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Fuel Cell Mass Transport
CHAPTER 5
Fuel Cell Mass Transport 5.1 Introduction In order to produce electricity, a fuel cell must be supplied continuously with fuel and oxidant. In addition, product water must be removed continually to insure proper fuel and oxidant at the catalyst layers to maintain high fuel cell efficiency. Voltage losses occur in the fuel cell due to activation losses (Chapter 3), ohmic losses (Chapter 4), and mass transport limitations~which is the topic of this chapter. Mass transport is the study of the flow of species, and can significantly affect fuel cell performance. Losses due to mass transport are also called "concentration losses," and can be minimized by optimizing mass transport in the flow field plates, gas diffusion layer, and catalyst layer. This chapter covers both the macro and micro aspects of mass transport. The specific topics to be covered are: 9 9 9 9 9
Fuel cell mass balances Convective mass transport from flow channels to electrode Diffusive mass transport in electrodes Convective mass transport in flow field plates Mass transport equations in the literature
In conventional fuel cells, the flow field plates have channels with dimensions in millimeters or centimeters. Due to the size of these channels, mass transport is dominated by convection and the laws of fluid dynamics. Convection is the movement of fluid flow due to density gradients or hydrodynamic transport, and is characterized by laminar or turbulent flow and stagnant regions. This type of flow dominates mass transfer in the flow channels. High fuel and oxidant flow rates sometimes insure good distribution of reactants, but if the flow rate is too high, the fuel may move too fast to diffuse through the GDL and catalyst layers. In addition, delicate fuel cell components such as the membrane can rupture. Mass transport in the fuel cell GDL and catalyst layers is dominated by diffusion due to the tiny pore sizes of these layers (4 to 10 microns). In
98
PEM Fuel Cell Modeling and Simulation Using MATLAB |
a flow channel, the velocity of the reactants is usually slower near the walls; therefore, this aids the flow change from convective to diffusive. The mass transport theory described in this chapter will help the reader to write mass balances, predict fuel cell flow rates, and calculate the mass transfer in the flow channels, electrodes, and membrane.
5.2 Fuel Cell Mass Balances Before convective and diffusive flows are covered, the overall mass flows through the fuel cell need to be discussed. These flow calculations, or mass balances, are critical for determining the correct flow rates for a fuel cell. In order to properly determine these mass flow rates, the mass that flows into and out of each process unit (or control volume)in the fuel cell subsystems, stack, or fuel cell layer need to be accounted for. The procedure for formulating a mass balance can be applied to any type of system, and is as follows: 1. A flow diagram must be drawn and labeled. Enough information should be included on the flow diagram to have a summary of each stream in the process. This includes known temperatures, pressures, mole fractions, flow rates, and phases. 2. The appropriate mass balance equation(s) must be written in order to determine the flow rates of all stream components and to solve for any desired quantities. An example flow diagram is shown in Figure 5-1. Hydrogen enters the cell at temperature, T, and pressure, P, with the mass flow rate, mw. Oxygen enters the fuel cell from the environment at a certain T, P, and mo2. The hydrogen and oxygen react completely in the cell to produce
Work,
H2(g) at T, P, mH2
Wel
t
~[~i~~ .:~:~~ ,:~~ .l
~i~.'~-::.. ~::~~.~:~ .,...-.,~!~:-'<.'~"~:~:.~y.:~ .,'.'..:~i~ O2(g) at T, P, mo2
H20(1) at T, P, mH20
.._Ii;~i;il~l~ ~. ~,~ ~.,..~
..~..
, ~ ' . ~ , ~ : ~ i t .~:~.,.,.:~4~.
FIGURE 5-1. Detailed flowchart to obtain mass balance equation.
Fuel Cell Mass Transport 99
water, which exits at a certain T, P, and mmo. This reaction can be described by: 2H21g) + O2(g)-+ 2H20(1)
Wel in Figure 5-1 is the work available through chemical availability. The generic mass balance for the fuel cell in this example is: mH2 + m o 2 - mH2o + Wel
15-11
The formal definition for material balances in a system (or control volume) can be written as: Input (enters through system boundaries) + Generation (producted within the system) -
Output (leaves through system boundaries)
- Consumption (consumed within the system) = Accumulation (buildup within the system) Generally, the fuel cell mass balance requires that the sum of all of the mass inputs is equal to the mass outputs, which can be expressed as: E(mi)in = E(mi)out
(5-2)
where i is the mass going into and out of the cell, and can be any species, including hydrogen, oxygen, and water. The flow rates at the inlet are proportional to the current and number of cells. The cell power output is: Wel = ncellVcellI
(5-3)
where ncen is the number of cells, Vce11is the cell voltage, and I is the current. All of the flows are proportional to the power output and inversely proportional to the cell voltage: I- ncell = Wel Vcell
(5-4}
The inlet flow rates for a PEM fuel cell are as follows: The hydrogen mass flow rate is
mH2,in = SH2 MH~ I'ncell 2F
15-5)
100 PEM Fuel Cell Modeling and Simulation Using MATLAB |
The oxygen mass flow rate (g/s) is
(5-61
m o 2 , i n - So2 M o 2 I'ncell
4F
The air mass flow rate (g/s) is mair, in =
SO2 Mair
ro2 4F
I'ncdl
(5-7)
The nitrogen mass flow rate (g/s) is mN2,in -- 802 MN-------!21 - ro2,in I. ncell 4F ro2,i n
(5-8)
Water vapor in the hydrogen inlet is MH20 mH2OinH2,in -- SH2 ~
2F
(PanPvs(Tan,in)
I. ncell
(5-9)
I. ncell
(5-10)
I'ncell
(5 11)
Pan - (Pan Pvs(Tan, in)
Water vapor in the oxygen inlet is mH2oinO2,in = 8o2
MH20
~OcaPvs(Tan,in)
4F
Pea - q)caPvs(Tan, in)
Water vapor in the air inlet (g/s) is mH2oinairin :
O P,,
So2 MH20
ro2
4F
--r
inl
Pea -- (/0caPvs(Tan,in)
The outlet flow rates for a PEM fuel cell are as follows"
The unused hydrogen flow rate is mH2,out -- (SH2 --
1)MH2 I" ncell 2F
(5-112,)
The oxygen flow rate at the outlet is equal to the oxygen supplied at the inlet minus the oxygen consumed in the fuel electrochemical reaction" mo2,out - (So2 - 1)Mo2 I" ncell
4F
(5-13)
The nitrogen flow rate at the exit is the same as the inlet because nitrogen does not participate in the fuel cell reaction:
Fuel Cell Mass Transport
mN2,out = mN2in = 802
MN21
-
4F
rozin
101
(5-14)
I. ncell
ro~in
The depleted air flow rate is then simply a sum of the oxygen and nitrogen flow rates: mair,out-
I(So2-1)Moz
+Sozr~1-I'ncellro2in MN2] 4F
(5-15)
The oxygen volume fraction at the outlet is much lower than the inlet volume fraction" 802 -- 1 S~ 1
r~176
15-16)
ro2,in
The additional outlet liquid and vapor water flow rates and balances for a PEM fuel cell are described by equations 5-17 through 5-20: The water vapor content at the anode outlet is the smaller of the total water flux:
[mH2oin, H2out,V =
t
I. ncell, mH2oin, H2out 1/
MH2O
P~(Wou~,an~
2F
Pan -- aPan - P~s~Wout,a~
mini(sin - 1)
/
(5-17)
APanis the pressure drop on the anode side. The amount of liquid water is the difference between the total water present and the water vapor:
where
MH2Oin,H2out,L = mH2oin, H2out
-
-
(5-18)
mH2oin, H2out,V
Water content in the cathode exhaust is equal to the amount of water brought into the cell, plus the water generated in the cell, along with the water transported across the membrane: mH2OinAirout ---- mH2OinAirin 4- mH2Ogen 4- mH2OED -- mH2OBD
(5-19)
The water vapor content at the cathode outlet is:
mH2oin'air~
= min I/8~
ro2i-, r~n
/ MH2~ 4F
Pvs(Tout, an}
P c a - APca -
PvslTout,anl
1
ncell' mH2oin'Air~ /
.J
(5-20)
102 PEM Fuel Cell Modeling and Simulation Using MATLAB |
EXAMPLE 5-1: Water Injection Flow Rate A hydrogen-air PEM fuel cell generates 500 watts (W) at 0.7 V. Dry hydrogen is supplied in a dead-end mode at 20 ~ The relative h u m i d i t y of the air at the fuel cell inlet is 50% at a pressure of 120 kPa. Liquid water is injected at the air inlet to help cool the fuel cell. The oxygen stoichiometric ratio is 2, and the outlet air is 100% saturated at 80~ and atmospheric pressure. What is the water injected flow rate (g/s)? The water mass balance is mH20_air_in + mH20_Inject + mH20_gen = mH20 in air_out
In order to calculate the a m o u n t of water in air, the saturation pressure needs to be calculated. To calculate the saturation pressure (in Pa)for any temperature between 0~ and 100~ 9 Pvs = eaT-1+b+cT+dT2+eT3+fln(T)
where a, b, c, d, e, and f are the coefficients. a =-5800.2206, b = 1.3914993, c =-0.048640239, d = 0.41764768 x 10-4, e =-0.14452093 x 10-7, and f = 6.5459673 with T = 293.15, pvs = 2.339 kPa. The a m o u n t of water in air can be calculated: mH20,airi n =
So2 MH2O
to2
nF
r162Pvs(Tca,in)
I 9ncell
P c a -- r
2 18.015 0.50*2.339 500W mH20,airin- 0.2095 (4,96,485As/mol) 120kPa - 0.50,2.339 0.7V mH20,airin = 0.00313g/s Water generated is mH20,gen = I M H 2 o
=
714.29 18.015=0.0667g/s 2x96,485
Water vapor in air out is mH2oi~'H2~
= [(S~ ro2,i - r~ n
] MH2~ 4F Pca -
Pvs(Tout,ca) I. ncell I APca - PvstTou,..,
Fuel Cell Mass Transport
First calculate the saturation pressure: T = 80~
= 353.15K, Pvs = 47.67kPa, Pca - APca = 101.325kPa --
mmoin, S~out,V-
I/2-0.2095) 18.015 47.67 500 ~ ' 1 0.2095 (4.96,485)(101.325-47.67) 0.7 mH2oin, airout -"
0.253g/s
The water balance is therefore" mH20_air_in 4- mH20_Inject 4- mH20__gen = m H 2 0 in air_out
0.003 13 + mH2oinject + 0.066 7 = 0.253 mH2oinject - - 0.223 17g/s Using MATLAB to solve:
]
103
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PEM Fuel Cell Modeling and Simulation Using MATLAB |
ii!iiii! ili!ii 84
EXAMPLE 5-2: Calculating Mass Flow Rates A PEM fuel cell with 100 cm ~ of active area is operating at 0.50 A/cm 2 at a voltage of 0.70. The operating temperature is 75 ~ and 1 atm with air supplied at a stoichiometric ratio of 2.5. The air is humidified by injecting hot water (75 ~ before the stack inlet. The ambient air conditions are 1 atm, 22 ~ and 70% relative humidity. Calculate the air flow rate, the amount of water required for 100% humidification of air at the inlet, and the heat required for humidification. The oxygen consumption is I No2
=
~
4F
0.50A/cm 2 x 100cm 2 =
4x96,485
= 0.129 x 10-3 mol/s
Fuel Cell Mass Transport
105
T h e oxygen flow rate at the cell inlet is No2 = SNo2,cons = 2.5 x 0.129 x 1 0 -3 - 0.324 x 10-3mol/s 1 Nai r - No2,act~
m a i r -- N a i r m a i r
-
ro~
0.324 x 10 -3 =
1.54 x 10 -3 mol/s
=
0.21
1.54 x 10-3mol/s x 28.85g/mol - 0.0445mol/s
T h e a m o u n t of w a t e r in air at the cell inlet w h e r e t p - 1 is
m H 2 0 -- X s m a i r
and
Xs =
mH20
Pvs
mair
P - Pvs
w h e r e pvs is the saturation pressure at 348.15 K, and P is the total pressure ( 101.325 kPa). Pvs =
eaT-l+b+cT+dT2+eT3+fln(T)
=
38.6kPa
Pvs _- 18 38.6 -- 0 . 3 8 4 g H 2 0 / g a i r mair P - Pvs 28.85 (101.325- 38.6)
Xs = m H 2 0
mH20 = Xsmair-
0.384gH20/gair X 0.128gair/S --0.0491gH20/S
T h e a m o u n t of w a t e r in a m b i e n t air at 70% RH and 295.15 K is Xs .
mH20 . mair
tppw . P - tPPvs
18 0.70 X 2.645 . . . 28.85101.325 - 0.7 x 2.645
0.011
gH20/gair
m H 2 0 = Xsmair -- 0 . 0 1 16gH20/gair X 0 . 1 2 8 g a i r / S -- 0 . 0 0 1 4 8 6 g H 2 0 / S
T h e a m o u n t of w a t e r needed for h u m i d i f i c a t i o n of air is m H 2 0 -- 0.0491 - 0.001486 = 0.047614gH20/S
T h e heat required for h u m i d i f i c a t i o n can be calculated from the heat balance. Hair, in 4- HH2o, in + Q = Hair, out
T h e e n t h a l p y of w e t / m o i s t air is hvair = Cp,airt + X(Cp,vt + hfg)
H u m i d i f i e d air:
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PEM Fuel Cell Modeling and Simulation Using MATLAB |
hvair- 1.01 x 75 + 0.384 x (1.87 x 75 + 2 5 0 0 ) = 1089.61J/g A m b i e n t air:
hvair = 1.01 x 22 + 0.011 6 x (1.87 x 22 + 2 5 0 0 ) - 51.70J/g Water: hH2o - 4.18 X 75 = 313.5J/g Q - 1089.61J/g x 0.128g/s - 51.70J/g x 0.128g/s - 313.5J/g x 0 . 0 1 5 7 g / s - 127.93w U s i n g M A T L A B to solve:
Fuel Cell Mass Transport
107
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PEM Fuel Cell Modeling and Simulation Using MATLAB |
5.3 Convective Mass Transport from Flow Channels to Electrode Figure 5-2 illustrates convective flow in the reactant flow channel and diffusive flow through the gas diffusion and catalyst layers. The reactant is supplied to the flow channel at a concentration Co, and it is transported from the flow channel to the concentration at the electrode surface, Cs, through convection. The rate of mass transfer is then: 1~ = Aelechm(Co - Cs)
(5-21)
where Aelec is the electrode surface area, and hm is the mass transfer coefficient. The value of hm is dependent upon the wall conditions, the channel geometry, and the physical properties of species i and j. Hm can be found from the Sherwood number: hm = Sh D~,j
(5-22)
Dh
where Sh is the Sherwood number, Dh is the hydraulic diameter, and Dij is the binary diffusion coefficient for species i and j. The Sherwood number depends upon channel geometry, and can be expressed as: Sh = hHDh k
(5-23)
where Sh = 5.39 for uniform surface mass flux (rh = constant), and Sh = 4.86 for uniform surface concentration (C~ = constant). The binary diffusion coefficient for hydrogen, oxygen, and water is given in Appendix G. If the binary diffusion coefficient needs to be calcu-
Fuel Cell Mass Transport
109
e m
T
Reactant Flow Channel
Gas Diffusion Layer
Convection
Diffusion
I
Electrolyte Layer Catalyst Layer (Carbon-Supported Catalyst) Diffusion and Reaction
FIGURE 5-2. Fuel cell layers (flow field, gas diffusion layer, catalyst layer) that have convective and diffusive mass transport. lated at a different temperature than what is shown in Appendix G, the following relation can be used: T ~3/2 Dj,j(T) = Di,j(Tref), ~ref j
(5-24)
110 PEM Fuel Cell Modeling and Simulation Using MATLAB | where T~ef is the temperature that the binary diffusion coefficient is given at, and T is the temperature of the fuel. There are several equations that are commonly used in the literature that govern the mass transfer to the electrode. One of these is the NernstPlanck equation. This equation for one-dimensional mass transfer along the x-axis is: Ji(x) ._ _Di o~Cxi,___________~~ Zi iF
~x
DiCi ..~.,x ,~d~l I 4- Civ(x )
RT
(5-25)
o~x
where Ji(x)is the flux of species i (mol/(s,cm ~) at a distance x from the surface, Di is the diffusion coefficient (cm2), 8Ci(x) is the concentration 8x gradient at distance x, 8q(x) is the potential gradient, zi and Ci are the 8x charge, and v(x) is the velocity (cm/s)with which a volume element in solution moves along the axis. The terms in Equation 5-25 represent the contributions to diffusion, migration, and convection, respectively, to the flUX.
5.4 Diffusive Mass Transport in Electrodes As shown in Figure 5-2, the diffusive flow occurs at the GDL and catalyst layer, where the mass transfer occurs at the microlevel. The electrochemical reaction in the catalyst layer can lead to reactant depletion, which can affect fuel cell performance through concentration losses. In turn, the reactant depletion will also cause activation losses. The difference in the catalyst layer reactant and product concentration from the bulk values determines the extent of the concentration loss. Using Fick's law, the rate of mass transfer by diffusion of the reactants to the catalyst layer (rh) can be calculated as shown in Equation 5-26: dC rh = - D ~ dx
(5-26)
where D is the bulk diffusion coefficient, and C is the concentration of reactants. The diffusional transport through the gas diffusion layer at steadystate is: rh = AdecDr
-Ci S
(5-27)
Fuel Cell Mass Transport
111
where C i is the reactant concentration at the GDL/catalyst interface, is the gas diffusion layer thickness, and D elf is the effective diffusion coefficient for the porous GDL, which is dependent upon the bulk diffusion coefficient D, and the pore structure. Assuming uniform pore size, and that gas diffusion layer is free from flooding of water, D elf can be defined as: D r = D~/2
(5-28)
where ~ is the electrode porosity. The total resistance to the transport of the reactant to the reaction sites can be expressed by combining Equations 5-21 and 5-27: Iil --
(
1
C~ - C i
S
/
(5-29)
hmAr162+ D r where
1
hmAelec
is the resistance to the convective mass transfer, and
L is the resistance to the diffusional mass transfer through the gas D elfAelec diffusion layer. When a fuel cell is started up, it begins producing electricity at a fixed current density, i. The reactant and product concentrations in the fuel cell are constant. When the current begins to be produced, the electrochemical reaction leads to the depletion of reactants at the catalyst layer. The flux of reactants and products will match the consumption/depletion rate of reactants and products at the catalyst layer, and can be described using the following equation: i=
nFm Aelec
(5-30)
where i is the fuel cell operating current density, F is the Faraday constant, n is the number of electrons transferred per mol of reactant consumed, and rh is the rate of mass transfer by diffusion of reactants to the catalyst layer. Substituting Equation 5-29 into Equation 5-30 yields: i =
-nF
Co - C i
The reactant concentration in the GDL/catalyst interface is less than the reactant concentration in the flow channels, which depends upon i, 6, and D elf. As the current density increases, the concentration losses become
112
PEM Fuel Cell Modeling and Simulation Using MATLAB |
greater. These concentration losses can be improved if the GDL thickness is reduced, or the porosity or effective diffusivity is increased. The limiting current density of the fuel cell occurs when the current density becomes so large the reactant concentration falls to zero. The limiting current density (iL)can be calculated if the minimum concentration at the GDL/catalyst layer interface is Ci = 0 as follows: C0
(5-32)
iL nr( m +
The limiting current density can be increased by insuring that Co is high through good flow field design, optimal GDL and catalyst layer porosity and thickness, and ideal operating conditions. The limiting current density is from 1 to 10 A/cm 2. The fuel cell cannot produce a higher current density than its limiting current density. However, the fuel cell voltage may fall to zero due to other types of losses before the limiting current density does. The Nernst equation introduced in Chapter 2 shows the relationship between the thermodynamic voltage of the fuel cell, and the reactant and product concentrations at the catalyst sites: E = E r - R T I n Hapir~ vi nF Hareactants
5
(-33)
In order to calculate the incremental voltage loss due to reactant depletion in the catalyst layer, the changes in Nernst potential using c~ values instead of c~ values are represented as follows: l)conc = Er, Nernst -- ENemst
l)conc =
Er
_ R T l n 1 _ ErIn nF Co nF
RT V~on~= ~ l n nF
Co Ci
(5-34) (5-351
(5-36)
where Er, Nerst is the Nernst voltage using Co values, and ENemst is the Nernst voltage using Ci values. Combining Equations 5-35 and 5-36: i = 1- C--L iL
(5-37)
Co
Therefore, the ratio C0/C~ (the concentration at the GDL/catalyst layer interface) can be written as:
Fuel Cell Mass Transport 113 Co
=
Ci
iL
iL -- i
(5-38)
Substituting Equation 5-38 into Equation 5-36 yields: 1)c~
RWln( iLI/ n--F-
iL - i
(5-39)
which is the expression for concentration losses, and is only valid for i < iL. The Butler-Volmer equation from Chapter 3 describes how the reaction kinetics affect concentration and fuel cell performance. The reaction kinetics are dependent upon the reactant and product concentrations at the reaction sites: *
c~
i = i 0co. cg exp(anFu~r
cO---;-exp(-(1- a)nFu~r
(5-40)
where c~ and c~ are arbitrary concentrations, and i0 is measured as the O* O* reference reactant and product concentration values CR and Cp . In the high current-density region, the second term in the Butler-Volmer equation drops out, and the expression then becomes: c__~ i = loco . exp (anFl)act/(RT)) .
(s-41)
In terms of activation overvoltage using c~ instead of c ~ l)c~
RT c ~ anF c~
(5-42)
The ratio can be written as: C~ ----;=
CR
iL
iL--i
(5-43)
The total concentration loss can be written as: =(RT~( 1 1)iL V~on~ \ - ~ J \ + iL--i
(5-44)
Fuel cell mass transport losses may be expressed by the following equation: Vconr = cln
iL iL--i
(5-45)
114 PEM Fuel Cell Modeling and Simulation Using MATLAB | where c is a constant and can have the approximate form: c=
W(5) nF
1+
(5-46)
In actuality, the fuel cell behavior often has a larger value than what the equation predicts. Due to this, c is often obtained empirically. The concentration losses appear at high current densities, and significant concentration losses can severely limit fuel cell performance.
5.5 Convective Mass Transport in Flow Field Plates The flow channels in fuel cell flow field plates are designed to evenly distribute reactants across a fuel cell to help keep mass transport losses to a minimum. Flow field designs are discussed in detail in Chapter 10. In the next section, the control volume method first introduced in Section 5.2 is used to calculate the mass transfer rates in the flow channels. 5.5.1 Mass T r a n s p o r t in Flow C h a n n e l s The mass transport in flow channels can be modeled using a control volume for reactant flow from the flow channel to the electrode layer as shown in Figure 5-3. The rate of convective mass transfer at the electrode surface (rh~) can be expressed as: rils = hm(Cm - Cs)
(5-47)
where Cm is the mean concentration of the reactant in the flow channel (averaged over the channel cross-section, and decreases along the flow direction, x), and Cs is the concentration at the electrode surface. As shown in Figure 5-3, the reactant moves at the molar flow rate, Ar at the position x, where Ac is the channel cross-sectional area and Vm is the mean flow velocity in the flow channel. This can be expressed as: - ~ ( A c C m v m ) - --rnsWelec
(5-48)
where Welec is the width of the electrode surface. If the flow in the channel is assumed to be steady, and the velocity and the concentration are constant, then:
d
C m ----
_rhs VmWflow
(5-49)
Fuel Cell Mass Transport
115
eFlow Channel Width, Wflow
I
dx
Gas Diffusion Layer
] Electrolyte Layer
Reactant Flow Channel Catalyst Layer FIGURE 5-3. Control volume for reactant flow from the flow channel to the electrode layer.
When the current density is small (i < 0.5 iL), it can be assumed i constant. Using Faraday's law, rh~ = ~ and integrating: nF
(i/
Cm(X ) = Cm,in(X)- ~ ~ F x VmWflow
(5-50)
where Cm,in is the m e a n concentration at the flow channel inlet. If the current density is large (i > 0.5 iL), the condition at the electrode surface can be approximated by assuming the concentration at the surface (Cs) is constant. This can be w r i t t e n as follows:
116 PEM Fuel Cell Modeling and Simulation Using MATLAB |
d(Cm
Cs) - h ~ m (Cm -- Cs) i
-
-
(5-51)
VmWflow
After integrating from the channel inlet to location x in the flow channel, Equation 5-51 becomes" Cm -Cs ( C m _ Cs)i n --
-hmx eXP~vmWflow
(5-52)
At the channel outlet, x = H, and Equation 5-52 becomes" Cm, out - C s Cm, in - C s
-hmH
= exp~
(5-53)
Vm Wflow
where Cm,outis the mean concentration at the flow channel outlet. A simple expression can be derived if the entire flow channel is assumed to be the control volume, as shown in Figure 5-4: r n s = VmWflowWelec(Cin - Cout)
(5-54)
r n s - VmWflowWelec(ACi n - ACout)
If C~ is constant, substituting for WflowWelec" 1~ s -- A h m A C 1 m
15-55)
AClm = ACin - AC~
(5-56)
where
ln( ACin ACout ) The local current density corresponding to the rate of mass transfer is: i(x) - nFhm(Cm- Cs)exp / VmWflo -hmxw /
(5-57)
The current density averaged over the electrode surface is: i = nFhmAClm
(5-58)
The limiting current density when Cs approaches 0 is" iL(X)= nFhmCmin , exp/ VmWflo -hmxw /
15-591
Fuel Cell Mass Transport
117
e_ Flow Channel Width, Wnow
Hydrogen
Gas Diffusion Layer
Electrolyte Layer
Reactant Flow Channel Catalyst Layer FIGURE 5-4. Entire channel as the control volume for reactant flow from the flow channel to the electrode layer.
= nFhm ACin - AC~ ln( ACm )
(5-60)
ACout
As seen from Equations 5-57 to 5-60, both the current density and limiting current density decrease exponentially along the channel length.
118
PEM Fuel Cell Modeling and Simulation Using M A T L A B |
EXAMPLE 5-3: Determine Current Density Distribution A fuel cell operating at 25 ~ and 1 atm uses bipolar plates with flow fields to distribute the fuel and oxidant to the electrode surface. The channels have a depth of 1.5 mm, with a distance of 1 m m apart. Air is fed parallel to the channel walls for distribution to the cathode electrode. The length of the flow channel is 18 cm, and the air travels at a velocity of 2 m/s. Determine the distribution of the current density due to the limitation of the convective mass transfer iL(X) and the average limiting current density iL. The Reynolds number can be calculated as follows: Re - pvmD _- ~vmD _- 2m/s * 2 * 1 x 10-3m -- 251 .73 p v 15.89 x 10-6m 2/s Since 251.73 is less than 2000, the flow is laminar. In order to calculate the limiting current density, the convective mass transfer coefficient and binary diffusivity coefficient need to be calculated:
( T ~3/2
Di,ilT) = Di, i(Tref)*k.~ref )
(298 ~s/2
- DO2-N21273)*
~-~)
= 0.21 x 10-4m2/s
hm = Sh Di,j _ 4.86, 0.21 x 10-4m2/s _ 0.0105m/s Dh 2"1 x 10-3m The concentration of 02 at the channel inlet, with a mole fraction of 02 is Xo2 = 0.21, is:
Co2,in- Xo2 *(-~T) - 0.21"
101,325 = 8.588mol/m 3 8.314J/molK 9 298K
The limiting current density based upon the rate of 02 transfer:
iL(X)= nFhmCmin , exp / VmWflo -hmx w / iL(X) = 4*96,487
C molO 2 *0.0105 m *8.588 * molO~ s m 3
exp (l*~*zm/0"0105m/s)~--1--;-~_~,x----,s 3.4802.1 04 x exp(-5.25x) The limiting current density will be 3.4802 A/cm ~ at the channel inlet (x = 0) and 3.3022 A/cm ~ at the channel outlet (x - 18 cm). In order
Fuel Cell Mass Transport
to calculate calculated:
iL,
the outlet concentration of oxygen needs to be
Cmout=Cmin*exp( ) , VmWflow h-mx = 8.588
mOlm 3
*exp
(-0.0105m/s*0.18m)l 91-~* ~Z-ln)S
= 3.338mol/m 3 Then,
lr = nFhm
119
ACin - ACout
In
ACout,]
4 96,487 C = * ~*0.0105mo102 A = 2.251~
cm2
Using MATLAB to solve:
(8.588-3.338) m~ ln()-8"588\ 3.338
120 PEM Fuel Cell Modeling and Simulation Using M A T L A B |
5.6 Mass Transport Equations in the Literature There are many equations used in the literature to determine mass flux and concentration losses. Knowing which equation to use is not as important as determining how to model the system effectively. Knowing how to solve the concentration gradients and species distributions requires knowledge of multicomponent diffusion, and can be a challenging task. In order to precisely solve the mass balance equations (especially in the electrode layers), the mass flux must be determined. The concentration losses are incorporated into a model as the reversible potential decreases due to a decrease in the reactant's partial pressure. There are three basic approaches for determining the mass flux (N): Fick's law, the StefanMaxwell equation, and the Dusty Gas Model.
Fuel Cell Mass Transport
121
5.6.1 Fick's Law The simplest diffusion model is Fick's law, which is used to describe diffusion processes involving two gas species. A form of Fick's law was introduced in Equation 5-26. The standard notation for Fick's law is the binary notation, and can be written as: (5-61)
Ni = -cDi, jVXi
A multicomponent version of Fick's law is shown in Equation 5-62: n
Ni = -cDi, mVXi +
Xi~Nj
(5-62)
j=l
where c is the total molar concentration. If three or more gas species are present, such as N2, 02, and H20, a multicomponent diffusion model such as the Maxwell-Stefan equation must be used, or the binary diffusion coefficients must be expanded to tertiary diffusion coefficients.
5.6.2 The Stefan-Maxwell Equation The Stefan-Maxwell equation is the only diffusion equation that separates diffusion from convection in a simple way. The flux equation is replaced by the difference in species velocities. The Stefan-Maxwell model is more rigorous, is commonly used in multicomponent species systems, and is employed quite extensively in the literature. The main disadvantage is that it is difficult to solve mathematically. It may be used to define the gradient in the mole fraction of components: Vyi = R T ~ y~Nj- yjNi pDe~f
(5-63)
where yi is the gas phase mol fraction of species i, and Ni is the superficial gas phase flux of species i averaged over a differential volume element, which is small with respect to the overall dimensions of the system, but large with respect to the pore size. De(f is the binary diffusion coefficient, and can be defined by: D~ff =
:/
T ~/TciTci,,
/b
/1 /
1 v2 (Pc'JPc'J)V3(Tc'iTc'i)s/12 +~Mj /~l.S
{5-64)
where Tc and pc are the critical temperature and pressure of species i and j, M is the molecular weight of species, A = 0.0002745 for diatomic gases, H2, 02, and N2, and a = 0.000364 for water vapor, and B = 1.832 for diatomic
122 PEM Fuel Cell ModeBng and Simulation Using MATLAB | gases, H2, 02, and N2, and b = 2.334 for water vapor. The Stefa-Maxwell Equation is discussed in more detail in Chapter 8. 5.6.3 The D u s t y Gas Model The Dusty Gas Model is also commonly used in the literature, and looks similar to the Stefan-Maxwell equation except that it also takes into account Knudsen diffusion. Knudsen diffusion occurs when a particle's meanfree-path is similar to, or larger than in size, the average pore diameter (and is discussed in greater detail in Chapter 8): ~VX i
=
Ni Di, k
+
+ z_,
XjNi-
j=l,i~l
XiNj
(5-651
cDi, i
where D~,i is the Knudsen diffusion coefficient for species i. The molecular diffusivity depends upon the temperature, pressure, and concentration. The effective diffusivity depends also upon the microstructural parameters such as porosity, pore size, particle size, and tortuosity. The molecular gas diffusivity must be corrected for the porous media. A large portion of the corrections are made using the ratio of porosity to tortuosity (E/T), although in some cases, the Bruggman model is used due to the lack of information for gas transport in porous media: fc'~
D eff ~/D t,1 "-\TJ
i,j
13eft = E I ' S D .-
~--"i,j
1,1
(5-66)
The Dusty Gas diffusion model requires Knudsen diffusivity to be solved, while Fick's law and the Stefan-Maxwell equation require more work to incorporate Knudsen diffusion. The Knudsen diffusion coefficient for gas species i can be calculated using Equation 5-67: 2T [8RT Dj,k = -~-~/~-[
(5-67)
where M is the molecular mass of species i, and r is the average pore radius.
EXAMPLE 5-4: Calculating the Diffusive Mass Flux Hydrogen gas is maintained at 2 bars and 1 bar on opposite sides of a Nation membrane that is 50 microns (/am) thick. The temperature is 20~ and the binary diffusion coefficient of hydrogen in Nation is 8.7 x 10-8 m2/s. The solubility of hydrogen in the membrane is 1.5 x 10-3 kmol/m 3 bar. What is the mass diffusive flux of hydrogen through
Fuel Cell Mass Transport
123
t h e m e m b r a n e ? W h a t are t h e m o l a r c o n c e n t r a t i o n s of h y d r o g e n i n t h e gas p h a s e ? First, c a l c u l a t e t h e s u r f a c e m o l a r c o n c e n t r a t i o n s of h y d r o g e n :
C A , sl = S * p A =
1.5 x 10 -a k m o l x 3 b a r s = 4.5 x 10 -3 k m o l m 3
= 1.5xl
C A sSl* = pA,
m 3
0_3 k m o l ma x l b a r s = 1 . 5 x 1 0 -3km~
C a l c u l a t e t h e m o l a r d i f f u s i v e flux: DAB
NA -- T ( C A ,
sl - C A , s2)
8.7 x 10 4 N A = 0.3 x 10 -3 (4.5 x 10 -3 - 1.5 x 10 -3) = 8.7 x 10 -~ k mm~~ s
C o n v e r t to a m a s s basis: nA = NA * MA - 8 . 7 X 10 -z k m o l . 2 k~g = 1 . 7 4 x 1 0 - 6 k g m ~ s kmol g C a l c u l a t e t h e m o l a r c o n c e n t r a t i o n s of h y d r o g e n in t h e gas phase"
C A __
p___~= 3 = 0.121~k-m~ RT 8.314 x 10 -9. 9 2 9 3 . 1 5 m3
C c = PA = 1 = 0 . 0 4 0 kmo_____~l RT 8.314x10 -~,293.15 m 3 U s i n g M A T L A B to solve:
ii(~ !i~.......~'
~i~ ..........i:i~.........
......... ..............9............~'~'~ . ~...................... ' <<~ ~'~,~~ .....,~'~,!?~!~v~% .. ~
ii! i
~iiiiii~'i;ii!i~i~i!iiii i~i;i~iiii~i~iiiilil~iii~i~ililiii~i~i'~ii~i;~i~)~~iii~i~i~!!
!
~i~i.!~i~iii~i~i~,;i~i~i~iiiii~ ~~:i~i:i;~i;i~ii~iii!i~!~ii~i~ ~G ;i~i~!i!~i!~ii~Jii~ii~i!~i~iiiii!i~iiii~(ii ~i~i~i~i~i!~i~i?~~~~i!~i~i~i~i~i~i~
ii
124 PEM Fuel Cell Modeling and Simulation Using MATLAB |
Chapter Summary The study of mass transport involves the supply of reactants and products in a fuel cell. Inadequate mass transport can result in poor fuel cell performance. In order to calculate the mass flows through the fuel cell, mass balances can be written to calculate the ideal flow rates and mole fractions of any unknown species. There are two main mass transport effects encountered in fuel cells: convection in the flow structures, and diffusion in the electrodes. Convective flow occurs in the flow channels due to hydrodynamic transport, and the relatively large-size channels (--1 mm to 1 cmJ. Diffusive transport occurs in the electrodes because of the tiny pore sizes. Mass transport losses in the fuel cell result in the depletion of reactants at the electrode, which affects the Nernstian cell voltage and the reaction rate. Commonly used mass transport equations in the literature include Fick's law, the Stefan-Maxwell equation and the Dusty Gas Model. This chapter provided the necessary background to create mass balances on any fuel-cell component with convective or diffusive transport.
Problems 9 A fuel cell is operating at 50~ and 1 atm. Humidified air is supplied with the mole fraction of water vapor equal to 0.2 in the cathode. If the
Fuel Cell Mass Transport
~ 9
9
9
125
channels are rectangular with a diameter of 1.2 mm, find the m a x i m u m velocity of air. For the fuel cell in the above problem, calculate the m a x i m u m velocity of air if the channels are circular. A fuel cell is operating at 50~ and 1 atm. The cathode is using pure oxygen, and there is no water vapor present. The diffusion layer is 400 microns with a porosity of 30%. Calculate the limiting current density. Calculate the limiting current density for a fuel cell operating at 80~ and 1 atm. The cathode is of the same construction as in the third problem. Under the conditions from the third problem, estimate the fuel cell area that can be operated at 0.7 A/cm 2. Assume a stoichiometric number of 2.5, and that the fuel cell is made of a single straight channel with a width of 1 m m and the rib width is 0.5 ram.
Bibliography Barbir, F. PEM Fuel Cells: Theory and Practice. 2005. Burlington, MA: Elsevier Academic Press. Beale, S.B. Calculation procedure for mass transfer in fuel cells. J. Power Sources. Vol. 128, 2004, pp. 185-192. Lin, B. 1999. Conceptual design and modeling of a fuel cell scooter for urban Asia. Princeton University, masters thesis. Li, X. Principles of Fuel Cells. 2006. New York: Taylor & Francis Group. Mench, M.M., C.-Y. Wang, and S.T. Tynell. An Introduction to Fuel Cells and Related Transport Phenomena. Department of Mechanical and Nuclear Engineering, Pennsylvania State University. Draft. Available at: http://mtrll.mne.psu .edu/Document/jtpoverview.pdf Accessed March 4, 2007. Mench, M.M., Z.H. Wang, K. Bhatia, and C.Y. Wang. 2001. Design of a Micro-Direct Methanol Fuel Cell. Electrochemical Engine Center, Department of Mechanical and Nuclear Engineering, Pennsylvania State University. Mennola T., et al. Mass transport in the cathode of a free-breathing polymer electrolyte membrane fuel cell. J. Appl. Electrochem. Vol. 33, 2003, pp. 979-987. O'Hayre, R., S.-W. Cha, W. Colella, and F.B. Prinz. 2006. Fuel Cell Fundamentals. New York: John Wiley & Sons. Rowe, A., and X. Li. Mathematical modeling of proton exchange membrane fuel cells. J. Power Sources. Vol. 102, 2001, pp. 82-96. Sousa, R., Jr., and E. Gonzalez. Mathematical modeling of polymer electrolyte fuel cells. J. Power Sources. Vol. 147, 2005, pp. 32-45. Springer et al. Polymer electrolyte fuel cell model. J. Electrochem. Soc. Vol. 138, No. 8, 1991, pp. 2334-2342. You, L., and H. Liu. A two-phase flow and transport model for PEM fuel cells. J. Power Sources. Vol. 155, 2006, pp. 219-230.
Fuel Cell Mass Transport 5.1 Introduction In order to produce electricity, a fuel cell must be supplied continuously with fuel and oxidant. In addition, product water must be removed continually to insure proper fuel and oxidant at the catalyst layers to maintain high fuel cell efficiency. Voltage losses occur in the fuel cell due to activation losses (Chapter 3), ohmic losses (Chapter 4), and mass transport limitations~which is the topic of this chapter. Mass transport is the study of the flow of species, and can significantly affect fuel cell performance. Losses due to mass transport are also called "concentration losses," and can be minimized by optimizing mass transport in the flow field plates, gas diffusion layer, and catalyst layer. This chapter covers both the macro and micro aspects of mass transport. The specific topics to be covered are: 9 9 9 9 9
Fuel cell mass balances Convective mass transport from flow channels to electrode Diffusive mass transport in electrodes Convective mass transport in flow field plates Mass transport equations in the literature
In conventional fuel cells, the flow field plates have channels with dimensions in millimeters or centimeters. Due to the size of these channels, mass transport is dominated by convection and the laws of fluid dynamics. Convection is the movement of fluid flow due to density gradients or hydrodynamic transport, and is characterized by laminar or turbulent flow and stagnant regions. This type of flow dominates mass transfer in the flow channels. High fuel and oxidant flow rates sometimes insure good distribution of reactants, but if the flow rate is too high, the fuel may move too fast to diffuse through the GDL and catalyst layers. In addition, delicate fuel cell components such as the membrane can rupture. Mass transport in the fuel cell GDL and catalyst layers is dominated by diffusion due to the tiny pore sizes of these layers (4 to 10 microns). In
98
PEM Fuel Cell Modeling and Simulation Using MATLAB |
a flow channel, the velocity of the reactants is usually slower near the walls; therefore, this aids the flow change from convective to diffusive. The mass transport theory described in this chapter will help the reader to write mass balances, predict fuel cell flow rates, and calculate the mass transfer in the flow channels, electrodes, and membrane.
5.2 Fuel Cell Mass Balances Before convective and diffusive flows are covered, the overall mass flows through the fuel cell need to be discussed. These flow calculations, or mass balances, are critical for determining the correct flow rates for a fuel cell. In order to properly determine these mass flow rates, the mass that flows into and out of each process unit (or control volume)in the fuel cell subsystems, stack, or fuel cell layer need to be accounted for. The procedure for formulating a mass balance can be applied to any type of system, and is as follows: 1. A flow diagram must be drawn and labeled. Enough information should be included on the flow diagram to have a summary of each stream in the process. This includes known temperatures, pressures, mole fractions, flow rates, and phases. 2. The appropriate mass balance equation(s) must be written in order to determine the flow rates of all stream components and to solve for any desired quantities. An example flow diagram is shown in Figure 5-1. Hydrogen enters the cell at temperature, T, and pressure, P, with the mass flow rate, mw. Oxygen enters the fuel cell from the environment at a certain T, P, and mo2. The hydrogen and oxygen react completely in the cell to produce
Work,
H2(g) at T, P, mH2
Wel
t
~[~i~~ .:~:~~ ,:~~ .l
~i~.'~-::.. ~::~~.~:~ .,...-.,~!~:-'<.'~"~:~:.~y.:~ .,'.'..:~i~ O2(g) at T, P, mo2
H20(1) at T, P, mH20
.._Ii;~i;il~l~ ~. ~,~ ~.,..~
..~..
, ~ ' . ~ , ~ : ~ i t .~:~.,.,.:~4~.
FIGURE 5-1. Detailed flowchart to obtain mass balance equation.
Fuel Cell Mass Transport 99
water, which exits at a certain T, P, and mmo. This reaction can be described by: 2H21g) + O2(g)-+ 2H20(1)
Wel in Figure 5-1 is the work available through chemical availability. The generic mass balance for the fuel cell in this example is: mH2 + m o 2 - mH2o + Wel
15-11
The formal definition for material balances in a system (or control volume) can be written as: Input (enters through system boundaries) + Generation (producted within the system) -
Output (leaves through system boundaries)
- Consumption (consumed within the system) = Accumulation (buildup within the system) Generally, the fuel cell mass balance requires that the sum of all of the mass inputs is equal to the mass outputs, which can be expressed as: E(mi)in = E(mi)out
(5-2)
where i is the mass going into and out of the cell, and can be any species, including hydrogen, oxygen, and water. The flow rates at the inlet are proportional to the current and number of cells. The cell power output is: Wel = ncellVcellI
(5-3)
where ncen is the number of cells, Vce11is the cell voltage, and I is the current. All of the flows are proportional to the power output and inversely proportional to the cell voltage: I- ncell = Wel Vcell
(5-4}
The inlet flow rates for a PEM fuel cell are as follows: The hydrogen mass flow rate is
mH2,in = SH2 MH~ I'ncell 2F
15-5)
100 PEM Fuel Cell Modeling and Simulation Using MATLAB |
The oxygen mass flow rate (g/s) is
(5-61
m o 2 , i n - So2 M o 2 I'ncell
4F
The air mass flow rate (g/s) is mair, in =
SO2 Mair
ro2 4F
I'ncdl
(5-7)
The nitrogen mass flow rate (g/s) is mN2,in -- 802 MN-------!21 - ro2,in I. ncell 4F ro2,i n
(5-8)
Water vapor in the hydrogen inlet is MH20 mH2OinH2,in -- SH2 ~
2F
(PanPvs(Tan,in)
I. ncell
(5-9)
I. ncell
(5-10)
I'ncell
(5 11)
Pan - (Pan Pvs(Tan, in)
Water vapor in the oxygen inlet is mH2oinO2,in = 8o2
MH20
~OcaPvs(Tan,in)
4F
Pea - q)caPvs(Tan, in)
Water vapor in the air inlet (g/s) is mH2oinairin :
O P,,
So2 MH20
ro2
4F
--r
inl
Pea -- (/0caPvs(Tan,in)
The outlet flow rates for a PEM fuel cell are as follows"
The unused hydrogen flow rate is mH2,out -- (SH2 --
1)MH2 I" ncell 2F
(5-112,)
The oxygen flow rate at the outlet is equal to the oxygen supplied at the inlet minus the oxygen consumed in the fuel electrochemical reaction" mo2,out - (So2 - 1)Mo2 I" ncell
4F
(5-13)
The nitrogen flow rate at the exit is the same as the inlet because nitrogen does not participate in the fuel cell reaction:
Fuel Cell Mass Transport
mN2,out = mN2in = 802
MN21
-
4F
rozin
101
(5-14)
I. ncell
ro~in
The depleted air flow rate is then simply a sum of the oxygen and nitrogen flow rates: mair,out-
I(So2-1)Moz
+Sozr~1-I'ncellro2in MN2] 4F
(5-15)
The oxygen volume fraction at the outlet is much lower than the inlet volume fraction" 802 -- 1 S~ 1
r~176
15-16)
ro2,in
The additional outlet liquid and vapor water flow rates and balances for a PEM fuel cell are described by equations 5-17 through 5-20: The water vapor content at the anode outlet is the smaller of the total water flux:
[mH2oin, H2out,V =
t
I. ncell, mH2oin, H2out 1/
MH2O
P~(Wou~,an~
2F
Pan -- aPan - P~s~Wout,a~
mini(sin - 1)
/
(5-17)
APanis the pressure drop on the anode side. The amount of liquid water is the difference between the total water present and the water vapor:
where
MH2Oin,H2out,L = mH2oin, H2out
-
-
(5-18)
mH2oin, H2out,V
Water content in the cathode exhaust is equal to the amount of water brought into the cell, plus the water generated in the cell, along with the water transported across the membrane: mH2OinAirout ---- mH2OinAirin 4- mH2Ogen 4- mH2OED -- mH2OBD
(5-19)
The water vapor content at the cathode outlet is:
mH2oin'air~
= min I/8~
ro2i-, r~n
/ MH2~ 4F
Pvs(Tout, an}
P c a - APca -
PvslTout,anl
1
ncell' mH2oin'Air~ /
.J
(5-20)
102 PEM Fuel Cell Modeling and Simulation Using MATLAB |
EXAMPLE 5-1: Water Injection Flow Rate A hydrogen-air PEM fuel cell generates 500 watts (W) at 0.7 V. Dry hydrogen is supplied in a dead-end mode at 20 ~ The relative h u m i d i t y of the air at the fuel cell inlet is 50% at a pressure of 120 kPa. Liquid water is injected at the air inlet to help cool the fuel cell. The oxygen stoichiometric ratio is 2, and the outlet air is 100% saturated at 80~ and atmospheric pressure. What is the water injected flow rate (g/s)? The water mass balance is mH20_air_in + mH20_Inject + mH20_gen = mH20 in air_out
In order to calculate the a m o u n t of water in air, the saturation pressure needs to be calculated. To calculate the saturation pressure (in Pa)for any temperature between 0~ and 100~ 9 Pvs = eaT-1+b+cT+dT2+eT3+fln(T)
where a, b, c, d, e, and f are the coefficients. a =-5800.2206, b = 1.3914993, c =-0.048640239, d = 0.41764768 x 10-4, e =-0.14452093 x 10-7, and f = 6.5459673 with T = 293.15, pvs = 2.339 kPa. The a m o u n t of water in air can be calculated: mH20,airi n =
So2 MH2O
to2
nF
r162Pvs(Tca,in)
I 9ncell
P c a -- r
2 18.015 0.50*2.339 500W mH20,airin- 0.2095 (4,96,485As/mol) 120kPa - 0.50,2.339 0.7V mH20,airin = 0.00313g/s Water generated is mH20,gen = I M H 2 o
=
714.29 18.015=0.0667g/s 2x96,485
Water vapor in air out is mH2oi~'H2~
= [(S~ ro2,i - r~ n
] MH2~ 4F Pca -
Pvs(Tout,ca) I. ncell I APca - PvstTou,..,
Fuel Cell Mass Transport
First calculate the saturation pressure: T = 80~
= 353.15K, Pvs = 47.67kPa, Pca - APca = 101.325kPa --
mmoin, S~out,V-
I/2-0.2095) 18.015 47.67 500 ~ ' 1 0.2095 (4.96,485)(101.325-47.67) 0.7 mH2oin, airout -"
0.253g/s
The water balance is therefore" mH20_air_in 4- mH20_Inject 4- mH20__gen = m H 2 0 in air_out
0.003 13 + mH2oinject + 0.066 7 = 0.253 mH2oinject - - 0.223 17g/s Using MATLAB to solve:
]
103
104
PEM Fuel Cell Modeling and Simulation Using MATLAB |
ii!iiii! ili!ii 84
EXAMPLE 5-2: Calculating Mass Flow Rates A PEM fuel cell with 100 cm ~ of active area is operating at 0.50 A/cm 2 at a voltage of 0.70. The operating temperature is 75 ~ and 1 atm with air supplied at a stoichiometric ratio of 2.5. The air is humidified by injecting hot water (75 ~ before the stack inlet. The ambient air conditions are 1 atm, 22 ~ and 70% relative humidity. Calculate the air flow rate, the amount of water required for 100% humidification of air at the inlet, and the heat required for humidification. The oxygen consumption is I No2
=
~
4F
0.50A/cm 2 x 100cm 2 =
4x96,485
= 0.129 x 10-3 mol/s
Fuel Cell Mass Transport
105
T h e oxygen flow rate at the cell inlet is No2 = SNo2,cons = 2.5 x 0.129 x 1 0 -3 - 0.324 x 10-3mol/s 1 Nai r - No2,act~
m a i r -- N a i r m a i r
-
ro~
0.324 x 10 -3 =
1.54 x 10 -3 mol/s
=
0.21
1.54 x 10-3mol/s x 28.85g/mol - 0.0445mol/s
T h e a m o u n t of w a t e r in air at the cell inlet w h e r e t p - 1 is
m H 2 0 -- X s m a i r
and
Xs =
mH20
Pvs
mair
P - Pvs
w h e r e pvs is the saturation pressure at 348.15 K, and P is the total pressure ( 101.325 kPa). Pvs =
eaT-l+b+cT+dT2+eT3+fln(T)
=
38.6kPa
Pvs _- 18 38.6 -- 0 . 3 8 4 g H 2 0 / g a i r mair P - Pvs 28.85 (101.325- 38.6)
Xs = m H 2 0
mH20 = Xsmair-
0.384gH20/gair X 0.128gair/S --0.0491gH20/S
T h e a m o u n t of w a t e r in a m b i e n t air at 70% RH and 295.15 K is Xs .
mH20 . mair
tppw . P - tPPvs
18 0.70 X 2.645 . . . 28.85101.325 - 0.7 x 2.645
0.011
gH20/gair
m H 2 0 = Xsmair -- 0 . 0 1 16gH20/gair X 0 . 1 2 8 g a i r / S -- 0 . 0 0 1 4 8 6 g H 2 0 / S
T h e a m o u n t of w a t e r needed for h u m i d i f i c a t i o n of air is m H 2 0 -- 0.0491 - 0.001486 = 0.047614gH20/S
T h e heat required for h u m i d i f i c a t i o n can be calculated from the heat balance. Hair, in 4- HH2o, in + Q = Hair, out
T h e e n t h a l p y of w e t / m o i s t air is hvair = Cp,airt + X(Cp,vt + hfg)
H u m i d i f i e d air:
106
PEM Fuel Cell Modeling and Simulation Using MATLAB |
hvair- 1.01 x 75 + 0.384 x (1.87 x 75 + 2 5 0 0 ) = 1089.61J/g A m b i e n t air:
hvair = 1.01 x 22 + 0.011 6 x (1.87 x 22 + 2 5 0 0 ) - 51.70J/g Water: hH2o - 4.18 X 75 = 313.5J/g Q - 1089.61J/g x 0.128g/s - 51.70J/g x 0.128g/s - 313.5J/g x 0 . 0 1 5 7 g / s - 127.93w U s i n g M A T L A B to solve:
Fuel Cell Mass Transport
107
108
PEM Fuel Cell Modeling and Simulation Using MATLAB |
5.3 Convective Mass Transport from Flow Channels to Electrode Figure 5-2 illustrates convective flow in the reactant flow channel and diffusive flow through the gas diffusion and catalyst layers. The reactant is supplied to the flow channel at a concentration Co, and it is transported from the flow channel to the concentration at the electrode surface, Cs, through convection. The rate of mass transfer is then: 1~ = Aelechm(Co - Cs)
(5-21)
where Aelec is the electrode surface area, and hm is the mass transfer coefficient. The value of hm is dependent upon the wall conditions, the channel geometry, and the physical properties of species i and j. Hm can be found from the Sherwood number: hm = Sh D~,j
(5-22)
Dh
where Sh is the Sherwood number, Dh is the hydraulic diameter, and Dij is the binary diffusion coefficient for species i and j. The Sherwood number depends upon channel geometry, and can be expressed as: Sh = hHDh k
(5-23)
where Sh = 5.39 for uniform surface mass flux (rh = constant), and Sh = 4.86 for uniform surface concentration (C~ = constant). The binary diffusion coefficient for hydrogen, oxygen, and water is given in Appendix G. If the binary diffusion coefficient needs to be calcu-
Fuel Cell Mass Transport
109
e m
T
Reactant Flow Channel
Gas Diffusion Layer
Convection
Diffusion
I
Electrolyte Layer Catalyst Layer (Carbon-Supported Catalyst) Diffusion and Reaction
FIGURE 5-2. Fuel cell layers (flow field, gas diffusion layer, catalyst layer) that have convective and diffusive mass transport. lated at a different temperature than what is shown in Appendix G, the following relation can be used: T ~3/2 Dj,j(T) = Di,j(Tref), ~ref j
(5-24)
110 PEM Fuel Cell Modeling and Simulation Using MATLAB | where T~ef is the temperature that the binary diffusion coefficient is given at, and T is the temperature of the fuel. There are several equations that are commonly used in the literature that govern the mass transfer to the electrode. One of these is the NernstPlanck equation. This equation for one-dimensional mass transfer along the x-axis is: Ji(x) ._ _Di o~Cxi,___________~~ Zi iF
~x
DiCi ..~.,x ,~d~l I 4- Civ(x )
RT
(5-25)
o~x
where Ji(x)is the flux of species i (mol/(s,cm ~) at a distance x from the surface, Di is the diffusion coefficient (cm2), 8Ci(x) is the concentration 8x gradient at distance x, 8q(x) is the potential gradient, zi and Ci are the 8x charge, and v(x) is the velocity (cm/s)with which a volume element in solution moves along the axis. The terms in Equation 5-25 represent the contributions to diffusion, migration, and convection, respectively, to the flUX.
5.4 Diffusive Mass Transport in Electrodes As shown in Figure 5-2, the diffusive flow occurs at the GDL and catalyst layer, where the mass transfer occurs at the microlevel. The electrochemical reaction in the catalyst layer can lead to reactant depletion, which can affect fuel cell performance through concentration losses. In turn, the reactant depletion will also cause activation losses. The difference in the catalyst layer reactant and product concentration from the bulk values determines the extent of the concentration loss. Using Fick's law, the rate of mass transfer by diffusion of the reactants to the catalyst layer (rh) can be calculated as shown in Equation 5-26: dC rh = - D ~ dx
(5-26)
where D is the bulk diffusion coefficient, and C is the concentration of reactants. The diffusional transport through the gas diffusion layer at steadystate is: rh = AdecDr
-Ci S
(5-27)
Fuel Cell Mass Transport
111
where C i is the reactant concentration at the GDL/catalyst interface, is the gas diffusion layer thickness, and D elf is the effective diffusion coefficient for the porous GDL, which is dependent upon the bulk diffusion coefficient D, and the pore structure. Assuming uniform pore size, and that gas diffusion layer is free from flooding of water, D elf can be defined as: D r = D~/2
(5-28)
where ~ is the electrode porosity. The total resistance to the transport of the reactant to the reaction sites can be expressed by combining Equations 5-21 and 5-27: Iil --
(
1
C~ - C i
S
/
(5-29)
hmAr162+ D r where
1
hmAelec
is the resistance to the convective mass transfer, and
L is the resistance to the diffusional mass transfer through the gas D elfAelec diffusion layer. When a fuel cell is started up, it begins producing electricity at a fixed current density, i. The reactant and product concentrations in the fuel cell are constant. When the current begins to be produced, the electrochemical reaction leads to the depletion of reactants at the catalyst layer. The flux of reactants and products will match the consumption/depletion rate of reactants and products at the catalyst layer, and can be described using the following equation: i=
nFm Aelec
(5-30)
where i is the fuel cell operating current density, F is the Faraday constant, n is the number of electrons transferred per mol of reactant consumed, and rh is the rate of mass transfer by diffusion of reactants to the catalyst layer. Substituting Equation 5-29 into Equation 5-30 yields: i =
-nF
Co - C i
The reactant concentration in the GDL/catalyst interface is less than the reactant concentration in the flow channels, which depends upon i, 6, and D elf. As the current density increases, the concentration losses become
112
PEM Fuel Cell Modeling and Simulation Using MATLAB |
greater. These concentration losses can be improved if the GDL thickness is reduced, or the porosity or effective diffusivity is increased. The limiting current density of the fuel cell occurs when the current density becomes so large the reactant concentration falls to zero. The limiting current density (iL)can be calculated if the minimum concentration at the GDL/catalyst layer interface is Ci = 0 as follows: C0
(5-32)
iL nr( m +
The limiting current density can be increased by insuring that Co is high through good flow field design, optimal GDL and catalyst layer porosity and thickness, and ideal operating conditions. The limiting current density is from 1 to 10 A/cm 2. The fuel cell cannot produce a higher current density than its limiting current density. However, the fuel cell voltage may fall to zero due to other types of losses before the limiting current density does. The Nernst equation introduced in Chapter 2 shows the relationship between the thermodynamic voltage of the fuel cell, and the reactant and product concentrations at the catalyst sites: E = E r - R T I n Hapir~ vi nF Hareactants
5
(-33)
In order to calculate the incremental voltage loss due to reactant depletion in the catalyst layer, the changes in Nernst potential using c~ values instead of c~ values are represented as follows: l)conc = Er, Nernst -- ENemst
l)conc =
Er
_ R T l n 1 _ ErIn nF Co nF
RT V~on~= ~ l n nF
Co Ci
(5-34) (5-351
(5-36)
where Er, Nerst is the Nernst voltage using Co values, and ENemst is the Nernst voltage using Ci values. Combining Equations 5-35 and 5-36: i = 1- C--L iL
(5-37)
Co
Therefore, the ratio C0/C~ (the concentration at the GDL/catalyst layer interface) can be written as:
Fuel Cell Mass Transport 113 Co
=
Ci
iL
iL -- i
(5-38)
Substituting Equation 5-38 into Equation 5-36 yields: 1)c~
RWln( iLI/ n--F-
iL - i
(5-39)
which is the expression for concentration losses, and is only valid for i < iL. The Butler-Volmer equation from Chapter 3 describes how the reaction kinetics affect concentration and fuel cell performance. The reaction kinetics are dependent upon the reactant and product concentrations at the reaction sites: *
c~
i = i 0co. cg exp(anFu~r
cO---;-exp(-(1- a)nFu~r
(5-40)
where c~ and c~ are arbitrary concentrations, and i0 is measured as the O* O* reference reactant and product concentration values CR and Cp . In the high current-density region, the second term in the Butler-Volmer equation drops out, and the expression then becomes: c__~ i = loco . exp (anFl)act/(RT)) .
(s-41)
In terms of activation overvoltage using c~ instead of c ~ l)c~
RT c ~ anF c~
(5-42)
The ratio can be written as: C~ ----;=
CR
iL
iL--i
(5-43)
The total concentration loss can be written as: =(RT~( 1 1)iL V~on~ \ - ~ J \ + iL--i
(5-44)
Fuel cell mass transport losses may be expressed by the following equation: Vconr = cln
iL iL--i
(5-45)
114 PEM Fuel Cell Modeling and Simulation Using MATLAB | where c is a constant and can have the approximate form: c=
W(5) nF
1+
(5-46)
In actuality, the fuel cell behavior often has a larger value than what the equation predicts. Due to this, c is often obtained empirically. The concentration losses appear at high current densities, and significant concentration losses can severely limit fuel cell performance.
5.5 Convective Mass Transport in Flow Field Plates The flow channels in fuel cell flow field plates are designed to evenly distribute reactants across a fuel cell to help keep mass transport losses to a minimum. Flow field designs are discussed in detail in Chapter 10. In the next section, the control volume method first introduced in Section 5.2 is used to calculate the mass transfer rates in the flow channels. 5.5.1 Mass T r a n s p o r t in Flow C h a n n e l s The mass transport in flow channels can be modeled using a control volume for reactant flow from the flow channel to the electrode layer as shown in Figure 5-3. The rate of convective mass transfer at the electrode surface (rh~) can be expressed as: rils = hm(Cm - Cs)
(5-47)
where Cm is the mean concentration of the reactant in the flow channel (averaged over the channel cross-section, and decreases along the flow direction, x), and Cs is the concentration at the electrode surface. As shown in Figure 5-3, the reactant moves at the molar flow rate, Ar at the position x, where Ac is the channel cross-sectional area and Vm is the mean flow velocity in the flow channel. This can be expressed as: - ~ ( A c C m v m ) - --rnsWelec
(5-48)
where Welec is the width of the electrode surface. If the flow in the channel is assumed to be steady, and the velocity and the concentration are constant, then:
d
C m ----
_rhs VmWflow
(5-49)
Fuel Cell Mass Transport
115
eFlow Channel Width, Wflow
I
dx
Gas Diffusion Layer
] Electrolyte Layer
Reactant Flow Channel Catalyst Layer FIGURE 5-3. Control volume for reactant flow from the flow channel to the electrode layer.
When the current density is small (i < 0.5 iL), it can be assumed i constant. Using Faraday's law, rh~ = ~ and integrating: nF
(i/
Cm(X ) = Cm,in(X)- ~ ~ F x VmWflow
(5-50)
where Cm,in is the m e a n concentration at the flow channel inlet. If the current density is large (i > 0.5 iL), the condition at the electrode surface can be approximated by assuming the concentration at the surface (Cs) is constant. This can be w r i t t e n as follows:
116 PEM Fuel Cell Modeling and Simulation Using MATLAB |
d(Cm
Cs) - h ~ m (Cm -- Cs) i
-
-
(5-51)
VmWflow
After integrating from the channel inlet to location x in the flow channel, Equation 5-51 becomes" Cm -Cs ( C m _ Cs)i n --
-hmx eXP~vmWflow
(5-52)
At the channel outlet, x = H, and Equation 5-52 becomes" Cm, out - C s Cm, in - C s
-hmH
= exp~
(5-53)
Vm Wflow
where Cm,outis the mean concentration at the flow channel outlet. A simple expression can be derived if the entire flow channel is assumed to be the control volume, as shown in Figure 5-4: r n s = VmWflowWelec(Cin - Cout)
(5-54)
r n s - VmWflowWelec(ACi n - ACout)
If C~ is constant, substituting for WflowWelec" 1~ s -- A h m A C 1 m
15-55)
AClm = ACin - AC~
(5-56)
where
ln( ACin ACout ) The local current density corresponding to the rate of mass transfer is: i(x) - nFhm(Cm- Cs)exp / VmWflo -hmxw /
(5-57)
The current density averaged over the electrode surface is: i = nFhmAClm
(5-58)
The limiting current density when Cs approaches 0 is" iL(X)= nFhmCmin , exp/ VmWflo -hmxw /
15-591
Fuel Cell Mass Transport
117
e_ Flow Channel Width, Wnow
Hydrogen
Gas Diffusion Layer
Electrolyte Layer
Reactant Flow Channel Catalyst Layer FIGURE 5-4. Entire channel as the control volume for reactant flow from the flow channel to the electrode layer.
= nFhm ACin - AC~ ln( ACm )
(5-60)
ACout
As seen from Equations 5-57 to 5-60, both the current density and limiting current density decrease exponentially along the channel length.
118
PEM Fuel Cell Modeling and Simulation Using M A T L A B |
EXAMPLE 5-3: Determine Current Density Distribution A fuel cell operating at 25 ~ and 1 atm uses bipolar plates with flow fields to distribute the fuel and oxidant to the electrode surface. The channels have a depth of 1.5 mm, with a distance of 1 m m apart. Air is fed parallel to the channel walls for distribution to the cathode electrode. The length of the flow channel is 18 cm, and the air travels at a velocity of 2 m/s. Determine the distribution of the current density due to the limitation of the convective mass transfer iL(X) and the average limiting current density iL. The Reynolds number can be calculated as follows: Re - pvmD _- ~vmD _- 2m/s * 2 * 1 x 10-3m -- 251 .73 p v 15.89 x 10-6m 2/s Since 251.73 is less than 2000, the flow is laminar. In order to calculate the limiting current density, the convective mass transfer coefficient and binary diffusivity coefficient need to be calculated:
( T ~3/2
Di,ilT) = Di, i(Tref)*k.~ref )
(298 ~s/2
- DO2-N21273)*
~-~)
= 0.21 x 10-4m2/s
hm = Sh Di,j _ 4.86, 0.21 x 10-4m2/s _ 0.0105m/s Dh 2"1 x 10-3m The concentration of 02 at the channel inlet, with a mole fraction of 02 is Xo2 = 0.21, is:
Co2,in- Xo2 *(-~T) - 0.21"
101,325 = 8.588mol/m 3 8.314J/molK 9 298K
The limiting current density based upon the rate of 02 transfer:
iL(X)= nFhmCmin , exp / VmWflo -hmx w / iL(X) = 4*96,487
C molO 2 *0.0105 m *8.588 * molO~ s m 3
exp (l*~*zm/0"0105m/s)~--1--;-~_~,x----,s 3.4802.1 04 x exp(-5.25x) The limiting current density will be 3.4802 A/cm ~ at the channel inlet (x = 0) and 3.3022 A/cm ~ at the channel outlet (x - 18 cm). In order
Fuel Cell Mass Transport
to calculate calculated:
iL,
the outlet concentration of oxygen needs to be
Cmout=Cmin*exp( ) , VmWflow h-mx = 8.588
mOlm 3
*exp
(-0.0105m/s*0.18m)l 91-~* ~Z-ln)S
= 3.338mol/m 3 Then,
lr = nFhm
119
ACin - ACout
In
ACout,]
4 96,487 C = * ~*0.0105mo102 A = 2.251~
cm2
Using MATLAB to solve:
(8.588-3.338) m~ ln()-8"588\ 3.338
120 PEM Fuel Cell Modeling and Simulation Using M A T L A B |
5.6 Mass Transport Equations in the Literature There are many equations used in the literature to determine mass flux and concentration losses. Knowing which equation to use is not as important as determining how to model the system effectively. Knowing how to solve the concentration gradients and species distributions requires knowledge of multicomponent diffusion, and can be a challenging task. In order to precisely solve the mass balance equations (especially in the electrode layers), the mass flux must be determined. The concentration losses are incorporated into a model as the reversible potential decreases due to a decrease in the reactant's partial pressure. There are three basic approaches for determining the mass flux (N): Fick's law, the StefanMaxwell equation, and the Dusty Gas Model.
Fuel Cell Mass Transport
121
5.6.1 Fick's Law The simplest diffusion model is Fick's law, which is used to describe diffusion processes involving two gas species. A form of Fick's law was introduced in Equation 5-26. The standard notation for Fick's law is the binary notation, and can be written as: (5-61)
Ni = -cDi, jVXi
A multicomponent version of Fick's law is shown in Equation 5-62: n
Ni = -cDi, mVXi +
Xi~Nj
(5-62)
j=l
where c is the total molar concentration. If three or more gas species are present, such as N2, 02, and H20, a multicomponent diffusion model such as the Maxwell-Stefan equation must be used, or the binary diffusion coefficients must be expanded to tertiary diffusion coefficients.
5.6.2 The Stefan-Maxwell Equation The Stefan-Maxwell equation is the only diffusion equation that separates diffusion from convection in a simple way. The flux equation is replaced by the difference in species velocities. The Stefan-Maxwell model is more rigorous, is commonly used in multicomponent species systems, and is employed quite extensively in the literature. The main disadvantage is that it is difficult to solve mathematically. It may be used to define the gradient in the mole fraction of components: Vyi = R T ~ y~Nj- yjNi pDe~f
(5-63)
where yi is the gas phase mol fraction of species i, and Ni is the superficial gas phase flux of species i averaged over a differential volume element, which is small with respect to the overall dimensions of the system, but large with respect to the pore size. De(f is the binary diffusion coefficient, and can be defined by: D~ff =
:/
T ~/TciTci,,
/b
/1 /
1 v2 (Pc'JPc'J)V3(Tc'iTc'i)s/12 +~Mj /~l.S
{5-64)
where Tc and pc are the critical temperature and pressure of species i and j, M is the molecular weight of species, A = 0.0002745 for diatomic gases, H2, 02, and N2, and a = 0.000364 for water vapor, and B = 1.832 for diatomic
122 PEM Fuel Cell ModeBng and Simulation Using MATLAB | gases, H2, 02, and N2, and b = 2.334 for water vapor. The Stefa-Maxwell Equation is discussed in more detail in Chapter 8. 5.6.3 The D u s t y Gas Model The Dusty Gas Model is also commonly used in the literature, and looks similar to the Stefan-Maxwell equation except that it also takes into account Knudsen diffusion. Knudsen diffusion occurs when a particle's meanfree-path is similar to, or larger than in size, the average pore diameter (and is discussed in greater detail in Chapter 8): ~VX i
=
Ni Di, k
+
+ z_,
XjNi-
j=l,i~l
XiNj
(5-651
cDi, i
where D~,i is the Knudsen diffusion coefficient for species i. The molecular diffusivity depends upon the temperature, pressure, and concentration. The effective diffusivity depends also upon the microstructural parameters such as porosity, pore size, particle size, and tortuosity. The molecular gas diffusivity must be corrected for the porous media. A large portion of the corrections are made using the ratio of porosity to tortuosity (E/T), although in some cases, the Bruggman model is used due to the lack of information for gas transport in porous media: fc'~
D eff ~/D t,1 "-\TJ
i,j
13eft = E I ' S D .-
~--"i,j
1,1
(5-66)
The Dusty Gas diffusion model requires Knudsen diffusivity to be solved, while Fick's law and the Stefan-Maxwell equation require more work to incorporate Knudsen diffusion. The Knudsen diffusion coefficient for gas species i can be calculated using Equation 5-67: 2T [8RT Dj,k = -~-~/~-[
(5-67)
where M is the molecular mass of species i, and r is the average pore radius.
EXAMPLE 5-4: Calculating the Diffusive Mass Flux Hydrogen gas is maintained at 2 bars and 1 bar on opposite sides of a Nation membrane that is 50 microns (/am) thick. The temperature is 20~ and the binary diffusion coefficient of hydrogen in Nation is 8.7 x 10-8 m2/s. The solubility of hydrogen in the membrane is 1.5 x 10-3 kmol/m 3 bar. What is the mass diffusive flux of hydrogen through
Fuel Cell Mass Transport
123
t h e m e m b r a n e ? W h a t are t h e m o l a r c o n c e n t r a t i o n s of h y d r o g e n i n t h e gas p h a s e ? First, c a l c u l a t e t h e s u r f a c e m o l a r c o n c e n t r a t i o n s of h y d r o g e n :
C A , sl = S * p A =
1.5 x 10 -a k m o l x 3 b a r s = 4.5 x 10 -3 k m o l m 3
= 1.5xl
C A sSl* = pA,
m 3
0_3 k m o l ma x l b a r s = 1 . 5 x 1 0 -3km~
C a l c u l a t e t h e m o l a r d i f f u s i v e flux: DAB
NA -- T ( C A ,
sl - C A , s2)
8.7 x 10 4 N A = 0.3 x 10 -3 (4.5 x 10 -3 - 1.5 x 10 -3) = 8.7 x 10 -~ k mm~~ s
C o n v e r t to a m a s s basis: nA = NA * MA - 8 . 7 X 10 -z k m o l . 2 k~g = 1 . 7 4 x 1 0 - 6 k g m ~ s kmol g C a l c u l a t e t h e m o l a r c o n c e n t r a t i o n s of h y d r o g e n in t h e gas phase"
C A __
p___~= 3 = 0.121~k-m~ RT 8.314 x 10 -9. 9 2 9 3 . 1 5 m3
C c = PA = 1 = 0 . 0 4 0 kmo_____~l RT 8.314x10 -~,293.15 m 3 U s i n g M A T L A B to solve:
ii(~ !i~.......~'
~i~ ..........i:i~.........
......... ..............9............~'~'~ . ~...................... ' <<~ ~'~,~~ .....,~'~,!?~!~v~% .. ~
ii! i
~iiiiii~'i;ii!i~i~i!iiii i~i;i~iiii~i~iiiilil~iii~i~ililiii~i~i'~ii~i;~i~)~~iii~i~i~!!
!
~i~i.!~i~iii~i~i~,;i~i~i~iiiii~ ~~:i~i:i;~i;i~ii~iii!i~!~ii~i~ ~G ;i~i~!i!~i!~ii~Jii~ii~i!~i~iiiii!i~iiii~(ii ~i~i~i~i~i!~i~i?~~~~i!~i~i~i~i~i~i~
ii
124 PEM Fuel Cell Modeling and Simulation Using MATLAB |
Chapter Summary The study of mass transport involves the supply of reactants and products in a fuel cell. Inadequate mass transport can result in poor fuel cell performance. In order to calculate the mass flows through the fuel cell, mass balances can be written to calculate the ideal flow rates and mole fractions of any unknown species. There are two main mass transport effects encountered in fuel cells: convection in the flow structures, and diffusion in the electrodes. Convective flow occurs in the flow channels due to hydrodynamic transport, and the relatively large-size channels (--1 mm to 1 cmJ. Diffusive transport occurs in the electrodes because of the tiny pore sizes. Mass transport losses in the fuel cell result in the depletion of reactants at the electrode, which affects the Nernstian cell voltage and the reaction rate. Commonly used mass transport equations in the literature include Fick's law, the Stefan-Maxwell equation and the Dusty Gas Model. This chapter provided the necessary background to create mass balances on any fuel-cell component with convective or diffusive transport.
Problems 9 A fuel cell is operating at 50~ and 1 atm. Humidified air is supplied with the mole fraction of water vapor equal to 0.2 in the cathode. If the
Fuel Cell Mass Transport
~ 9
9
9
125
channels are rectangular with a diameter of 1.2 mm, find the m a x i m u m velocity of air. For the fuel cell in the above problem, calculate the m a x i m u m velocity of air if the channels are circular. A fuel cell is operating at 50~ and 1 atm. The cathode is using pure oxygen, and there is no water vapor present. The diffusion layer is 400 microns with a porosity of 30%. Calculate the limiting current density. Calculate the limiting current density for a fuel cell operating at 80~ and 1 atm. The cathode is of the same construction as in the third problem. Under the conditions from the third problem, estimate the fuel cell area that can be operated at 0.7 A/cm 2. Assume a stoichiometric number of 2.5, and that the fuel cell is made of a single straight channel with a width of 1 m m and the rib width is 0.5 ram.
Bibliography Barbir, F. PEM Fuel Cells: Theory and Practice. 2005. Burlington, MA: Elsevier Academic Press. Beale, S.B. Calculation procedure for mass transfer in fuel cells. J. Power Sources. Vol. 128, 2004, pp. 185-192. Lin, B. 1999. Conceptual design and modeling of a fuel cell scooter for urban Asia. Princeton University, masters thesis. Li, X. Principles of Fuel Cells. 2006. New York: Taylor & Francis Group. Mench, M.M., C.-Y. Wang, and S.T. Tynell. An Introduction to Fuel Cells and Related Transport Phenomena. Department of Mechanical and Nuclear Engineering, Pennsylvania State University. Draft. Available at: http://mtrll.mne.psu .edu/Document/jtpoverview.pdf Accessed March 4, 2007. Mench, M.M., Z.H. Wang, K. Bhatia, and C.Y. Wang. 2001. Design of a Micro-Direct Methanol Fuel Cell. Electrochemical Engine Center, Department of Mechanical and Nuclear Engineering, Pennsylvania State University. Mennola T., et al. Mass transport in the cathode of a free-breathing polymer electrolyte membrane fuel cell. J. Appl. Electrochem. Vol. 33, 2003, pp. 979-987. O'Hayre, R., S.-W. Cha, W. Colella, and F.B. Prinz. 2006. Fuel Cell Fundamentals. New York: John Wiley & Sons. Rowe, A., and X. Li. Mathematical modeling of proton exchange membrane fuel cells. J. Power Sources. Vol. 102, 2001, pp. 82-96. Sousa, R., Jr., and E. Gonzalez. Mathematical modeling of polymer electrolyte fuel cells. J. Power Sources. Vol. 147, 2005, pp. 32-45. Springer et al. Polymer electrolyte fuel cell model. J. Electrochem. Soc. Vol. 138, No. 8, 1991, pp. 2334-2342. You, L., and H. Liu. A two-phase flow and transport model for PEM fuel cells. J. Power Sources. Vol. 155, 2006, pp. 219-230.