Dynamic model of a PEMFC stack suitable for component ... - CDER

WHEC 16 / 13-16 June 2006 – Lyon France

Dynamic model of a PEMFC stack suitable for component level modeling of a fuel cell based generator Michaël Fourniera, Kodjo Agbossoua*, Alain Poulinb, Yves Dubéa, Guillaume Simarda a

Institut de recherche sur l’hydrogène, Université du Québec à Trois-Rivières, C.P. 500, Trois-Rivières, (QC), G9A 5H7, Canada.

b

LTE, Hydro-Québec, 600 av. de la Montagne, Shawinigan, (QC), G9N 7N5, Canada

ABSTRACT: Dynamic fuel cell stack models offering a good trade-off between computation time and realistic transient behavior are required for the optimized control of fuel cell based generators. This paper describes a model which dynamically considers the membrane dehydration by computing its hydration level and the flooding/mass transport phenomena caused by the gradual accumulation of liquid water in the cathode “gas diffusion layer”. The model’s output is shown to better fit the experimental stack central cell voltage than the averaged stack voltage, the error being less than 2%.

Additional experimental results are required to

properly validate the model water management behavior. Part of the voltage difference between the stack central cells and top/bottom cells has been attributed to the temperature variation along the stack. A threedimensional stack thermal model is being developed to take this temperature distribution into account.

KEYWORDS : PEMFC, stack, dynamic, model, water management.

1. Introduction Fuel cells are seen as promising power conversion devices for their relatively high efficiencies and low pollution levels [1]. As a result of this hype, abundant literature has been written describing either empirical PEMFC (Proton Exchange Membrane Fuel Cell) stack models [2,3] or complex cell models using Computer Fluid Dynamics (CFD) [4,5] . However, few [6,7] offer a fair trade-off between computation time and realistic transient results for more than a specific set of fuel cell and operating conditions. These two factors are of the foremost importance when designing the control of fuel cell based generators.

To allow the study of fuel cell based generators control, a detailed dynamic parametric PEMFC stack model is proposed. This model is to be part of a fuel cell based generator library written for the Matlab/Simulink® environment. For convenience of use, the model should be kept light enough to run faster than real time.

The proposed model has first been described by Poulin et al [8]. Its main capabilities are to dynamically consider the membrane dehydration by computation of the hydration level across the membrane’s thickness and the flooding/mass transport phenomena by gradual accumulation of liquid water in the cathode “gas diffusion layer”. As primary assumption, the stack is supposed to be made of identical cells exposed to the

*

Corresponding author : phone: (819) 376-5011 # 3911, fax: (819) 376-5164, [email protected], www.irh.uqtr.ca

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same set of operating conditions. Simulating a single cell and then multiplying the output voltage by the number of cells in the stack gives the stack output voltage.

The following section describes the governing principles of the single cell model. The model’s outputs are compared to experimental polarisation curves in section 3. Further attempts to consider the temperature distribution along the stack are also described.

2. Single cell model Figure 1 shows the single cell components accounted for in the model. The membrane, within witch the anode and cathode are supposed to lay, is pressed between two identical Gas Diffusion Layers (GDL). The anode gas channels are represented as a single channel on the proper GDL surface. The cathode gas channels are represented the same way. Every transport phenomenon of the model is computed in 1-D along a Figure 1 : Components of a single cell

single axis. As some of these axes are

perpendicular to each other, the model lies in the plane shown on figure 1 and is said quasi 2-D.

2.1 Cell voltage The cell voltage is the sum of three terms: act ohm , Vcell = Ethermo − η tot − η tot

where E thermo is the thermodynamic potential (V),

η totact

(1)

is the activation overvoltage (V) and

η totohm

is the

ohmic overvoltage (V).

2.1.1 Thermodynamic potential For an air-hydrogen fuel cell, the thermodynamic potential is given by:

⎛ ∆S o Ethermo = E 0o + (Top − T0 )⎜⎜ ⎝ nF

⎞ RTop ⎡ ⎟⎟ + ln pO2 ⎠ nF ⎢⎣

( ) ( p )⎤⎥⎦ , 1

2

(2)

H2

o

where E0 is the thermodynamic potential at the standard conditions (V), Top is the average cell operating temperature (K) given by the thermal model, To is the standard temperature (=298,15 K), ∆S

o

is the reaction

entropy change at To, n is the number of transferred electrons (=2), F is the Faraday constant (=96485 C/eqmol) and p H 2 and p O2 are respectively the hydrogen and oxygen effective partial pressures at the corresponding electrode (atm). 2/12

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2.1.2 Activation overvoltage Using Amphlett’s parametric approach [9], the sum of the anode and the cathode activation overvoltages can be expressed as :

η totact = ξ1 + ξ 2Top + ξ 3Top [ln (cO

)] + ξ T [ln(I )]

(3)

where cO2 is the effective oxygen concentration at the cathode (mol/cm³) and

I is the drawn current (A).

2

The parameters

ξ1..4 are determined to fit experimental data.

4 op

All these parameters are supposed uniform in

the cell plane.

2.1.3 Ohmic overvoltage The ohmic overvoltage, supposed to be caused solely by the membrane’s resistance to the proton’s flow, is given by: ohm η tot = Rm I

(4)

where Rm is the membrane’s resistance given the following integration along the membrane thickness:

Rm =

Lm

0

In this equation,

σT

op

dx

∫Aσ m

Top

( x)

.

(5)

is the membrane conductivity at Top (S.m-1), Am is the membrane effective area (m²)

and Lm is the membrane thickness (m).

Combining the work of Clerghorn et al [10] and Hinatsu et al [11], the membrane conductivity can be expressed as a function of the temperature of operation and of the membrane hydration level,

λ , expressed

as the molar ratio H2O/SO3- :

⎛

σ T = 0.25 * exp⎜ 4.1932 − op

⎜ ⎝

⎛ 1 7.6138 1.9796 1 ⎞⎟ ⎞⎟ ⎜ + − 1892 − ⎜T ⎟⎟ λ λ2 ⎝ op 353.15 ⎠ ⎠

(6)

A dryer membrane leads to higher ohmic losses.

2.2 Mean gas concentrations in the gas channels The effective reactants pressures used to compute the cell voltage are supposed uniform on the membrane’s surface. They are derived from the mean gas concentrations in the gas channels. Along their progression in the gas channels, species are exchanged between the GDL and the channels. To get the average concentration for the different species in the channel these exchange flows should be computed.

Gas flows reach their steady state [12] much faster than temperature or membrane hydration do. Steady state flow can thus be assumed to compute the gas concentrations along the gas channel. As shown on figure 2 for the anode side, the channels are divided into m control volumes around witch mass and energy 3/12

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balances are computed. Reactants inlets conditions are applied to the first control volume while the last control volume conditions correspond to the electrode outlet conditions. As the current density is supposed uniform, the reactant x concentration in the

(i + 1)th control

volume

C xi +1 is given by:

C xi +1 = C xi −

∆t =

I ∆t nF Vchannel

(7)

Lchannel mv g

(8)

where ∆t is the time required for the gas flow to get across the control volume, Figure 2 : Gas concentration along the channel

Vchannel is the volume of the channel (m³), Lchannel is the length

of the channel (m) and

v g is the gas flow speed along the channel

(m/s).

The flow of water vapor (mol/s) transferred from the electrode to the gas channel in the

(i + 1)th

control volume is computed with the following

expression:

Qwi +1 =

Dw− N 2 Ad k

2 Dw− N 2 + LGDL

(C k

i +1 w

− C we

)

(9)

where Dw− N 2 is the water vapor diffusion coefficient in air, Figure 3 : Water vapor transport

membrane area encompassed by the control volume, transport coefficient (m/s),

Ad is the

k is the mass

LGDL is the GDL thickness (m), and C we is

the water vapor concentration at the electrode (mol/m³) as shown on figure 3. This last value depends on the membrane hydration level at the membrane/GDL interface. In this model, the water concentration is interpolated from the water activity computed at 30°C [13] and 80°C [11].

Equation 9 accounts for two phenomena: the convective flow of the gases from the channel center to the porous gas diffusion layer (GDL) and the diffusion flow of the gas through the GDL to reach the reaction site. It should be noted that, on the cathode side,

LGDL should be replaced by LGDL (1 − S ) . This is required to

account for the shorter distance the water vapor has to travel from the cathode GDL liquid water front to the GDL-channel interface when a fraction S of the pores are filled with liquid vapor, the liquid water being assumed to stay as close to the membrane as possible. The water vapor concentration in the next control volume is then given by:

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C

i +1 w

Qwi +1 ∆t =C − Vchannel / m i w

(10)

As it is assumed that there is no liquid water outside the membrane and the cathode GDL, the water vapor flow from the electrode to the channel must be limited so as not to get a water vapor pressure exceeding the saturation vapor pressure. Determining the saturation pressure implies finding the gas outlet temperature of the control volume.

The gas outlet temperature at the

(i + 1)th control volume, T i +1 (K), is computed from: hA(Top − T i ) +

T

i +1

=

∑ m& Cp T ∑ m& Cp

n ,vc =i

n

n ,vc = i +1

i

n

n

−

∑ m& Cp T

n ,vc = i

n

n op

(11)

n

where the first term corresponds to the heat exchanged with the gas channel walls and the three summations respectively run over the species entering the volume, the species leaving the

(i + 1)th

(i + 1)th control

control volume for the next one.

(i + 1)th control volume from the previous control

volume through the GDL and the species leaving the

The heat transfer coefficient h (W/m²K) is computed from

correlations for a laminar flow within a rectangular channel, A is the total area of the channels walls within

& n is the molar flow rate (kg/s) and Cp n is the heat capacity (J/kg.K) of the n th the control volume (m²), m specie. In the cathode case, Top should be replaced by Tcathode , the cathode temperature, taken as:

Tcathode = Top + I

(12)

14

As the saturation vapor pressure of the channel gas mixture, which limits the water removal from the electrode, depends itself on the water vapor flow from the electrode, the computation of the temperature and water flow from the electrode is done iteratively until the temperature variation between two consecutive iterations becomes negligible.

When the reactants concentration profile has been obtained, the

concentration values are averaged to be used to compute effective reactant partial pressure at the reaction cathode

site. Water vapor flows from the electrodes are summed and converted to give either N w

anode

or N w

(mol/m².s), the total water vapor fluxes from the electrode to the gas channels.

2.3 Effective reactants partial pressure at the reaction site To compute the cell voltage, one needs to know the effective partial pressure/concentration at the reaction mean

site. Given the mean reactant x concentration C x

(mol/m³) in the channel, the effective concentration

C x can be computed from the following equation (similarly to (9)): C xe = C xmean −

I 2 D x + LGDL k nF D x Am k 5/12

(13)

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where n takes the values 2 and 4 respectively for hydrogen and oxygen and D x is the diffusion coefficient. The diffusion coefficients D x are computed using the Slattery and Bird’s approach [14] and a Bruggemann correction term to account for the porosity of the GDL [13].

D AB

1 ⎛ Top = c⎜⎜ P ⎝ d

where D AB is the diffusion coefficient for the

b

⎞ 3/ 2 ⎟⎟ ε ⎠

(14)

AB binary gas mixture (m²/s), A being the reactant

(hydrogen or oxygen) and B the diluent (carbon dioxide or nitrogen), P is the mixture pressure (atm),

ε

is

the porosity of the electrode and b , c and d are constants for the gas pair. On the cathode side only,

ε

is

replaced by

ε (1 − S ) to account for the proportion S

of the pore’s volume filled with liquid water. This is the

only place in the model where liquid water can exist outside the membrane. The higher is the S , the lower is the oxygen partial pressure at the cathode and the lower is the cell voltage. Details on S computation will be given in section 2.4. The reactants x partial pressures px can then be computed from the equation:

p x = C x RTop

(15)

2.4 Water balance and membrane’s hydration level The flow of water vapor from the electrodes to the gas channel has been described in section 2.2. This section presents how these flows relate to the water production rate and the water transport mechanisms to determine the hydration level of the membrane.

This problem,

shown on figure 4, is treated in one dimension according to a simplified version of Eaton’s work [15]. Figure 4 : Water balance model

The cathode is assumed to be

part of the membrane and of a measurable thickness. The water mass balance gives:

∂C w ∂N w ⎧ RH 2O =⎨ + ∂x ∂t ⎩0

in the cathode elsewhere in the membrane

(16)

where N w is the water molar flow (mol/m³) and R H 2O is the water generation rate (mol/m³.s) given by:

R H 2O =

i 2 Lcathode F 6/12

, i=

I Am

(17)

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where Lcathode is the cathode thickness (m) and i is the current density (A/m²).

The water flow in the membrane is caused by the three phenomena shown on figure 4: diffusion, electroosmotic drag and convection. They are respectively expressed as [15]:

N w = − Dw

∂C w i + η d + Cw vw ∂x F

where Dw is the water diffusion coefficient (m²/s),

ηd

(18)

is the electro-osmotic drag coefficient (-) and v w is

the water speed (m/s). The water diffusion coefficient depends on the operating temperature and on the membrane hydration level [4,15].

The water drag coefficient expresses the number of water molecules pulled by each proton from the anode to the cathode. For Nafion® membranes the following expressions are often used [13]:

ηd =

2.5 EW Cw λ, λ= 22 ρm

(19, 20)

where λ is the membrane hydration level (mol H2O/mol SO3-), EW is the equivalent membrane weight (membrane mass (kg)/mol SO3-), and

ρm

is the density of the dry membrane (kg/m3). The membrane

swelling is neglected.

Neglecting the gravity and assuming a linear pressure drop across the membrane, the water speed is given by [15]:

v=−

K ∂P K ⎛ Pa − Pc = ⎜ µ ∂x µ ⎜⎝ Lm

⎞ ⎟⎟ ⎠

(21)

where K is the membrane permeability (m2), µ is the water viscosity (Pa⋅s), Pa and Pc are respectively the absolute pressure on the anode and cathode side of the membrane (Pa) and Lm is the membrane thickness (m).

By replacing (18) to (21) in (16), one gets a second order partial derivative equation:

∂C w ∂C wm ⎧ 0 ∂C w ∂C w ⎞ ∂ ⎛ ⎟ ⎜ −α −β +⎨ = ⎜ Dw ∂x ∂x ∂t ∂x ⎝ ∂x ⎟⎠ ⎩ R H 2O 2.5 i EW K ⎛ P − Pc ⎞ ⎟ with α = and β = ⎜⎜ a 22 F ρ m µ ⎝ Lm + Lint ⎟⎠

(22)

Equation 22 is then solved using an implicit upwind [16] finite difference scheme [15] using the water vapor anode

flow N w

cathode

and N w

as boundary conditions. The results are the water concentration in the membrane

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C w as well as the water flow N wint at the membrane/cathode interface. The water concentrations are converted to hydration levels using equation 20.

If the hydration level in the cathode reaches its maximum value and if the sum of the water flows outside the cathode is less than the water production rate, the excess water is allowed to accumulate in liquid form in the elec

cathode GDL, at the membrane interface. The amount of liquid water n w,liq (mol) in the GDL is computed from: t

n welec ,liq = ∫ 0

(

)

I + Am N wint − N wcathode dt 2F

(23)

As long as there is liquid water in the GDL, the water concentration in the cathode is assumed to be corresponding to

λ = λmax .

The fraction of the cathode GDL pores volume filled with liquid water is given by:

S=

M w nwelec , liq

ερ w Am Lelec

where M w is the molar mass of water (kg/mol) and

ρw

(24)

is the density of water (kg/m3). The S factor is used

to lower the oxygen diffusion coefficient from the gas channel to the cathode.

2.5 Single cell thermal model In the model, the operating temperature of the cell corresponds to its average temperature. For the single cell model, the cell is considered to be a solid body with uniform thermal properties. Its thermal behavior is thus described by the equation [17]:

dTop dE = (mC p )stack = Q& th − Q& elec + Q& sens − Q& h dt dt

(

Where mC p

)

stack

(25)

& is the thermodynamic is the product of the mass and specific heat of the cell (J/K), Q th

& & & theoretical power (W), Q elec is the electrical power (W), Q sens is the sensible heat variation (W) and Q h is the thermal power (W) taken out of the cell by convection on its front and rear faces.

The theoretical power and electrical power are respectively given by:

Q& th = Ethermo ⋅ I , Q& e = Vcell ⋅ I .

(26, 27)

The sensible heat variation is the difference between the thermal energy of the gases entering the fuel cell and the thermal energy of the gases leaving the fuel cell [17]:

Q& sens = Q& sens _ in − Q& sens _ out for the entering gases 8/12

(28)

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(

ano w ano 2 Q& sens _ in = Tinano C pH 2 QHano2 ,in + C CO p QCO2 ,in + C p Q w,in

(

)

+ Tincat C Op2 QOcat2 ,in + C pN 2 Q Ncat2 ,in + C pw Qwcat,in

)

(29)

and for the exiting gases

(

)

ano ano w ano 2 Q& sens _ out = Tout C pH 2 QHano2 ,out + C CO p QCO2 ,out + C p Q w, out +

(

cat Tout C Op2 QOcat2 ,out + C pN 2 Q Ncat2 ,out + C pw Qwcat,out

The modeled stack is air-cooled by fans operating in on-off mode.

)

.

(30)

The thermal power taken out by

convection is then:

Q& h = δ hA(Ts − Tamb ) , with δ = 1 if (Ts − Tamb ) ≥ 0.1 and else δ = 0 .

(31)

where hA is a convection coefficient (W/K), Tamb is the ambient temperature (K) and Ts is the cell surface temperature (K). This temperature is linked to the average cell temperature by the following equation, assuming steady-state conditions, uniform temperature on the cooled surfaces and an internal parabolic temperature profile:

Ts = Top

(Q& −

th

)

2 − Q& elec + Q& sens Lc 3kV

(32)

where Lc is the length of the air-cooled side of the cell (m), k is the cell thermal conductivity (W/m.K) and

V is the cell volume (m³). In transient operation, the surface temperature is interpolated between the nearest steady-state values using an exponential curve.

3. Results The model output voltage has been compared to experimental measurements.

The fuel cell used to get the experimental

results, shown on figure 5, was part of a 500 W Uninterruptible Power Supply (UPS) produced by H-Power. It has 65 cells, each with an active area of about 0.0078 m². It operates on ambient air at less than 1 psig and on dry hydrogen at 8 psig, the latter being in dead-end configuration. The vertical stack measures 0,24 x 0,16 x 0.06 m and is located between two fans. finned.

The collector plates faces exposed to the fans are A thermocouple located near the stack lateral face

center is used as input signal to keep the stack at the desired temperature. The cell voltages are recorded in pairs as well as the total stack voltage. Figure 5 : Stack setup (lateral view) Figure 6 shows polarization curves obtained at 40°C. The filled points correspond to the stack total voltage divided by the number of cells while the square ones have been obtained from the two cells located near the stack center. The triangles and crosses are respectively the cell voltages of the top and bottom cells. The 9/12

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curve represents the simulated model results.

Voltage (V)

0,85

The model was

coded in Matlab/Simulink® in a modular way using mostly C S-

0,75

0,65

Averaged

functions. The model is fast; it took

Top

less than a minute on a P4 1,7 GHz

Center

PC to simulate 83 minutes of fuel

Bottom

cell operation.

Simulated 0,55 0

2

4

6

8

10

12

From this figure, it can be seen that the

Current (A)

simulated

averaged Figure 2 : Polarization curves

and

data

follow

central

the

voltage

measures quite adequately.

The

standard deviations of the errors are respectively of 1.8 and 1.5%. The model error is thus less for the central cell than the averaged stack, which is expected since the temperature setpoint (used also by the single cell thermal model) is located near the stack center. The simulated results are within ±2% of the center cell voltage (as shown on figure 6). This is satisfying as the voltage difference between day to day runs has been determined to be of about 2.5%. It is understood that these results are not complete enough to validate the whole cell model, especially the water management effects. Nevertheless, it can be taken as a good starting point.

The figure 6 also shows the voltages of the top and bottom cells. These voltages are significantly lower than the averaged stack voltage. For control modeling purposes, one should consider these cells to be as important as any other one because their failure would be as catastrophic as would be the failure of any other cell of the stack. One should thus investigate the causes of these discrepancies.

Figure 7 shows a thermal image of the fuel cell in operation at 10 A when maintained at 40°C by actuating the fans. As shown by the plotted temperature profiles, the temperature distribution is far from being uniform. This non-uniformity (up to 6°C on the lateral stack surface) is believed to yield to diverging behaviors in terms of water management between the stack center and its top and bottom portions. To study this temperature distribution along the stack axis, a stack thermal model is being developed.

This dynamic stack thermal model is three-dimensional and allows anisotropic thermal properties.

The modeled body is

divided into horizontal slices, each of them corresponding to a certain number of cells. A different quantity of heat can be injected in each slice, corresponding to the heat (losses) having to be dissipated by convection. The model then computes the 10/12

Figure 7 : Stack thermal image

WHEC 16 / 13-16 June 2006 – Lyon France

new three-dimensional temperature distribution. The temperature of each slice is averaged and used as the operational temperature of an instance of the electrochemical model described in the section 2.

4. Conclusion Dynamic fuel cell stack models offering a good trade-off between computation time and realistic transient behavior are required for the optimized control of fuel cell based generators. This paper has described such a model which considers the progressive drying of the membrane and the flooding/mass transport limitation caused by the accumulation of liquid water at the cathode surface. These considerations should allow the model to capture the behavior of a PEMFC exposed to real conditions, including unbalanced water management.

The model computes concentration profiles for both the anode and the cathode gas channels. These values are then averaged to compute the species concentration at the electrodes. These values are used as input to compute the effective cell voltage.

Water balance equations are used to determine the membrane

hydration levels across its thickness. The latter are inputs to the computation of the membrane’s ohmic resistance. Excess liquid water is allowed to accumulate in the cathode GDL, affecting the cell voltage by lowering the oxygen partial pressure at the cathode. Keeping in mind that this model is control oriented, it has been coded in Matlab/Simulink® in a modular way using mostly C S-functions. It will form the basic block of a library of components for PEMFC based power generator modeling. The model runs fast; it executes about 80 times faster than real time.

Simulated results have been compared to experimental measures. The model’s output voltage has been shown to better fit the stack central cell voltage than the mean stack voltage, the relative error being less than 2%. Additional experimental results will be required to properly validate the model water management behavior. A significant voltage difference has also been observed between the stack center cells and the top and bottom cells. Part of this difference has been attributed to the temperature variation along the stack. A stack three-dimensional thermal model is being developed to take this temperature distribution into account.

5. Acknowledgements This work has been supported by Hydro-Québec, Natural Resources Canada and the Natural Sciences and Engineering Research Council of Canada. We also acknowledge the advices of Michel Dostie from the LTE Hydro-Québec.

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