APPENDIX A:
The following relations are useful to express the
governing equations in (1-9). Using linearized nozzle
equation, the outlet air rate is defined as:
W SM ,out  k SM ,out ( pSM  pCa )
(A.1)
These parameters are calculated using the ideal gas
relations [1]. The mass flow rate of reacted oxygen,
hydrogen and the water vapor generated in the cathode
are respectivly defined through the following equations:
nI
(A.2)
W O2 , react  M o 2  st
4F
nI
(A.3)
W H 2 , react  M H 2  st
2F
nI
(A.4)
W  ,Ca , gen  M   st
2F
Where W  , A n ,in and W  , An ,out stand for the entering and
leaving water mass flow rate of vapor to and from the
anode [19] respectively. Furthermore the water mass
flow rate through the fuel cell membrane W  ,m , is
defined using the membrane hydration model [19]. The
compressor torque  CM , and that of the steady-state  Cp
are also described by:
k
CM  CM t ( CM  k  Cp )
RCM
(A.5)
V
W
ε
λ
τ
ω
Table 2: The used subscripts for modelling the PEMFC [19]
Subscripts
An
atm
c
Ca
CM
Cp
d
gen
H2
in
I
m
Net
N2
Out
O2
react
RM
st
SM
 1


C p T atm  pSM  
W
 Cp 

1
(A.6)


 Cp
Cp Cp  patm 


Using the thermodynamic principles, the compressor air
temperature equation is expressed by:
 1


T atm  pSM  

TCp  T atm 
(A.7)

  1

Cp  patm 


The air temperature in the supply manifold T SM, is
deriveded from the mSM, pSM and VSM using the ideal gas
law [19]. The return manifold air temperature TRM, is
supposed as a constant parameters and equal to the
temperature of the FC stack. The cathode outlet air flow
rate WCa, out, is depended to the cathode pressure and
also the return manifold pressure which is obtained from
the linearized nozzle equation similar to that in (10).
The relation corresponds to the return manifold outlet
air flow rate W RM ,out , is again defined using a nonlinearized nozzle relation [1].
v
w
max
o
opt
Table 1: The notations in the PEMFC modelling [19]
NOTATION
I
K
m
p
t
T
V
current (A)
gain factor
mass (kg)
pressure (Pa)
time (s)
temperature (K)
voltage (V)
volume (m3)
flow rate (kg/s)
Eigenvalues of the linearized model
excess ratio
torque (Nm)
rotational speed (rad/s)
anode-related quantity
quantity at 1 atm pressure
canonical form of the quantity
cathode-related quantity
compressor-motor-related quantity
compressor-related quantity
desired value of a quantity
quantity generated through electrochemical
reaction
hydrogen-related quantity
inlet quantity
integral-control quantity
membrane-related quantity
net quantity
nitrogen-related quantity
outlet quantity
oxygen-related quantity
quantity consumed in electrochemical reaction
quantity associated with the return manifold for
cathode
fuel cell stack-related quantity
quantity associated with the supply manifold for
cathode
water-vapour-related quantity
liquid-water-related quantity
maximum value
nominal value
optimal value
Table 3:
The used parameters during modeling and simulation
of PEMFC [19]
Parameter
Symbol SI units
Value
Atmospheric pressure
patm
Pa
1.013×105
Atmospheric temperature
Air-specific heat ratio
Air-specific heat
Air density
Universal gas constant
Air gas constant
Oxygen gas constant
Nitrogen gas constant
Vapour gas constant
Hydrogen gas constant
Molar mass of air
Tatm
γ
Cp
ρa
R
Ra
RO2
RN2
Rv
R H2
Ma
K
–
J/kg/K
kg/m3
J/mol/K
J/kg/K
J/kg/K
J/kg/K
J/kg/K
J/kg/K
kg/mol
298.15
1.4
1004
1.23
8.314
286.9
259.8
296.8
461.5
4124.3
Molar mass of oxygen
MO2
kg/mol
32.0×10−3
Molar mass of nitrogen
MN2
kg/mol
28.0×10−3
Molar mass of vapour
Mv
kg/mol
18.02×10−3
Molar mass of hydrogen
MH2
kg/mol
Faraday’s constant
F
Temperature of the FC
Motor constant
Motor constant
Motor constant
Compressor efficiency
Compressor
motor
Tfc
kt
RCM
kv
gCp
Gcm
A.s/mol
K
Nm/A
ohm
V/(rad/s)
–
–
2.0×10−3
96 487
28.97×10−3
353
0.0153
0.82
0.0153
0.80
0.98
mechanical efficiency
Number of cells in FC
stack
FC active area
n
–
Afc
m2
Supply manifold volume
Single
stack
cathode
volume
Single stack anode volume
Return manifold volume
Supply manifold outlet
orifice constant
Cathode outlet orifice
constant
Membrane dry density
VSM
VCa
m3
m3
VAn
VRM
kSM, out
m3
m3
kg/s/Pa
0.005
0.005
kCa, out
kg/s/Pa
0.2177×10−5
rm, dry
kg/m
Membrane dry equivalent
weight
Membrane thickness
Mm, dry
kg/mol
3 2×103
1.1
tm
m
Compressor diameter
Compressor and motor
inertia
Return manifold throttle
discharge coefficient
Return manifold throttle
area
Average
ambient
air
relative humidity
Oxygen mole fraction at
cathode inlet
Hydrogen mole fraction at
anode inlet
dCp
JCp
m
1.275×10−4
0.2286
kg.m2
2 5×10−5
CD
–
0.0124
Aτ
m2
0.002
Φatm
–
0.5
xO2, in
–
0.21
xH2, in
–
1.0
381
280×10−4
0.02
0.01
0.3629×10−5