. Lemmon, M. O. McLinden, and D. G. Friend,
"Thermophysical properties of fluid systems," in NIST
chemistry WebBook, NIST standard reference database
number 69, P. J. Linstrom and W. G. Mallard, Eds. Gaithersburg
MD: National Institute of Standards and Technology.
[40] Weber AZ, Newman J. Modeling transport in polymerelectrolyte fuel cells. Chem Rev 2004;104(10):4679e726.
[41] Weber AZ, Newman J. Transport in polymer-electrolyte
membranes : II. Mathematical model. J Electrochem Soc
2004;151(2):A311.
[42] Goshtasbi A, Pence BL, Ersal T. Computationally efficient
Pseudo-2D non-isothermal modeling of polymer electrolyte
membrane fuel cells with two-phase phenomena. J
Electrochem Soc 2016;163(13):F1412e32.
[43] Liso V, Simon Araya S, Olesen AC, Nielsen MP, Kær SK.
Modeling and experimental validation of water mass balance
in a PEM fuel cell stack. Int J Hydrogen Energy
2016;41(4):3079e92.
[44] Ge S, Li X, Yi B, Hsing IM. Absorption, desorption, and
transport of water in polymer electrolyte membranes for fuel
cells. J Electrochem Soc 2005;152(6):A1149.
[45] Hinatsu JT, Mizuhata M, Takenaka H. Water uptake of
perfluorosulfonic acid membranes from liquid water and
water vapor. J Electrochem Soc 1994;141(6):1493e8.
[46] Weber AZ, et al. A critical review of modeling transport
phenomena in polymer-electrolyte fuel cells. J Electrochem
Soc 2014;161(12):F1254e99.
[47] Fuller EN, Schettler PD, Giddings JC. New method for
prediction of binary gas-phase diffusion coefficients. Ind Eng
Chem 1966;58(5):18e27.
[48] Qin CZ, Hassanizadeh SM. A new approach to modelling
water flooding in a polymer electrolyte fuel cell. Int J
Hydrogen Energy 2015;40(8):3348e58.
33240
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 e n e r g y 4 6 ( 2 0 2 1 ) 3 3 2 1 8 e3 3 2 4 0
[49] Kumbur EC, Sharp KV, Mench MM. On the effectiveness of
Leverett approach for describing the water transport in fuel
cell diffusion media. J Power Sources 2007;168(2):356e68.
[50] Weber AZ. Improved modeling and understanding of
diffusion-media wettability on polymer-electrolyte-fuel-cell
performance. J Power Sources 2010;195(16):5292e304.
[51] Kumbur EC, Sharp KV, Mench MM. Validated Leverett
approach for multiphase flow in PEFC diffusion media. J
Electrochem Soc 2007;154(12):B1295.
[52] Vargaftik NB, Volkov BN, Voljak LD. International tables of
the surface tension of water. J Phys Chem Ref Data
1983;12(3):817e20.
[53] Gostick JT, Ioannidis MA, Fowler MW, Pritzker MD.
Wettability and capillary behavior of fibrous gas diffusion
media for polymer electrolyte membrane fuel cells. J Power
Sources 2009;194(1):433e44.
[54] Pasaogullari U, Wang CY. Liquid water transport in gas
diffusion layer of polymer electrolyte fuel cells. J
Electrochem Soc 2004;151(3):A399.
[55] Weber AZ. Effective diffusion-medium thickness for
simplified polymer-electrolyte-fuel-cell modeling.
Electrochim Acta 2008;54(2):311e5.
[56] Zohuri B. Heat exchanger types and classifications. In:
Compact heat exchangers. Cham: Springer; 2017.
[57] Zawodzinski TA, Davey J, Valerio J, Gottesfeld S. The water
content dependence of electro-osmotic drag in protonconducting polymer electrolytes. Electrochim Acta
1995;40(3):297e302.
[58] Powercell Sweden AB. Datasheet PowerCellution P stack.
Available: https://www.datocms-assets.com/36080/
1611437833-p-stack.pdf. [Accessed 3 March 2021].
[59] ElringKlinger AG. Datasheet PEMFC stacks NM12. Available:
https://www.elringklinger.de/sites/default/files/brochures/
downloads/ek_factsheets_pemfc_stack_nm12_en_a4.pdf.
[Accessed 3 March 2021].
[60] Montaner Rı́os G, Schirmer J, Gentner C, Kallo J. Efficient
thermal management strategies for cold starts of a proton
exchange membrane fuel cell system. Appl Energy
2020;279:115813.
[61] Becker F, Pillath F, Kallo J. Cathode exhaust gas recirculation
for polymer electrolyte fuel cell stack. Fuel Cell
2018;18(5):568e75.
[62] Becker F, Gentner C, Rı́os GM, Sokolov I, Kallo J. Steigerung
€ higkeit von Brennstoffzellensystemen für
der Leistungsfa
Luftfahrttechnische Anwendungen. In: Presented at the
deutscher luft-und raumfahrt kongress. Germany:
Friedrichshafen; 2018.
[63] Fischer AG. Datasheet EMTCT-120k air. Available: https://
shop.fischerspindle.com/WebRoot/Store/Shops/fp/5813/
D763/DDD2/25F2/C853/B2FA/1909/C010/137901_data_map_
sheet_EMTCT_120k_Air.pdf. [Accessed 3 March 2021].
[64] Fumatech Bwt GmbH. Membrane humidifiers. Available:
https://www.fumatech.com/NR/rdonlyres/0B9A1C7F-5BA64409-A003-5C4E79CD61AB/0/FUMATECH_BWT_
GmbHMembrane_Humidifiers.pdf. [Accessed 3 March 2021].
[65] Zanger J, Krummrein T, Siebel T, Roth J. Characterization of
an aircraft auxiliary power unit test rig for cycle optimization
studies. J Eng Gas Turbines Power 2018;141(1).
[66] Giannakakis P, Laskaridis P, Pilidis P. Effects of off-takes for
aircraft secondary-power systems on jet engine efficiency. J
Propul Power 2011;27(5):1024e31.
[67] Dollmayer J, Bundschuh N, Carl UB. Fuel mass penalty due to
generators and fuel cells as energy source of the all-electric
aircraft. Aero Sci Technol 2006;10(8):686e94.
w. Our products/APS 5000
[68] Pratt&Whitney AreoPower Rzeszo
for boeing 787 dreamliner. Available: http://pwaeropower.
com/en/apu-engine-center/our-products. [Accessed 3 March
2021].
[69] Powercell Sweden Ab. Datasheet PowerCellution power
generation system 100. Available: https://www.datocmsassets.com/36080/1611437574-power-generation-system100.pdf. [Accessed 3 March 2021].
[70] German Aerospace Center (Dlr). Conceptual study for
environment-friendly flight. Available: https://www.dlr.de/
content/en/articles/news/2020/02/20200504_conceptualstudy-for-environment-friendly-flight.html. [Accessed 3
March 2021].
[71] Struchtrup H. Thermodynamics and energy conversion. 1st
ed. Berlin Heidelberg: Springer-Verlag; 2014.
[72] Lombardi A, Ferrari D, Santos L. Aircraft air inlet design
optimization via surrogate-assisted evolutionary
computation. In: Gaspar-Cunha A, Henggeler Antunes C,
Coello C, editors. Evolutionary multi-criterion optimization.
EMO 2015. Lecture notes in computer science, vol. 9019.
Cham: Springer; 2015.
[73] Ji SW, Myung NS, Kim TS. Analysis of operating
characteristics of a polymer electrolyte membrane fuel cell
coupled with an air supply system. J Mech Sci Technol
2011;25(4):945e55.
[74] Zhao H, Hou Y, Zhu Y, Chen L, Chen S. Experimental study
on the performance of an aircraft environmental control
system. Appl Therm Eng 2009;29(16):3284e8.
[75] Freudenberg Filtration Technologies Se & Co Kg. Datasheet
fuel cell filter type FC F-0630-N. Available: https://www.
freudenberg-filter.com/en/world-of-automotive/products/
fuel-cell-solutions/. [Accessed 6 April 2021].
[76] Carlucci AP, Ficarella A, Laforgia D, Renna A. Supercharging
system behavior for high altitude operation of an aircraft 2stroke Diesel engine. Energy Convers Manag
2015;101:470e80.
[77] Mecrow BC, Jack AG. Efficiency trends in electric machines
and drives. Energy Pol 2008;36(12):4336e41.