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Experimental investigation and mathematical modeling of triode PEM fuel cells
Accepted Manuscript Title: Experimental investigation and mathematical modeling of triode PEM fuel cells Authors: E. Martino, G. Koilias, M. Athanasiou, A. Katsaounis, Y. Dimakopoulos, J. Tsamopoulos, C.G. Vayenas PII: DOI: Reference:
S0013-4686(17)31608-0 http://dx.doi.org/doi:10.1016/j.electacta.2017.07.168 EA 29985
To appear in:
Electrochimica Acta
Received date: Revised date: Accepted date:
13-3-2017 25-7-2017 27-7-2017
Please cite this article as: E.Martino, G.Koilias, M.Athanasiou, A.Katsaounis, Y.Dimakopoulos, J.Tsamopoulos, C.G.Vayenas, Experimental investigation and mathematical modeling of triode PEM fuel cells, Electrochimica Actahttp://dx.doi.org/10.1016/j.electacta.2017.07.168 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
MARKED COPY Experimental investigation and mathematical modeling of triode PEM fuel cells E. Martino1, G. Koilias1, M. Athanasiou1, A. Katsaounis1, Y. Dimakopoulos1, J. Tsamopoulos1 and C.G. Vayenas1,2, 1
Department of Chemical Engineering, University of Patras, Caratheodory 1 St, GR-26504 Patras,
Greece, 2
Academy of Athens, Panepistimiou 28 Ave., GR-10679 Athens, Greece
Corresponding author: [email protected] (Costas G. Vayenas)
EO17-1450R1 Highlights
The triode fuel cell operation was tested using novel comb-type electrode designs Triode operation enhances the PEMFC power output by up to 500% Power output enhancement exceeds auxiliary power by up to 20% Good agreement with mathematical model based on the laws of Kirchhoff Proton fluxes in the membrane found via solution of the Nernst Planck equation
Abstract The triode operation of humidified PEM fuel cells has been investigated both with pure H2 and with CO poisoned H2 feed over commercial Vulcan supported Pt(30%)-Ru(15%) anodes. It was found that triode operation, which involves the use of a third, auxiliary, electrode, leads to up to 400% power output increase with the same CO poisoned H2 gas feed. At low current densities, the power increase is accompanied by an increase in overall thermodynamic efficiency. A mathematical model, based on Kirchhoff’s laws, has been developed which is in reasonably good agreement with the experimental results. In order to gain some additional insight into the mechanism of triode operation, the model has been also extended to describe the potential distribution inside the Nafion membrane via the numerical solution of the Nernst-Planck equation. Both model and experiment have shown the critical role of minimizing the auxiliary-anode or auxiliary-cathode resistance, and this has led to improved comb-shaped anode or cathode electrode geometries.
1. Introduction A major problem of low-temperature Proton Exchange Membrane Fuel Cells (PEMFCs) is the observed susceptibility of Pt-based carbon supported catalysts to CO poisoning. This process leads to the degradation of the anode and consequently to a severe drop in the cell performance, as CO is strongly adsorbed on Pt and blocks any further H2 adsorption [1-3]. Since H2 is the most often used fuel in PEMFC applications and is mainly produced by reforming processes of hydrocarbons or liquid alcohols, it is very common that the anode fuel feed contains significant amounts of CO. In this paper, we examine how the triode design of PEM fuel cells can enhance the cell performance both with pure H2 feed and under CO poisoning conditions. The triode fuel cell operation, first proposed a few years ago [4-8], is an alternative method for the enhancement of the power output and thermodynamic efficiency of electrochemical power producing devices. The triode fuel cell design introduces, in addition to the anode and the cathode, a third, auxiliary, electrode in contact with the solid electrolyte, e.g. polymer electrolyte membrane in the case of PEMFCs. This electrode forms together with the anode or cathode, a second, auxiliary, electric circuit operating in parallel with the conventional main circuit of the cell (Fig. 1). The auxiliary circuit runs in the electrolytic mode, pumping ions (i.e. protons in the case of a PEMFC) from the cathode to the auxiliary electrode. In this way, imposition of a potential difference between the auxiliary electrode and the cathode permits the primary circuit of the fuel cell to operate under previously inaccessible, i.e. larger than Uo=1.23 V, anode-cathode potentials. This introduces a new controllable variable in fuel cell operation and can lead to significant reduction of the anodic and cathodic overpotentials and thus to a significant increase in the power output of a fuel cell [4-7]. It can also over certain ranges of design and operational conditions lead to an enhancement of the overall thermodynamic efficiency. Two parameters are used to quantify the performance of a triode cell, i.e. the power enhancement ratio, , and the power gain ratio, , defined by:
Pfc / Pfco
(1)
Pfc / Paux (Pfc Pfco ) / Paux
(2)
and
where Pfc and Pfco
are the power outputs in triode mode operation and conventional mode of
operation respectively, Paux is the power sacrificed in the auxiliary circuit and Pfc is the increase in the fuel cell power output. The overall thermodynamic efficiency is enhanced when 1 . In more recent papers the triode concept has been applied also to enhance electrolysis [8], to study a prototype anode supported solid oxide fuel cell, SOFC, under hydrogen and steam
reforming conditions [9] and to enhance the performance of H2S – poisoned SOFCs operated under CH4 – H2O mixtures [10]. On the modeling side, there has been so far only one reference [11]
reporting an
electrochemical model of a triode SOFC based on a two-dimensional commercial software, COMSOL Multiphysics. In this paper we first present a simple mathematical model based on Kirchhoff's laws, describing conventional fuel cell operation (CFCO) of triode fuel cells which enables one to extract the ohmic resistance values between the three electrodes which are all in electrolytic contact. Subsequently, an extension of this model, also based on Kirchhoff's laws, is presented which can describe the triode fuel cell operation (TFCO), using the resistance values extracted from the CFCO model and allowing for the computation of the power enhancement ratio, , and the gain ratio, , as a function of the pre-calculated resistances, the cell geometry and the operating conditions. This model also provides the boundary conditions for solving, via finite elements, the Nernst-Planck equation inside the electrolyte and thus obtaining the spatial distribution of ionic current and local potential in the electrolyte during triode operation. Both the model and the experiments show the crucial role of minimizing the ionic resistance between anode and auxiliary electrode for successful triode operation. This led to the development of a novel and more efficient triode design than the classical triode design used in previous studies [4-7]. Both the classical design and the novel one were tested both with humidified hydrogen (3% H2O) fuel as well as under humidified CO poisoning mixtures of hydrogen (3% H2O) with CO concentrations up to 120 ppm.
2. Experimental Apparatus a. Graphite casing design The graphite casing triode PEMFC design used in previous studies [4-7] is shown schematically in Fig. 2 and has been already described in detail [4-7]. Anode and auxiliary electrodes were commercial E-TEK Pt(30%)–Ru(15%) supported on Vulcan XC-72 carbon deposited on E-TEK carbon cloth with a total catalyst loading of 0.5 mg/cm2. The cathode was Pt supported on Vulcan XC-72 carbon deposited on carbon cloth (0.5 mg/cm2). The electrode geometry is shown in Fig. 2b. The flow and gas analysis system has been described elsewhere [47]. The superficial surface area of the cathode (Pt) was 5.29 cm2, of the anode (Pt-Ru) was 3.85 cm2 and of the auxiliary electrode (Pt-Ru) was 0.49 cm2 (Figure 1b). The cathode was a square and the auxiliary electrode was a smaller square located in the center of the hollow square anode (Fig. 1b). The membrane was Nafion 117 with nominal thickness 185m. The membrane electrode assembly
(MEA) was prepared by hot pressing in a model C Carver hot press at 120oC and under pressure of 1 metric ton for 3 min. Preliminary investigation showed that the applied auxiliary potential, Uaux , should not exceed 1.9 V, as this leads to CO2 formation at the cathode via oxidation of the carbon support. The gas feeds to the cathode and anode compartments (the latter includes also the auxiliary electrode, Fig. 1) were continuously humidified using thermostated gas saturators. The cell temperature was typically set at the same temperature (25oC) with the gas saturators. The anode compartment gas feed was Air Liquide certified gas mixtures of 490 ppm CO/He, which could be further diluted with Linde (N4.5) H2. The cathode feed was humidified Air Liquide synthetic air. The fuel cell circuit included a decade resistance Box (Time Electronics Ltd 1051) in order to vary the external load. The current and the potential were measured by three-digital multimeters (Metex ME 21). For the fixed triode operation, constant potentials or currents in the auxiliary circuit were applied using an AMEL 2053 Potentiostat – Galvanostat. It is evident from Fig. 1a that all three electrodes operate in a corrosion-type mode with part of their surface used for oxidation and part of their surface used for reduction. Despite this rather intense mode of operation no performance deterioration was observed during operation for several weeks.
b. Teflon casing design Figure 3 shows the configuration of the novel Teflon (Polytetrafluoroethylene, (PTFE)) cased triode PEM fuel cell developed in this study. The anode-auxiliary side of the new MEA consisted of two alternating comb-like electrodes with the comb teeth in close proximity (0.5 mm) to each other. The comb shape of the anode and auxiliary electrodes ensures better proton conduction between the two electrodes, which is achieved via the minimization of the electrolyte resistance between them. The need to minimize this resistance is underlined both by the experimental results and by the mathematical model presented in the next section. On the other side of the membrane is placed a square type cathode (Figure 3a). Figure 3b shows the geometry of the bottom part of the Teflon casing, which was used to replace the common graphite casing, so that the comb-type structured anode and auxiliary electrodes of Figure 3a can be utilized without any electronic shortcircuiting problems. Figure 3b shows the engraved gas flow channels and the grooves for the silicon rubber gasket to provide gas tight sealing. It also shows two detachable graphite blocks embedded in the Teflon casing which are in contact with the electrode edges outside the gas flow region and which in combination with a 0.025 mm thick Au foil, act as current collectors.
The gas feed unit and electrical measurement unit were the same with those utilized in the graphite casing experiments.
c. MEA preparation and cell operation With both designs (graphite or Teflon casing) the same fully humidified Nafion 117 (DuPont) membranes with a nominal thickness of 185 μm were used as proton conducting electrolyte. The membrane was treated according to the standard procedure of preparation analyzed in detail in previous studies [5-8]. The MEA was prepared by hot pressing the electrodes and electrolyte in a model Carver heated press using pressure steps of 1 metric ton. Every step had a duration of 3 min and corresponded to a 10 °C rise up to 80°C and then to a 5°C rise up to the final temperature of 120°C. Table 1 shows the details of all anodic-auxiliary electrodes which were prepared and used in this study. The first letter of the name shows the fuel cell casing material (G: Graphite, T: Teflon), the second letter indicates the geometry of the anode-auxiliary electrodes (S: Square, C: Comb) and the third letter indicates the number of the sample. All current and potentials were measured by a series of digital multimeters. The anode – auxiliary compartment fuel gas feed was humidified ultrapure H2 (Linde Gas certified 99.999% H2) and humidified H2/CO mixtures (Air Liquide certified 490ppm CO in He) further diluted in He (Air Liquide). The cathode feed was (Air Liquide) synthetic air. In order to humidify the inlet gas feeds, the gases passed continuously through two thermostated gas saturators operated at 25oC. The concentrations of CO and CO2 in the anode – auxiliary feed and effluent were monitored using a Fuji Electric IR CO/CO2 analyzer. All the current-potential curves were obtained under a constant total volumetric flow rate of 300 cm3STP/min both at the anode and at the cathode compartments, with CO concentration varying between 5 and 120 ppm. In the conventional mode, the Ifc-Ufc curves were obtained by varying the external resistance, Rex, from 0 up to 9 MΩ in specific steps, such that the Ufc varied by approximately 50 mV in each step. In the case of triode operation various potentials, Uaux, were applied to the auxiliary circuit via the galvanostat / potentiostat in the range of 0-1.6 V.
3. Mathematical modeling a. Conventional Fuel cell operation modeling and comparison with experiment In this section we present a simple mathematical model based on Kirchhoff’s laws which describes conventional fuel cell operation of a triode cell, i.e. without the application of auxiliary
current or potential. The model permits the extraction of the three ohmic resistances for proton conduction between anode and cathode (denoted R1), auxiliary electrode and cathode (denoted R2) and anode-auxiliary (denoted R3) from current-potential data. All the equations apply for the case of conventional mode operation, but obviously the extracted resistance values are valid for triode cell operation as well. Fig. 4 shows the electrical circuit of a triode fuel cell under conventional mode operation, including all the parameters of the model, which are the values of the three internal resistances, R1, R2 and R3, the open circuit potential Uo, the fuel cell current, Ifc, flowing through the main circuit and the external resistance of the fuel cell circuit, Rex. We first note that in the conventional fuel cell operation of the triode cell, the potential of the auxiliary electrode, Uaux, takes a value between the potentials, Ua and Uc, of the anode and cathode. This implies that in addition to the anode-cathode current (proton) pathway, some current, denoted I3, flows from the anode to the auxiliary electrode and a current I2, equal to I3, flows from the auxiliary electrode to the cathode. Thus part of the auxiliary electrode acts as a cathode, reducing protons to H2, and part of the same electrode serves as an anode, oxidizing H2 to protons. We then examine the loop ABCD (see Fig. 4) and we apply Kirchhoff's second law. Note that this closed loop does not contain the resistance Rex. Following the path of charge along the loop, we note that (a) the two potential source terms (Uo and, along the path, - Uo) cancel out and (b) that the currents I2 and I3 have the same sign which is opposite to that of I1. Note that Kirchhoff's laws are valid regardless of the nature of the charge carrier, which in the present case is electrons in the electrodes and protons in the membrane. Thus, application of Kirchhoff's 2nd law to the loop ABCD (Figure 4) gives: I2R 2 I3R 3 I1R1 0
(3)
and since I2 I3 it follows,
I3 R1 I1 R 2 R 3
I3 R1 I1 I3 R1 R 2 R 3
;
(4)
Application of Kirchhoff's first law at point A gives I1 I3 Ifc
(5)
and thus from eqs. (4) and (5) it follows:
(R 2 R 3 ) I1 Ifc R1 R 2 R 3
;
I3 R1 Ifc R1 R 2 R 3
(6)
Denoting by U3 and U1 the products U3 I3R 3
;
U1 I1R1
(7)
where U3 is the potential difference between the anode and the auxiliary electrode, and U1 ( Uo Ufc ) is the total fuel cell overpotential, assumed to be ohmic, it follows from equations
(4-7) that U1 R1 (R 2 R 3 ) R1 Ifc R1 R 2 R 3
;
U3 R1R 3 R*3 Ifc R1 R 2 R 3
(8)
The values of R1 and R *3 can be obtained experimentally from the I-U plots during conventional fuel cell operation as shown in Fig 5. A third equation for obtaining R1, R2 and R3 is obtained by assuming that the ratio, , of the resistances R1 and R2 of the anode and of the auxiliary circuit can be estimated from the surface areas, S1 and S2, of the two electrodes using the expression for the resistance of a cylindrical conductor and assuming uniform current density, i.e. R1 L / S1 S2 R 2 L / S2 S1
(9)
where L is the thickness of the electrolyte and is the specific electrical resistance of the electrolyte, which equals to 1 , where is the ionic conductivity of the hydrated Nafion membrane. There is a rich literature on the conductivity of Nafion which is of the order of 0.02 S cm-1 at temperatures 20o to 80 oC [12 - 15] Denoting S2 / S1 ;
and
R1* R *3
1
it follows
1 1 R 3 R *3 1 ;
R 2 R 3
;
R1 R 3
(10)
As shown in Figure 6, which compares the values of R1, R2 and R3 computed for three different geometries of the anode and auxiliary electrodes, the value of R1 is not very sensitive to electrode geometry which, however, strongly affects the values of R2 and R3. Thus R2 decreases from 14.2 to 1.3 and to 1.1 while R3 decreases from ~50 to ~10 and to ~1.8 upon replacing the square electrode geometry of Figure 6a to the comb-like electrode geometries of Figures 6b and 6c. The decrease in R2 is due to the increase in the surface area of the auxiliary electrode, while the decrease in R3 is additionally due to the increase in the three-phase-boundaries length gas-electrode-electrolyte as further discussed below.
b. Triode fuel cell operation modeling via the equations of Kirchhoff and comparison with experiment As already noted, the electrical potentials, Uc, Ua and Uaux, denote respectively the assumed uniform potentials of the cathode, anode and auxiliary electrode.
It thus follows Ufc Uc Ua 0
(11)
The applied electrolytic auxiliary potential Uaux is defined from
Uaux Uaux,o Uaux Uaux Uaux,o 0
(12)
where the subscript “o” denotes open auxiliary circuit operation, i.e. conventional fuel cell operation. In the case of triode operation (Fig. 7a) an electrolytic potential difference, Uaux , is applied between the auxiliary and cathodic electrode and therefore electrolytic current, Iaux , flows through the auxiliary circuit. In this way, protons are pumped from the cathode, but also from the anode, to the auxiliary electrode where some H2 evolution occurs (Fig. 7a). Consequently, the potential difference Uc-Ua is forced to increase and the power output of the fuel cell circuit also increases. Application of Kirchhoff's laws in both circuits in the triode mode of operation (Fig. 7a) gives the following fundamental equations: U o Ifc R ex Ifar R1 U o U aux I fc (R ex R 3 )
(13)
( U o U aux ) U o I el R 2
which are obtained from the 2nd law of Kirchhoff in loops ABCA, ABCDEFA and FEDF respectively. Also applying the 1st Kirchhoff’s law at point A one obtains Iaux Ifar Ifc
(14)
and using Kirchhoff’s 1st law at point F it follows
Iaux Iel IP/G
(15)
Also, Ohm’s law for line ABC gives Ufc Ifc R ex
(16)
and Kirchhoff’s 2nd law in loop FEDF yields U o Iel R 2 U o U aux 0 Iel
Uaux R2
(17)
where Ifar is the net faradaic current of fuel consumption, Ifc is the electric current flowing through the main (fuel cell) circuit in triode mode, Iaux is the current flowing through the secondary (auxiliary) circuit, U fc is the potential that develops between anode-cathode in the triode mode,
Uaux the potential imposed between the auxiliary electrode and the cathode, U o is the open circuit
potential of the two electrochemical cells, R ex is the external variable resistance of the main circuit and R 3 is the internal electrolyte resistance between the two electrodes on the same side, i.e. between the anode and the auxiliary electrode. Upon combining the second equation (13) with equation (16) one obtains U fc
U o U aux 1 R 3 / R ex
; Ifc
U o U aux R ex R 3
(18)
This equation is plotted in Figure 7b as a function of the external resistance R ex for various values of R 3 . One observes that U fc reaches the value Uo Uaux of the applied potential for R ex values higher than R 3 . Since R ex values less than 1 are necessary to obtain current and power densities of the order of 1 A/cm2 and 1 W/cm2 respectively, it follows that R3 values less than 5 are necessary for this purpose as shown in Figure 7b. The very good agreement between equation (19) and experiment is shown in Figure 8. This agreement confirms the need for minimizing the resistance R3 for successful triode operation. Furthermore by using the first equation (13) one can compute the Faradaic current Ifar, i.e. Ifar
Uo Ifc R ex R1
(19)
Using equation (18) one obtains
Ifar
U o 1 Uaux / U o 1 R1 1 R 3 / R ex
Iaux Ifc Ifar Uo R 3 R ex
(20)
U o U aux U o 1 U aux / U o – 1 R 3 R ex R1 1 R 3 / R ex
R 3 U aux R ex 1 1 R1 Uo R1
(21)
Consequently, also accounting for equations (15) and (17), the power enhancement ratio, , and the gain ratio, , can be expressed as a function of Uaux , Uo, R1 , R 3 and R ex via the equations
(U o U aux )2 2 Pfc R ex (1 R 3 / R ex )2 (U o U aux )2 R1 R ex o Pfc U 2o R ex U o2 R 3 R ex (R1 R ex )2
U o Uaux 2
Pfc Paux
U 2o R ex (R1 R ex )2
R ex (1 R 3 / R ex )2 R 3 U aux R ex U aux Uo U aux 1 1 R1 U o R1 R 2 (R 3 R ex )
(22)
(23)
Equation (23) shows again the crucial role of the resistance between the anode and the auxiliary electrode, R3, and of the applied potential Uaux on the fuel cell potential U fc . This dependence was also displayed in Fig. 8. It is evident that as R 3 decreases, the effect of the applied potential on the auxiliary circuit becomes more pronounced. This observation regarding the role of R 3 led to the new cell design of Figure 3 and to the introduction of comb-shaped electrodes. This new electrode geometry leads to a significant decrease in R 3 as already shown in Fig. 6. This is due to the pronounced decrease in the distance, d, between neighboring comb teeth and to the concomitant increase in three phase boundary length (
tpb ).
The dependence of R 3 on d and
tpb can
be
approximated by
d ltpb 0 R 3 R 30 d 0 ltpb
(24)
where R 30 ( 50 ) , d0 5 103 m and
tpb 0 (
0.1 m) corresponds to the graphite casing design
of Figure 6a. Therefore, the need to minimize the resistance R 3 leads to the need to minimize the distance d and to maximize
tpb ,
which dictates the comb type geometry of Figure 6c.
Figures 9a and 9b present measured and values as a function of Uaux and compare them with those predicted from equations (22) and (23). In this comparison between the model and the experimental data the imposed potential, Uaux , was varied with a step of 10-50 mV. For small
Uaux values, the power gain ratio, , exceeds unity, because the power consumed in the auxiliary circuit is very small. On the other hand, the power enhancement ratio, , increases with increasing imposed potential. The experimental data shown in Figures 9a and 9b are the same and have been obtained with TC3 for which it is R3 > R1. These data show that increases significantly, up to 4.5, with increasing Uaux , while reaches quite high values, up to 8, for very small Uaux values. When the model is run with the experimental R1, R2 and R3 values (R1=1.50 , R2=2.00 , R3=1.95 , respectively, Fig. 9a) then it fails to predict the pronounced increase in with decreasing Uaux and instead predicts a maximum. If, however, the model is run with an R3 value smaller than R1 (e.g. R3=1.2 , Fig. 9b) then the model is in excellent agreement with the experimental vs Uaux behavior. However in this case it does not describe well the effect of R3 on the vs Uaux behavior. It is worth noting that equation (22) suggests that the value of can be enhanced not only by decreasing R3 but, in the case R3>R1, by increasing the external resistance Rex. Although this may not be practically important, since in general power output is maximized for Rex values comparable
to R1 and R3, still one can observe in Figure 9a that the highest R ex value leads to the maximum experimental and values and to the highest model predicted values. One may overall conclude from Figures 8 and 9 that there is a reasonable qualitative agreement between model and experiment. However, the model, in its present form, does not account explicitly for activation and diffusion overpotentials although, to a first approximation, one might model activation overpotential by decreasing the open-circuit potential.
4. Experimental results and Discussion The performance of the fuel cell was investigated both for conventional and for triode operation. In the latter case, the triode operation was tested both in absence and in presence of CO.
4.1 Pure H2 feed Figure 10 shows typical transient and steady state results obtained upon imposition of an electrolytic potential Uo Uaux 1.6 V in absence of CO at the anode. One observes in figure 10a that application of a potential of Uo Uaux 1.6 V leads to an 180% increase in the power output of the fuel cell ( 2.8) . The power output increase (80 mW) is 5% larger than the power sacrificed in the auxiliary circuit, therefore =1.05, which implies a 5% increase in the overall thermodynamic efficiency. As shown in Figure 10b, application of an electrolytic potential of 1.6 V leads to enhanced fuel cell performance over the entire current density range. The open circuit potential almost doubles (Fig. 10b) and this leads to values above 2.8 (Fig. 10a), in good qualitative agreement with equation (22).
4.2 CO poisoned feed Figure 11 shows some typical transient and steady state results obtained upon imposition of an electrolytic potential Uo Uaux 1.6 V in presence of 90 ppm CO added to the H2 feed at the anode. Application of the same electrolytic potential of 1.6 V leads to a 380% increase in power output (4.8). The power output increase (~60 mW) is roughly 10% lower than the power sacrificed in the auxiliary circuit, thus 0.9. One observes in Figure 11b that triode operation enhances very significantly the power output of the Teflon supported fuel cell with values of the order of 4 – 5. Actually, for small current values (< 100 mA), the triode performance of the CO poisoned cell is better than the typical performance of the pure H2 fed cell.
Figure 12 shows the dependence of the triode fuel cell performance indicators ρ and Λ on Ifc , under different CO concentrations (30, 60, 120, 160 ppm) for three different Uaux values (200, 500 and 900 mV). One observes that upon increasing Uaux there is a pronounced increase in , particularly under more severe poisoning conditions, and a significant decrease in . Values of significantly larger than unity are obtained only at lower Uaux values (Fig.12a). The rate enhancement ratio reaches values up to =5.5 for larger Uaux values and interestingly passes through a maximum at intermediate Ifc values (Figure 12c). Figures 13 and 14 present similar data for the graphite-supported PEMC which exhibits selfsustained current and potential oscillations (and thus also power output oscillations). Increasing the level of CO at the anode (from 50 ppm in Fig. 13 to 90 ppm in Fig. 14) causes an increase in the time-averaged value of , (denoted ) from ~2.5 to 3.5 and a concomitant decrease in the value of the time-averaged value (denoted ) from 0.8 to 0.55. It is worth noting that self-sustained oscillations of the type shown in Figures 13 and 14 were not observed in the Teflon-supported cell. This may be due to the presence of the Au current collecting grid in the latter case which is also in contact with the aqueous phase. Thus part of the anodic charge transfer reaction occurs at the Au-solution-Nafion three-phase-boundaries instead of the (Pt-Ru)-solution-Nafion tpb which is well-known [2, 17-22] to be responsible for the current and potential oscillations. It is interesting that from a practical point of view the presence of Au current collecting grid is advantageous, as it eliminates the oscillations and thus simplifies the cell operation and control.
5.
Nernst-Planck model Particles, such as protons, move due to gradients in their electrochemical potential (in
J mol 1 ). The latter is defined from
eF
(25)
where is the chemical potential of the protons in the membrane, e 1 is the proton charge, F is Faraday’s constant (96487 C mol 1 ) and (in V ) is the local electrical potential in the membrane. The chemical potential, , is commonly expressed as
o T RT lna
(26)
where o T is the standard chemical potential of protons in Nafion at atmospheric pressure and a is the proton thermodynamic activity in the membrane.
For relatively dilute solutions, a is proportional to the concentration, c , (in mol m3 ), i.e. c a co ao
(27)
where co is the concentration corresponding to a ao 1 , i.e. to pure Nafion and PH2 1bar . In this case, the chemical potential is given as c co
o (T) RT ln
(28)
while the electrochemical as c eF co
o (T) RT ln
(29)
If we take the gradient of the electrochemical potential, eqs. (29), it follows
RT lnc eF
d dlnc d RT eF dx dx dx
;
(30)
which is the driving force for diffusion and ion migration. If we now multiply eq. (30) with the concentration, c , and the mobility J
Deff RT
c Deff c
Deff RT
Deff RT
, we get the flux vector J (in mol m2 s 1 )
eFc
;
(31)
where Deff is the effective proton diffusivity in the membrane (in m2 s 1 ), R (in 8.314 J mol 1 K 1 ) stands for the ideal gas constant, and T (in K ) for the temperature of the system. This value of the effective diffusivity, Deff , can be computed via the Nernst-Einstein equation: H
F2 Deff CH RT
(32)
where CH is the proton concentration in the membrane (in mol m3 ). For an ionic conductivity 2 S m-1 and a proton concentration CH = 900 mol/m-3 [12-16], one computes at room temperature (300 K) an effective proton diffusivity, Deff , equal to 610-10 m2 s-1. In one dimension, eq.(31) takes the following form Jx
Deff RT
c
d dc Deff d Deff eFc dx dx RT dx
If no electrical field is applied J x Deff
(33) dc dx
In order to gain more detailed insight into the mechanism of the triode operation we have calculated the local proton ( H ) concentration, c , and/or its flux in the solid electrolyte, in both conventional and triode operation by using the transport law for the protons [23-24]:
Dc J Dt
In Eqs. (31) & (34),
(34) D is the gradient v is the material derivative, e x e y ez Dt t x y z
vector. Under steady operation and in absence of any flow field, i.e. v 0 , the left-hand side of eq. (31) is equal to zero, and the transport law for H becomes: Deff J c 0 RT
(35)
If we assume that the concentration which is the prefactor of the gradient of the electrochemical potential is a constant equal to c0 (the average proton concentration in the membrane), we can linearize eq. (35) and get the electrochemical potential field by solving numerically a Laplace equation: 0 2
(36)
The current in the membrane due to the motion H can be easily expressed: i FeJ
(37)
where i is the current density expressed in A m2 , and F e is again the charge per mole. The imposed potential across the membrane is subject to Gauss’s law for electricity D
(38)
under the assumptions that the corresponding electric field E is irrotational, the membrane has a uniform composition, and the local variation in ion concentration even close to electrodes little affects the spatial variation of the externally imposed . The first assumption states that E and the other two that Fz c 0 (“quasi-neutral” (QN) assumption). If we consider that the electric displacement D is related to the electric field through D E , and substitute the last two expressions into eq. (38), we get that 2 0
(39)
where is the relative electric permittivity (in C V 1m1 ) of the membrane [25]. Figures 15a and 15b show the geometric model used for the membrane and the three electrodes, indicating spatial symmetry in all variables and the cartesian coordinate system. A spatial unit of the electrolyte or the membrane ignoring the thickness of the electrodes is considered to be a cube of height H 0.2 mm and length x3 , with the cathode covering its entire bottom side, while the anode is placed symmetrically on its top with length x1 and the auxiliary electrode is located between x2 and x3 , respectively.
Fig. 15c and 15d show the fluxes, Ja, J1, J2, Jaux,in, Jaux,out, Jel, Jfar, Jtotal between the three electrodes in conventional and triode operation, respectively. In both fig. 15c and 15d the H flux contributions are depicted along with the imposed or computed electric potentials on each electrode and the currents between them.
5.1. Conventional fuel cell operation In the conventional operation, protons are produced at the anode and consumed at the cathode introducing a concentration gradient between the two. These electrodes are connected through an external resistance, Rex , allowing an equal electron current through it to close the circuit. No external potential is imposed on the auxiliary electrode. Thus 0 throughout the membrane, the electrochemical potential is equal to the chemical potential , U aux is, in this case, an unknown, and a base value, Uo ( 1V) , is induced to it by the presence of H2 and O2 on the two sides of the membrane. In view of eq.(28) the chemical potential can be also expressed by an effective Nernst potential via Ec Eco
RT c ln F co
(40)
Governing equation for this system is eq. (36) divided by F Ec 0 2
(41)
subject to the potential in the two electrodes: Ec (0 x x1 , y H ) U fc
(42)
Ec ( 0 x x3 , y 0) Uc 0
(43)
The high conductivity of the auxiliary electrode guarantees that it is in a uniform, but unknown scaled electrochemical potential. Imposing no overall flux through this electrode generates the condition needed to determine this potential: x3
x2
Ec y
dx 0
(44)
yH
This condition necessitates the existence of a zero-flux point on the auxiliary electrode, irrespective of its distance from the anode, i.e. a point at x xzfp , y H where
Ec y
0 . No proton flow yH
can take place through the side surface of the membrane, or its surface between the anode and the auxiliary electrode: Ec ( x x3 , y ) 0 x
(45)
Ec ( x1 x x2 , y H ) 0 y
(46)
Finally, the potential field is insulated along surface x 0 : Ec ( x 0, y ) 0 x
(47)
5.2 Triode fuel cell operation In this case U aux Uc 0 in the region x 2 x x3 inducing a proton current from the cathode to the auxiliary electrode through a second external circuit, while U fc which is the boundary value along the anode is computed from eq.(18) of the Kirchhoff model. To this end, eq.(36) is solved again, but with boundary conditions on the electrodes modified as follows: / F Ec (0 x x1, y H ) U fc
(48)
/ F (x 2 x x3 , y H ) Uo Uaux
(49)
/ F Ec (0 x x3 , y 0) 0
(50)
The rest of the boundary conditions are same as in section 5.1.
5.3 Solution method The method of finite elements is used to solve eq.(35) in Cartesian geometry (e.g. [25]) in a similar geometry, along with the two sets of boundary conditions, each one for the different operation of the cell. According to it, the part of the membrane depicted in fig. 15(c) is tessellated in orthogonal and equal elements. Accuracy tests demonstrated that 600 elements in both directions were sufficient, resulting in 361,201 unknowns. Bilinear basis functions, i , are used to locally approximate the chemical potential field, Ec (or the electrochemical potential field, ): Ec i i ( x, y )
(51)
i
Galerkin’s method is used to compute the unknown coefficients, i , as follows: This equation is multiplied by a basis function, the divergence theorem is used, and the relevant boundary conditions are introduced. The integrals are computed via a 3-point Gauss quadrature in each direction. The final set of linear algebraic equations is solved using the LAPACK library. The total currents in each electrode are computed in post processing the solution by computing the gradient of the potential on the respective interface and integrating via Gauss quadrature and the location of the zero-flux point.
6. Computed potential fields 6.1 Conventional operation First, we examined the proton distribution in the membrane. As representative values, we used for the length of the membrane element x3 1.5mm , for the potential at the anode U a 1V and kept in all cases the length of the auxiliary electrode at x3 x2 0.5mm . We first examined the importance of the distance between the anode and the auxiliary electrode, d x2 x1 , while maintaining
the
auxiliary
electrode
constant.
The
resulting
equipotential
lines
for
d 0.05mm,0.2 mm,0.5 mm, and 0.9 mm are shown in Fig. 16. Close inspection of these lines yields
several interesting and general observations. These contours are clearly normal to the membrane surface between the anode and the auxiliary electrode, attesting to the accuracy of our computations, which imposed no flux there. In the vicinity of the anode and the cathode they are nearly parallel to the respective electrodes. A proton current exists from the anode to the cathode, I1, from the anode to the auxiliary electrode, I3 and from the auxiliary electrode to the cathode, I2, all in agreement with Figs 4 and 15c. The fact that currents in opposite directions are related with the auxiliary electrode should have been anticipated given that eq. (39) has imposed on it zero total flux. The zero-flux point is always located closer to the anode. Although the auxiliary electrode is in a uniform potential U aux , which is computed to have a value in between those in the anode and the cathode, the contour lines in the immediate proximity of the auxiliary electrode are not parallel to it. The further away and the smaller the anode is, the more uniform and closer to the cathode value the potential in the membrane between the auxiliary electrode and the cathode becomes. Having computed the three current densities in A m-2 (through eq. (31) and eq. (37) using for the proton diffusivity, Deff, the calculated from eq. (32) value of 610-10 m2 s-1) and the equilibrium potential of the auxiliary electrode, Uaux, one can readily determine the values for the resistances between these electrodes. For the case of TC1 (Figure 6c), the three resistances can be computed from eq. (47) taking into account the given in Figure 16 current densities and the geometric surfaces of the anodic and auxiliary electrode. R1
U fc I1
, R2
U fc U aux U c U aux , R3 I2 I3
(52)
The calculations are shown in more detail in the Supplementary material. As shown in Figure 17, decreasing d, causes a pronounced decrease in R3 , while causing a small decrease in R1 and leaving R2 practically unaffected. This dependence on d could have been anticipated given that a resistance
is proportional to the distance between electrodes and inversely proportional to the cross section
through which a current pass. Both are constant for R2 , while the former increases and the later decreases for R3 .
6.2 Triode operation In triode operation the total potential can be assumed as the superposition of two potentials (alternative form of eq. (25)): the effective Nernst potential, Ec, (eq. 40) and the electrical potential, ( Uaux Uo ) . The membrane geometry remains the same as in 6.1.
Fig. 18a shows the contour lines of the Nernst potential Ec for conventional operation (diffusive process) for U fc 0.5V (calculated from Eq. 18 for U o 0.5V , Uaux 0 V and R3 Raux 2 ), while Fig. 18b depicts the corresponding contour lines during triode operation for
the diffusive operation alone and the corresponding directions of proton diffusion. Fig. 18c shows the corresponding variables generated only by the electrical potential, ( Uaux Uo ) . Now a much more intense flow of protons takes place in the opposite direction. The contour lines are parallel to the electrodes, except for a short area just to the left of the auxiliary electrode, where they turn sharply upwards to satisfy the no-flux condition for
x1 x x2 . Fig. 18d shows the
superposition of the previous two fields. The flux of protons now is from the anode to the cathode in the left portion and more intensive from the cathode to the entire auxiliary electrode in the right part of the membrane. These two fluxes are separated by a sharp front of contour lines. For the above set of parameters, the Nernst-Planck model predicts a value equal to 2.5 and a value equal to 1.1 which are both very close to the experimental values (Figure 9). The calculations are shown in more detail in the Supplementary material.
7. Conclusions The triode operation of PEM fuel cells was investigated and modeled using the laws of Kirchhoff and the Nernst-Planck equation. Both model and experiment showed the importance of minimizing the resistance between auxiliary electrode and anode or cathode. This led to a novel comb-type electrode geometry and significantly enhanced triode performance, with power enhancements ratios up to 6, i.e. enhanced power up to 500% and power gain ratios up to 1.2, i.e. 20% enhancement in thermodynamic efficiency. The model provides good semiquantitative estimates of the performance indicators and as a function of the external resistance of the fuel cell circuit and of the total imposed auxiliary potential. Overall the present results have shown that triode operation of PEM fuel cells can be quite advantageous and with proper fuel cell design may lead to practically useful results. The crucial
design problem is the minimization of the resistance between auxiliary and anode or cathode and this has been demonstrated both by model and by experiment utilizing novel comb-type electrode geometries which minimize distance and maximize three phase boundary length. This type of electrode geometries may be easily adapted for SOFCs and for monolithic electropromoted reactors [20,26]. Triode operation appears to be quite promising for poisoned anodes or cathodes and the explicit inclusion of activation and diffusion overpotentials to the model presented here will be necessary for obtaining closer agreement between model and experiment.
Acknowledgements This work was supported by the European Union’s Seventh Framework Programme (FP7/20072013) for the Fuel Cells and Hydrogen Joint Technology Initiative, under the project T-CELL (G.A. Number: 298300).
REFERENCES [1] S. Ye, CO-tolerant catalysts, in: J. Zhang (Ed.) PEM Fuel Cell Electrocatalysts and Catalyst Layers: Fundamentals and Applications, Springer, London, 2008, pp. 759–834. [2] A.H. Thomason, T.R. Lalk, A.J. Appleby, Journal of Power Sources, 135 (2004) 204-211. [3] C.G. Farrell, C.L. Gardner, M. Ternan, Journal of Power Sources, 171 (2007) 282-293. [4] S.P. Balomenou, C.G. Vayenas, J. Electrochem. Soc., 151 (2004) A1874-A1877. [5] S.P. Balomenou, F. Sapountzi, D. Presvytes, M. Tsampas, C.G. Vayenas, Solid State Ionics, 177 (2006) 2023-2027. [6] F.M. Sapountzi, S.C. Divane, M.N. Tsampas, C.G. Vayenas, Electrochim. Acta, 56 (2011) 6966-6975. [7] M.N. Tsampas, F.M. Sapountzi, S. Divane, E.I. Papaioannou, C.G. Vayenas, Solid State Ionics, 225 (2012) 272-276. [8] C.R. Cloutier, D.P. Wilkinson, Triode operation of a Proton Exchange Membrane (PEM) electrolyser, in: ECS Transactions, 2010, pp. 47-57. [9] D. Montinaro, A. Dellai, I. Tudorancea, A. Abdoun, Application of the triode concept to anode supported solid oxide fuel cells, in: EFC 2013 - Proceedings of the 5th European Fuel Cell Piero Lunghi Conference, 2013, pp. 131-132. [10] F.M. Sapountzi, M.N. Tsampas, C. Zhao, A. Boreave, L. Retailleau, D. Montinaro, P. Vernoux, Solid State Ionics, 277 (2015) 65-71. [11] P. Caliandro, S. Diethelm, A. Nakajo, J. Van Herle, Electrochemical model of a triode solid oxide fuel cell, in: ECS Transactions, 2015, pp. 2387-2396. [12] P. Choi, N. H. Jalani, R. Datta, J Electrochem. Soc. 152 (2005) E123-E130 [13] MN Tsampas, A Pikos, S Brosda, A Katsaounis, CG Vayenas, Electrochim. Acta 51 (2006) 2743-2755 [14] C.G. Vayenas, M.N. Tsampas, A. Katsaounis, Electrochim. Acta 52 (2007) 2244-2256 [15] S. Ochi, O. Kamishima, J. Mizusaki, J. Kawamura, Solid State Ionics 180 (2009) 580-584 [16] O. Sel, L. To Thi Kim, C. Debiemme-Chouvy, C. Gabrielli, C. Laberty-Robert and H. Perrot, Langmuir 29 (2013) 13655-13660 [17] H. Lu, L. Rihko-Struckmann, R. Hanke-Rauschenbach, K. Sundmacher, Topics in Catalysis, 51 (2008) 89-97. [18] A. Mota, P.P. Lopes, E.A. Ticianelli, E.R. Gonzalez, H. Varela, Journal of the Electrochemical Society, 157 (2010). [19] S. Kirsch, R. Hanke-Rauschenbach, B. Stein, R. Kraume, K. Sundmacher, Journal of the Electrochemical Society, 160 (2013). [20] C.G. Vayenas, S. Bebelis, C. Pliangos, S. Brosda, D. Tsiplakides, Electrochemical Activation of Catalysis: Promotion, Electrochemical Promotion and Metal-Support Interactions, Kluwer Academic/Plenum Publishers, New York, 2001. [21] H.-G. Lintz, C.G. Vayenas, Angewandte Chemie - International Edition in English, 28 (1989) 708-715. [22] C.G. Vayenas, S. Bebelis, I.V. Yentekakis, P. Tsiakaras, H. Karasali, Platinum Metals Review, 34 (1990) 122-130. [23] J. Papaioannou, G. Karapetsas, Y. Dimakopoulos and J. Tsamopoulos, "Injection of a viscoplastic material inside a tube or between parallel disks: conditions for wall detachment of the advancing front”, J. Rheol. 53(5), 1155-1191 (2009). [24] Electrochemical Systems, Third Edition, by John Newman and Karen E. Thomas-Alyea ISBN 0-471-47756-7, John Wiley & Sons, Inc (2004). [25] N.A. Pelekasis, K. Economou, J.A. Tsamopoulos, Linear oscillations and stability of a liquid bridge in an axial electric field, Physics of Fluids, 13 (12), 3564-3581 (2001). [26] Ph. Vernoux, L. Lizzaraga, M.N. Tsampas, F.M. Sapountzi, A. De Lucas-Consuegra, J.-L. Valverde, S. Souentie, C.G. Vayenas, D. Tsiplakides, S. Balomenou, E.A. Baranova, Chemical Reviews, 113, 8192-8260 (2013).
Figure Captions Figure 1: Schematic of the triode fuel cell concept, showing the fuel cell circuit and the auxiliary circuit; P/G: potentiostat - galvanostat Figure 2: Schematic of the triode fuel cell concept, showing (a) the geometry of the triode PEM fuel cell and the electrical circuits and (b) the geometry of the membrane electrode assembly. The fuel cell cathode acts simultaneously as an electrode of the auxiliary circuit. P/G: potentiostat– galvanostat. (a) Graphite casing design, (b) Electrode geometry. Figure 3: (a) Layout of the new comb-type triode membrane electrode assembly (MEA) electrode design for decreasing the resistance between the anode and the auxiliary electrode. (b) Polytetrafluoroethylene (PTFE,Teflon) custom-made plate with two detachable graphite blocks acting as current collectors. Figure 4: Schematic of the electrical circuit of a triode PEM fuel cell operating in the conventional mode without imposition of auxiliary potential. Uo is the open-circuit potential. Figure 5: Typical dependence of fuel cell potential, Ufc, and of anode-auxiliary electrode potential difference, U3, on fuel cell current, Ifc, and current density, jfc, showing the definition of R1* and
R *3 ; pure H2 feed; TC1, R3=1.8 Ω, conventional operation (Iaux=0 A). Figure 6: Ifc-Ufc and Ifc-U3 curves under conventional mode (Iaux=0 mA) of fuel cell operation, and corresponding computed R1, R2 and R3 values for cells (a) GS1, (b) GC1 and (c) TC1. Figure 7: (a) Schematic of the electrical circuit of a triode PEM fuel cell operating in the triode mode via auxiliary potential (Uaux) application. (b) Plot of Eq. 18 showing the contribution of the applied Uaux, to the potential of the fuel cell, Ufc, as a function of the external resistance, Rex, for three different values of the resistance R3 (=5, 50 and 100 Ω) and Uo=1V. Figure 8: Comparison of experimentally measured and model predicted (equation 18) dependence of the fuel cell potential, Ufc, on the external resistance, Rex, and on the imposed auxiliary potential (Uo to Uaux=1 to 1.6 V) (a) for cell GS1 (R3=50 ) and (b) for case TC2 (R3=2 ); Anode feed: 100%H2. Figure 9: Comparison of experimentally measured and model predicted (equations (22) and (23)) dependence of the power enhancement ratio, (top), and of the power gain ratio, (bottom) on the imposed auxiliary potential Uaux, using the triode cell TC3. Experimental resistance values: R1=1.50 , R2=2.00 , R3=1.95 . Experimental Rex values shown in the Figures and also used in the model: Rex=1.75 , 4 , and 15 . Electrolyte resistance values used in the model: R1=1.50 , R2=2.00 , R3=1.95 (same as the experimental values, Fig. 9a) and R1=1.50 , R2=2.00 , R3=1.2 , Fig. 9b, see text for discussion. Figure 10: (a) Top left: Time variation of imposed auxiliary potential ΔUaux and corresponding auxiliary current Iaux and auxiliary power Paux. Lower left: Corresponding variation of Ufc, Ifc and power output Pfc. Top right: Corresponding variation of power enhancement ratio and power gain ratio. Anode feed: 100% H2 ; external resistance Rex=15 Ω; TC1, R3 =1.8 Ω. (b) Ifc - Ufc curves of the fuel cell circuit obtained during conventional operation (Iaux=0) and triode operation (Uo+ΔUaux=1.6V) by varying Rex; anode feed 100% H2; TC1; R3 =1.8 Ω. Figure 11: (a) Top left: Time variation of imposed auxiliary potential Uaux and of corresponding auxiliary current Iaux and auxiliary power Paux. Lower left: corresponding variation of Ufc, Ifc and power output Pfc. Top right: corresponding variation of power enhancement ratio ( Pfc Pfc0 ) and power gain ratio ( Pfc Paux ) . Anode feed: 90 ppm CO, 81.6% H2. (b) Ifc–Ufc curves of the fuel cell circuit obtained during conventional operation (Iaux = 0 mA) for the case of 100% H2 feed for
the anode (filled circles) and for the case of 90 ppm CO/81.6% H2 feed for the anode (open diamonds) and Ifc–Ufc curve obtained during triode operation (Uo+Uaux=1.6V) for the case of 90 ppm CO/81.6% H2 feed for the anode (filled diamonds) by varying the external resistive load, R ex; TC1, R3=1.8 Ω. Figure 12: Experimental values of ratios ρ and Λ, as a function of Ifc, under various CO concentrations (30, 60, 120, 160 ppm) and applied potential Uaux of 200 mV (a), 500 mV (b) and 900 mV (c); TC2, R3=2.0 . Figure 13: (a) Time variation of fuel cell power output, Pfc, (dense line), fuel cell current, Ifc, (lighter line) and fuel cell potential, Ufc, (dashed line) on the lower left, sacrificed power and current in the auxiliary circuit, Paux and I aux respectively on the top left, power enhancement ratio, ρ, on the top right and power gain ratio, Λ, on the lower right and time-averaged values for Pfc, ρ and Λ for triode operation during imposition of a constant auxiliary potential, U o+Uaux =1.9V, under a fixed external resistive load (Rex = 3 Ω). Anode feed: 50 ppm CO/16% H2. (b) Ifc–Ufc curves of the fuel cell circuit obtained during conventional operation (Iaux = 0 mA) for the case of 16% H2 feed for the anode (filled circles) and for the case of 50 ppm CO/16% H2 feed for the anode (open diamonds) and Ifc–Ufc curve obtained during triode operation mode (Uo+Uaux=1.9V) for the case of 50 ppm CO/16% H2 feed for the anode (filled diamonds) by varying the external resistive load, R ex; GS1, R3=50 Ω. Figure 14: (a) Time variation of fuel cell power output, Pfc, (dense line), fuel cell current, Ifc, (lighter line) and fuel cell potential, Ufc, (dashed line) on the lower left, sacrificed power and current in the auxiliary circuit, Paux and I aux respectively on the top left, power enhancement ratio, ρ, on the top right and power gain ratio, Λ, on the lower right and time-averaged values for Pfc, ρ and Λ for triode operation during imposition of a constant auxiliary potential, U o+Uaux =1.6 V, under a fixed external resistive load (Rex = 1.2 Ω). Anode feed: 90 ppm CO/81.6% H2. (b) Ifc–Ufc curves of the fuel cell circuit obtained during conventional operation (Iaux = 0) for the case of 100% H2 feed for the anode (filled circles) and for the case of 90 ppm CO/81.6% H2 feed for the anode (open diamonds) and Ifc–Ufc curve obtained during triode operation mode (Uo+Uaux=1.6V) for the case of 90 ppm CO/81.6% H2 feed for the anode (filled diamonds) by varying the external resistive load, Rex; GC1, R3=10.1 Ω. Figure 15: Schematic of (a) the comb-type structured anode and auxiliary electrodes and (b) the spatial unit of the electrolyte (membrane) used for the Nernst-Planck mode solution. Current fluxes due to (c) diffusion (conventional operation-chemical potential gradient driven) and (d) diffusion and electrolysis process (triode operation-chemical potential and electric potential gradients driven). Figure 16: Potential distribution in the electrolyte for different anode-auxiliary electrode distances (d=0.9, 0.5, 0.2, 0.05 mm). Rex =2 . Figure 17: Effect of the anode - auxiliary distance, d on the resistances R1, R2, R3 computed via the Nernst-Planck model during conventional operation. Figure 18: Potential distribution inside the electrolyte taking into account (a) the diffusion process during conventional operation (b) the diffusion process during triode operation, (c) the electrolytic process during triode operation and (d) the two processes together (for the triode operation) according to the Nernst-Planck Equation, R3=Rex=2 Ω, d=0.2 mm , Uaux=0.6 V, S2 is the auxiliary electrode surface area.
Table 1: Geometric characteristics of the anodic and auxiliary electrodes used in the graphite and Teflon casing designs (G: Graphite, T: Teflon, S: Square, C: Comb, #: number of sample)
Name
GS1 GC1 TC1 TC2 TC3
number of comb teeth 4 6 6 6
Anode Surface area/ cm2 3.85 2.35 8.5 9.42 8.90
number of comb teeth 3 6 5 5
Auxiliary Surface area/ cm2
R3/
0.49 1.96 5.88 6.71 6.50
50 10.1 1.8 2.0 1.95
Figure 1
(a)
(b) Figure 2
(a)
(b) Figure 3
Figure 4
Figure 5
R1=1.8Ω R2=14.2Ω R3=49.9Ω
Figure 6 (a)
R1=1.1Ω R2=1.3Ω R3=10.1Ω
Figure 6 (b)
R1=0.7Ω R2=1.1Ω R3=1.8Ω
Figure 6 (c)
(a)
(b)
Figure 7
(a)
(b) Figure 8
Figure 9 (a)
Figure 9 (b)
(a)
(b) Figure 10
(a)
(b) Figure 11
(a)
Figure 12
(b)
Figure 12
(c) Figure 12
(a)
(b) Figure 13
(a)
(b) Figure 14
(a)
(b)
(c)
(d)
Figure 15
Figure 16
Figure 17
(a)
(b)
(c)
(d)
Figure 18
S0013-4686(17)31608-0 http://dx.doi.org/doi:10.1016/j.electacta.2017.07.168 EA 29985
To appear in:
Electrochimica Acta
Received date: Revised date: Accepted date:
13-3-2017 25-7-2017 27-7-2017
Please cite this article as: E.Martino, G.Koilias, M.Athanasiou, A.Katsaounis, Y.Dimakopoulos, J.Tsamopoulos, C.G.Vayenas, Experimental investigation and mathematical modeling of triode PEM fuel cells, Electrochimica Actahttp://dx.doi.org/10.1016/j.electacta.2017.07.168 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
MARKED COPY Experimental investigation and mathematical modeling of triode PEM fuel cells E. Martino1, G. Koilias1, M. Athanasiou1, A. Katsaounis1, Y. Dimakopoulos1, J. Tsamopoulos1 and C.G. Vayenas1,2, 1
Department of Chemical Engineering, University of Patras, Caratheodory 1 St, GR-26504 Patras,
Greece, 2
Academy of Athens, Panepistimiou 28 Ave., GR-10679 Athens, Greece
Corresponding author: [email protected] (Costas G. Vayenas)
EO17-1450R1 Highlights
The triode fuel cell operation was tested using novel comb-type electrode designs Triode operation enhances the PEMFC power output by up to 500% Power output enhancement exceeds auxiliary power by up to 20% Good agreement with mathematical model based on the laws of Kirchhoff Proton fluxes in the membrane found via solution of the Nernst Planck equation
Abstract The triode operation of humidified PEM fuel cells has been investigated both with pure H2 and with CO poisoned H2 feed over commercial Vulcan supported Pt(30%)-Ru(15%) anodes. It was found that triode operation, which involves the use of a third, auxiliary, electrode, leads to up to 400% power output increase with the same CO poisoned H2 gas feed. At low current densities, the power increase is accompanied by an increase in overall thermodynamic efficiency. A mathematical model, based on Kirchhoff’s laws, has been developed which is in reasonably good agreement with the experimental results. In order to gain some additional insight into the mechanism of triode operation, the model has been also extended to describe the potential distribution inside the Nafion membrane via the numerical solution of the Nernst-Planck equation. Both model and experiment have shown the critical role of minimizing the auxiliary-anode or auxiliary-cathode resistance, and this has led to improved comb-shaped anode or cathode electrode geometries.
1. Introduction A major problem of low-temperature Proton Exchange Membrane Fuel Cells (PEMFCs) is the observed susceptibility of Pt-based carbon supported catalysts to CO poisoning. This process leads to the degradation of the anode and consequently to a severe drop in the cell performance, as CO is strongly adsorbed on Pt and blocks any further H2 adsorption [1-3]. Since H2 is the most often used fuel in PEMFC applications and is mainly produced by reforming processes of hydrocarbons or liquid alcohols, it is very common that the anode fuel feed contains significant amounts of CO. In this paper, we examine how the triode design of PEM fuel cells can enhance the cell performance both with pure H2 feed and under CO poisoning conditions. The triode fuel cell operation, first proposed a few years ago [4-8], is an alternative method for the enhancement of the power output and thermodynamic efficiency of electrochemical power producing devices. The triode fuel cell design introduces, in addition to the anode and the cathode, a third, auxiliary, electrode in contact with the solid electrolyte, e.g. polymer electrolyte membrane in the case of PEMFCs. This electrode forms together with the anode or cathode, a second, auxiliary, electric circuit operating in parallel with the conventional main circuit of the cell (Fig. 1). The auxiliary circuit runs in the electrolytic mode, pumping ions (i.e. protons in the case of a PEMFC) from the cathode to the auxiliary electrode. In this way, imposition of a potential difference between the auxiliary electrode and the cathode permits the primary circuit of the fuel cell to operate under previously inaccessible, i.e. larger than Uo=1.23 V, anode-cathode potentials. This introduces a new controllable variable in fuel cell operation and can lead to significant reduction of the anodic and cathodic overpotentials and thus to a significant increase in the power output of a fuel cell [4-7]. It can also over certain ranges of design and operational conditions lead to an enhancement of the overall thermodynamic efficiency. Two parameters are used to quantify the performance of a triode cell, i.e. the power enhancement ratio, , and the power gain ratio, , defined by:
Pfc / Pfco
(1)
Pfc / Paux (Pfc Pfco ) / Paux
(2)
and
where Pfc and Pfco
are the power outputs in triode mode operation and conventional mode of
operation respectively, Paux is the power sacrificed in the auxiliary circuit and Pfc is the increase in the fuel cell power output. The overall thermodynamic efficiency is enhanced when 1 . In more recent papers the triode concept has been applied also to enhance electrolysis [8], to study a prototype anode supported solid oxide fuel cell, SOFC, under hydrogen and steam
reforming conditions [9] and to enhance the performance of H2S – poisoned SOFCs operated under CH4 – H2O mixtures [10]. On the modeling side, there has been so far only one reference [11]
reporting an
electrochemical model of a triode SOFC based on a two-dimensional commercial software, COMSOL Multiphysics. In this paper we first present a simple mathematical model based on Kirchhoff's laws, describing conventional fuel cell operation (CFCO) of triode fuel cells which enables one to extract the ohmic resistance values between the three electrodes which are all in electrolytic contact. Subsequently, an extension of this model, also based on Kirchhoff's laws, is presented which can describe the triode fuel cell operation (TFCO), using the resistance values extracted from the CFCO model and allowing for the computation of the power enhancement ratio, , and the gain ratio, , as a function of the pre-calculated resistances, the cell geometry and the operating conditions. This model also provides the boundary conditions for solving, via finite elements, the Nernst-Planck equation inside the electrolyte and thus obtaining the spatial distribution of ionic current and local potential in the electrolyte during triode operation. Both the model and the experiments show the crucial role of minimizing the ionic resistance between anode and auxiliary electrode for successful triode operation. This led to the development of a novel and more efficient triode design than the classical triode design used in previous studies [4-7]. Both the classical design and the novel one were tested both with humidified hydrogen (3% H2O) fuel as well as under humidified CO poisoning mixtures of hydrogen (3% H2O) with CO concentrations up to 120 ppm.
2. Experimental Apparatus a. Graphite casing design The graphite casing triode PEMFC design used in previous studies [4-7] is shown schematically in Fig. 2 and has been already described in detail [4-7]. Anode and auxiliary electrodes were commercial E-TEK Pt(30%)–Ru(15%) supported on Vulcan XC-72 carbon deposited on E-TEK carbon cloth with a total catalyst loading of 0.5 mg/cm2. The cathode was Pt supported on Vulcan XC-72 carbon deposited on carbon cloth (0.5 mg/cm2). The electrode geometry is shown in Fig. 2b. The flow and gas analysis system has been described elsewhere [47]. The superficial surface area of the cathode (Pt) was 5.29 cm2, of the anode (Pt-Ru) was 3.85 cm2 and of the auxiliary electrode (Pt-Ru) was 0.49 cm2 (Figure 1b). The cathode was a square and the auxiliary electrode was a smaller square located in the center of the hollow square anode (Fig. 1b). The membrane was Nafion 117 with nominal thickness 185m. The membrane electrode assembly
(MEA) was prepared by hot pressing in a model C Carver hot press at 120oC and under pressure of 1 metric ton for 3 min. Preliminary investigation showed that the applied auxiliary potential, Uaux , should not exceed 1.9 V, as this leads to CO2 formation at the cathode via oxidation of the carbon support. The gas feeds to the cathode and anode compartments (the latter includes also the auxiliary electrode, Fig. 1) were continuously humidified using thermostated gas saturators. The cell temperature was typically set at the same temperature (25oC) with the gas saturators. The anode compartment gas feed was Air Liquide certified gas mixtures of 490 ppm CO/He, which could be further diluted with Linde (N4.5) H2. The cathode feed was humidified Air Liquide synthetic air. The fuel cell circuit included a decade resistance Box (Time Electronics Ltd 1051) in order to vary the external load. The current and the potential were measured by three-digital multimeters (Metex ME 21). For the fixed triode operation, constant potentials or currents in the auxiliary circuit were applied using an AMEL 2053 Potentiostat – Galvanostat. It is evident from Fig. 1a that all three electrodes operate in a corrosion-type mode with part of their surface used for oxidation and part of their surface used for reduction. Despite this rather intense mode of operation no performance deterioration was observed during operation for several weeks.
b. Teflon casing design Figure 3 shows the configuration of the novel Teflon (Polytetrafluoroethylene, (PTFE)) cased triode PEM fuel cell developed in this study. The anode-auxiliary side of the new MEA consisted of two alternating comb-like electrodes with the comb teeth in close proximity (0.5 mm) to each other. The comb shape of the anode and auxiliary electrodes ensures better proton conduction between the two electrodes, which is achieved via the minimization of the electrolyte resistance between them. The need to minimize this resistance is underlined both by the experimental results and by the mathematical model presented in the next section. On the other side of the membrane is placed a square type cathode (Figure 3a). Figure 3b shows the geometry of the bottom part of the Teflon casing, which was used to replace the common graphite casing, so that the comb-type structured anode and auxiliary electrodes of Figure 3a can be utilized without any electronic shortcircuiting problems. Figure 3b shows the engraved gas flow channels and the grooves for the silicon rubber gasket to provide gas tight sealing. It also shows two detachable graphite blocks embedded in the Teflon casing which are in contact with the electrode edges outside the gas flow region and which in combination with a 0.025 mm thick Au foil, act as current collectors.
The gas feed unit and electrical measurement unit were the same with those utilized in the graphite casing experiments.
c. MEA preparation and cell operation With both designs (graphite or Teflon casing) the same fully humidified Nafion 117 (DuPont) membranes with a nominal thickness of 185 μm were used as proton conducting electrolyte. The membrane was treated according to the standard procedure of preparation analyzed in detail in previous studies [5-8]. The MEA was prepared by hot pressing the electrodes and electrolyte in a model Carver heated press using pressure steps of 1 metric ton. Every step had a duration of 3 min and corresponded to a 10 °C rise up to 80°C and then to a 5°C rise up to the final temperature of 120°C. Table 1 shows the details of all anodic-auxiliary electrodes which were prepared and used in this study. The first letter of the name shows the fuel cell casing material (G: Graphite, T: Teflon), the second letter indicates the geometry of the anode-auxiliary electrodes (S: Square, C: Comb) and the third letter indicates the number of the sample. All current and potentials were measured by a series of digital multimeters. The anode – auxiliary compartment fuel gas feed was humidified ultrapure H2 (Linde Gas certified 99.999% H2) and humidified H2/CO mixtures (Air Liquide certified 490ppm CO in He) further diluted in He (Air Liquide). The cathode feed was (Air Liquide) synthetic air. In order to humidify the inlet gas feeds, the gases passed continuously through two thermostated gas saturators operated at 25oC. The concentrations of CO and CO2 in the anode – auxiliary feed and effluent were monitored using a Fuji Electric IR CO/CO2 analyzer. All the current-potential curves were obtained under a constant total volumetric flow rate of 300 cm3STP/min both at the anode and at the cathode compartments, with CO concentration varying between 5 and 120 ppm. In the conventional mode, the Ifc-Ufc curves were obtained by varying the external resistance, Rex, from 0 up to 9 MΩ in specific steps, such that the Ufc varied by approximately 50 mV in each step. In the case of triode operation various potentials, Uaux, were applied to the auxiliary circuit via the galvanostat / potentiostat in the range of 0-1.6 V.
3. Mathematical modeling a. Conventional Fuel cell operation modeling and comparison with experiment In this section we present a simple mathematical model based on Kirchhoff’s laws which describes conventional fuel cell operation of a triode cell, i.e. without the application of auxiliary
current or potential. The model permits the extraction of the three ohmic resistances for proton conduction between anode and cathode (denoted R1), auxiliary electrode and cathode (denoted R2) and anode-auxiliary (denoted R3) from current-potential data. All the equations apply for the case of conventional mode operation, but obviously the extracted resistance values are valid for triode cell operation as well. Fig. 4 shows the electrical circuit of a triode fuel cell under conventional mode operation, including all the parameters of the model, which are the values of the three internal resistances, R1, R2 and R3, the open circuit potential Uo, the fuel cell current, Ifc, flowing through the main circuit and the external resistance of the fuel cell circuit, Rex. We first note that in the conventional fuel cell operation of the triode cell, the potential of the auxiliary electrode, Uaux, takes a value between the potentials, Ua and Uc, of the anode and cathode. This implies that in addition to the anode-cathode current (proton) pathway, some current, denoted I3, flows from the anode to the auxiliary electrode and a current I2, equal to I3, flows from the auxiliary electrode to the cathode. Thus part of the auxiliary electrode acts as a cathode, reducing protons to H2, and part of the same electrode serves as an anode, oxidizing H2 to protons. We then examine the loop ABCD (see Fig. 4) and we apply Kirchhoff's second law. Note that this closed loop does not contain the resistance Rex. Following the path of charge along the loop, we note that (a) the two potential source terms (Uo and, along the path, - Uo) cancel out and (b) that the currents I2 and I3 have the same sign which is opposite to that of I1. Note that Kirchhoff's laws are valid regardless of the nature of the charge carrier, which in the present case is electrons in the electrodes and protons in the membrane. Thus, application of Kirchhoff's 2nd law to the loop ABCD (Figure 4) gives: I2R 2 I3R 3 I1R1 0
(3)
and since I2 I3 it follows,
I3 R1 I1 R 2 R 3
I3 R1 I1 I3 R1 R 2 R 3
;
(4)
Application of Kirchhoff's first law at point A gives I1 I3 Ifc
(5)
and thus from eqs. (4) and (5) it follows:
(R 2 R 3 ) I1 Ifc R1 R 2 R 3
;
I3 R1 Ifc R1 R 2 R 3
(6)
Denoting by U3 and U1 the products U3 I3R 3
;
U1 I1R1
(7)
where U3 is the potential difference between the anode and the auxiliary electrode, and U1 ( Uo Ufc ) is the total fuel cell overpotential, assumed to be ohmic, it follows from equations
(4-7) that U1 R1 (R 2 R 3 ) R1 Ifc R1 R 2 R 3
;
U3 R1R 3 R*3 Ifc R1 R 2 R 3
(8)
The values of R1 and R *3 can be obtained experimentally from the I-U plots during conventional fuel cell operation as shown in Fig 5. A third equation for obtaining R1, R2 and R3 is obtained by assuming that the ratio, , of the resistances R1 and R2 of the anode and of the auxiliary circuit can be estimated from the surface areas, S1 and S2, of the two electrodes using the expression for the resistance of a cylindrical conductor and assuming uniform current density, i.e. R1 L / S1 S2 R 2 L / S2 S1
(9)
where L is the thickness of the electrolyte and is the specific electrical resistance of the electrolyte, which equals to 1 , where is the ionic conductivity of the hydrated Nafion membrane. There is a rich literature on the conductivity of Nafion which is of the order of 0.02 S cm-1 at temperatures 20o to 80 oC [12 - 15] Denoting S2 / S1 ;
and
R1* R *3
1
it follows
1 1 R 3 R *3 1 ;
R 2 R 3
;
R1 R 3
(10)
As shown in Figure 6, which compares the values of R1, R2 and R3 computed for three different geometries of the anode and auxiliary electrodes, the value of R1 is not very sensitive to electrode geometry which, however, strongly affects the values of R2 and R3. Thus R2 decreases from 14.2 to 1.3 and to 1.1 while R3 decreases from ~50 to ~10 and to ~1.8 upon replacing the square electrode geometry of Figure 6a to the comb-like electrode geometries of Figures 6b and 6c. The decrease in R2 is due to the increase in the surface area of the auxiliary electrode, while the decrease in R3 is additionally due to the increase in the three-phase-boundaries length gas-electrode-electrolyte as further discussed below.
b. Triode fuel cell operation modeling via the equations of Kirchhoff and comparison with experiment As already noted, the electrical potentials, Uc, Ua and Uaux, denote respectively the assumed uniform potentials of the cathode, anode and auxiliary electrode.
It thus follows Ufc Uc Ua 0
(11)
The applied electrolytic auxiliary potential Uaux is defined from
Uaux Uaux,o Uaux Uaux Uaux,o 0
(12)
where the subscript “o” denotes open auxiliary circuit operation, i.e. conventional fuel cell operation. In the case of triode operation (Fig. 7a) an electrolytic potential difference, Uaux , is applied between the auxiliary and cathodic electrode and therefore electrolytic current, Iaux , flows through the auxiliary circuit. In this way, protons are pumped from the cathode, but also from the anode, to the auxiliary electrode where some H2 evolution occurs (Fig. 7a). Consequently, the potential difference Uc-Ua is forced to increase and the power output of the fuel cell circuit also increases. Application of Kirchhoff's laws in both circuits in the triode mode of operation (Fig. 7a) gives the following fundamental equations: U o Ifc R ex Ifar R1 U o U aux I fc (R ex R 3 )
(13)
( U o U aux ) U o I el R 2
which are obtained from the 2nd law of Kirchhoff in loops ABCA, ABCDEFA and FEDF respectively. Also applying the 1st Kirchhoff’s law at point A one obtains Iaux Ifar Ifc
(14)
and using Kirchhoff’s 1st law at point F it follows
Iaux Iel IP/G
(15)
Also, Ohm’s law for line ABC gives Ufc Ifc R ex
(16)
and Kirchhoff’s 2nd law in loop FEDF yields U o Iel R 2 U o U aux 0 Iel
Uaux R2
(17)
where Ifar is the net faradaic current of fuel consumption, Ifc is the electric current flowing through the main (fuel cell) circuit in triode mode, Iaux is the current flowing through the secondary (auxiliary) circuit, U fc is the potential that develops between anode-cathode in the triode mode,
Uaux the potential imposed between the auxiliary electrode and the cathode, U o is the open circuit
potential of the two electrochemical cells, R ex is the external variable resistance of the main circuit and R 3 is the internal electrolyte resistance between the two electrodes on the same side, i.e. between the anode and the auxiliary electrode. Upon combining the second equation (13) with equation (16) one obtains U fc
U o U aux 1 R 3 / R ex
; Ifc
U o U aux R ex R 3
(18)
This equation is plotted in Figure 7b as a function of the external resistance R ex for various values of R 3 . One observes that U fc reaches the value Uo Uaux of the applied potential for R ex values higher than R 3 . Since R ex values less than 1 are necessary to obtain current and power densities of the order of 1 A/cm2 and 1 W/cm2 respectively, it follows that R3 values less than 5 are necessary for this purpose as shown in Figure 7b. The very good agreement between equation (19) and experiment is shown in Figure 8. This agreement confirms the need for minimizing the resistance R3 for successful triode operation. Furthermore by using the first equation (13) one can compute the Faradaic current Ifar, i.e. Ifar
Uo Ifc R ex R1
(19)
Using equation (18) one obtains
Ifar
U o 1 Uaux / U o 1 R1 1 R 3 / R ex
Iaux Ifc Ifar Uo R 3 R ex
(20)
U o U aux U o 1 U aux / U o – 1 R 3 R ex R1 1 R 3 / R ex
R 3 U aux R ex 1 1 R1 Uo R1
(21)
Consequently, also accounting for equations (15) and (17), the power enhancement ratio, , and the gain ratio, , can be expressed as a function of Uaux , Uo, R1 , R 3 and R ex via the equations
(U o U aux )2 2 Pfc R ex (1 R 3 / R ex )2 (U o U aux )2 R1 R ex o Pfc U 2o R ex U o2 R 3 R ex (R1 R ex )2
U o Uaux 2
Pfc Paux
U 2o R ex (R1 R ex )2
R ex (1 R 3 / R ex )2 R 3 U aux R ex U aux Uo U aux 1 1 R1 U o R1 R 2 (R 3 R ex )
(22)
(23)
Equation (23) shows again the crucial role of the resistance between the anode and the auxiliary electrode, R3, and of the applied potential Uaux on the fuel cell potential U fc . This dependence was also displayed in Fig. 8. It is evident that as R 3 decreases, the effect of the applied potential on the auxiliary circuit becomes more pronounced. This observation regarding the role of R 3 led to the new cell design of Figure 3 and to the introduction of comb-shaped electrodes. This new electrode geometry leads to a significant decrease in R 3 as already shown in Fig. 6. This is due to the pronounced decrease in the distance, d, between neighboring comb teeth and to the concomitant increase in three phase boundary length (
tpb ).
The dependence of R 3 on d and
tpb can
be
approximated by
d ltpb 0 R 3 R 30 d 0 ltpb
(24)
where R 30 ( 50 ) , d0 5 103 m and
tpb 0 (
0.1 m) corresponds to the graphite casing design
of Figure 6a. Therefore, the need to minimize the resistance R 3 leads to the need to minimize the distance d and to maximize
tpb ,
which dictates the comb type geometry of Figure 6c.
Figures 9a and 9b present measured and values as a function of Uaux and compare them with those predicted from equations (22) and (23). In this comparison between the model and the experimental data the imposed potential, Uaux , was varied with a step of 10-50 mV. For small
Uaux values, the power gain ratio, , exceeds unity, because the power consumed in the auxiliary circuit is very small. On the other hand, the power enhancement ratio, , increases with increasing imposed potential. The experimental data shown in Figures 9a and 9b are the same and have been obtained with TC3 for which it is R3 > R1. These data show that increases significantly, up to 4.5, with increasing Uaux , while reaches quite high values, up to 8, for very small Uaux values. When the model is run with the experimental R1, R2 and R3 values (R1=1.50 , R2=2.00 , R3=1.95 , respectively, Fig. 9a) then it fails to predict the pronounced increase in with decreasing Uaux and instead predicts a maximum. If, however, the model is run with an R3 value smaller than R1 (e.g. R3=1.2 , Fig. 9b) then the model is in excellent agreement with the experimental vs Uaux behavior. However in this case it does not describe well the effect of R3 on the vs Uaux behavior. It is worth noting that equation (22) suggests that the value of can be enhanced not only by decreasing R3 but, in the case R3>R1, by increasing the external resistance Rex. Although this may not be practically important, since in general power output is maximized for Rex values comparable
to R1 and R3, still one can observe in Figure 9a that the highest R ex value leads to the maximum experimental and values and to the highest model predicted values. One may overall conclude from Figures 8 and 9 that there is a reasonable qualitative agreement between model and experiment. However, the model, in its present form, does not account explicitly for activation and diffusion overpotentials although, to a first approximation, one might model activation overpotential by decreasing the open-circuit potential.
4. Experimental results and Discussion The performance of the fuel cell was investigated both for conventional and for triode operation. In the latter case, the triode operation was tested both in absence and in presence of CO.
4.1 Pure H2 feed Figure 10 shows typical transient and steady state results obtained upon imposition of an electrolytic potential Uo Uaux 1.6 V in absence of CO at the anode. One observes in figure 10a that application of a potential of Uo Uaux 1.6 V leads to an 180% increase in the power output of the fuel cell ( 2.8) . The power output increase (80 mW) is 5% larger than the power sacrificed in the auxiliary circuit, therefore =1.05, which implies a 5% increase in the overall thermodynamic efficiency. As shown in Figure 10b, application of an electrolytic potential of 1.6 V leads to enhanced fuel cell performance over the entire current density range. The open circuit potential almost doubles (Fig. 10b) and this leads to values above 2.8 (Fig. 10a), in good qualitative agreement with equation (22).
4.2 CO poisoned feed Figure 11 shows some typical transient and steady state results obtained upon imposition of an electrolytic potential Uo Uaux 1.6 V in presence of 90 ppm CO added to the H2 feed at the anode. Application of the same electrolytic potential of 1.6 V leads to a 380% increase in power output (4.8). The power output increase (~60 mW) is roughly 10% lower than the power sacrificed in the auxiliary circuit, thus 0.9. One observes in Figure 11b that triode operation enhances very significantly the power output of the Teflon supported fuel cell with values of the order of 4 – 5. Actually, for small current values (< 100 mA), the triode performance of the CO poisoned cell is better than the typical performance of the pure H2 fed cell.
Figure 12 shows the dependence of the triode fuel cell performance indicators ρ and Λ on Ifc , under different CO concentrations (30, 60, 120, 160 ppm) for three different Uaux values (200, 500 and 900 mV). One observes that upon increasing Uaux there is a pronounced increase in , particularly under more severe poisoning conditions, and a significant decrease in . Values of significantly larger than unity are obtained only at lower Uaux values (Fig.12a). The rate enhancement ratio reaches values up to =5.5 for larger Uaux values and interestingly passes through a maximum at intermediate Ifc values (Figure 12c). Figures 13 and 14 present similar data for the graphite-supported PEMC which exhibits selfsustained current and potential oscillations (and thus also power output oscillations). Increasing the level of CO at the anode (from 50 ppm in Fig. 13 to 90 ppm in Fig. 14) causes an increase in the time-averaged value of , (denoted ) from ~2.5 to 3.5 and a concomitant decrease in the value of the time-averaged value (denoted ) from 0.8 to 0.55. It is worth noting that self-sustained oscillations of the type shown in Figures 13 and 14 were not observed in the Teflon-supported cell. This may be due to the presence of the Au current collecting grid in the latter case which is also in contact with the aqueous phase. Thus part of the anodic charge transfer reaction occurs at the Au-solution-Nafion three-phase-boundaries instead of the (Pt-Ru)-solution-Nafion tpb which is well-known [2, 17-22] to be responsible for the current and potential oscillations. It is interesting that from a practical point of view the presence of Au current collecting grid is advantageous, as it eliminates the oscillations and thus simplifies the cell operation and control.
5.
Nernst-Planck model Particles, such as protons, move due to gradients in their electrochemical potential (in
J mol 1 ). The latter is defined from
eF
(25)
where is the chemical potential of the protons in the membrane, e 1 is the proton charge, F is Faraday’s constant (96487 C mol 1 ) and (in V ) is the local electrical potential in the membrane. The chemical potential, , is commonly expressed as
o T RT lna
(26)
where o T is the standard chemical potential of protons in Nafion at atmospheric pressure and a is the proton thermodynamic activity in the membrane.
For relatively dilute solutions, a is proportional to the concentration, c , (in mol m3 ), i.e. c a co ao
(27)
where co is the concentration corresponding to a ao 1 , i.e. to pure Nafion and PH2 1bar . In this case, the chemical potential is given as c co
o (T) RT ln
(28)
while the electrochemical as c eF co
o (T) RT ln
(29)
If we take the gradient of the electrochemical potential, eqs. (29), it follows
RT lnc eF
d dlnc d RT eF dx dx dx
;
(30)
which is the driving force for diffusion and ion migration. If we now multiply eq. (30) with the concentration, c , and the mobility J
Deff RT
c Deff c
Deff RT
Deff RT
, we get the flux vector J (in mol m2 s 1 )
eFc
;
(31)
where Deff is the effective proton diffusivity in the membrane (in m2 s 1 ), R (in 8.314 J mol 1 K 1 ) stands for the ideal gas constant, and T (in K ) for the temperature of the system. This value of the effective diffusivity, Deff , can be computed via the Nernst-Einstein equation: H
F2 Deff CH RT
(32)
where CH is the proton concentration in the membrane (in mol m3 ). For an ionic conductivity 2 S m-1 and a proton concentration CH = 900 mol/m-3 [12-16], one computes at room temperature (300 K) an effective proton diffusivity, Deff , equal to 610-10 m2 s-1. In one dimension, eq.(31) takes the following form Jx
Deff RT
c
d dc Deff d Deff eFc dx dx RT dx
If no electrical field is applied J x Deff
(33) dc dx
In order to gain more detailed insight into the mechanism of the triode operation we have calculated the local proton ( H ) concentration, c , and/or its flux in the solid electrolyte, in both conventional and triode operation by using the transport law for the protons [23-24]:
Dc J Dt
In Eqs. (31) & (34),
(34) D is the gradient v is the material derivative, e x e y ez Dt t x y z
vector. Under steady operation and in absence of any flow field, i.e. v 0 , the left-hand side of eq. (31) is equal to zero, and the transport law for H becomes: Deff J c 0 RT
(35)
If we assume that the concentration which is the prefactor of the gradient of the electrochemical potential is a constant equal to c0 (the average proton concentration in the membrane), we can linearize eq. (35) and get the electrochemical potential field by solving numerically a Laplace equation: 0 2
(36)
The current in the membrane due to the motion H can be easily expressed: i FeJ
(37)
where i is the current density expressed in A m2 , and F e is again the charge per mole. The imposed potential across the membrane is subject to Gauss’s law for electricity D
(38)
under the assumptions that the corresponding electric field E is irrotational, the membrane has a uniform composition, and the local variation in ion concentration even close to electrodes little affects the spatial variation of the externally imposed . The first assumption states that E and the other two that Fz c 0 (“quasi-neutral” (QN) assumption). If we consider that the electric displacement D is related to the electric field through D E , and substitute the last two expressions into eq. (38), we get that 2 0
(39)
where is the relative electric permittivity (in C V 1m1 ) of the membrane [25]. Figures 15a and 15b show the geometric model used for the membrane and the three electrodes, indicating spatial symmetry in all variables and the cartesian coordinate system. A spatial unit of the electrolyte or the membrane ignoring the thickness of the electrodes is considered to be a cube of height H 0.2 mm and length x3 , with the cathode covering its entire bottom side, while the anode is placed symmetrically on its top with length x1 and the auxiliary electrode is located between x2 and x3 , respectively.
Fig. 15c and 15d show the fluxes, Ja, J1, J2, Jaux,in, Jaux,out, Jel, Jfar, Jtotal between the three electrodes in conventional and triode operation, respectively. In both fig. 15c and 15d the H flux contributions are depicted along with the imposed or computed electric potentials on each electrode and the currents between them.
5.1. Conventional fuel cell operation In the conventional operation, protons are produced at the anode and consumed at the cathode introducing a concentration gradient between the two. These electrodes are connected through an external resistance, Rex , allowing an equal electron current through it to close the circuit. No external potential is imposed on the auxiliary electrode. Thus 0 throughout the membrane, the electrochemical potential is equal to the chemical potential , U aux is, in this case, an unknown, and a base value, Uo ( 1V) , is induced to it by the presence of H2 and O2 on the two sides of the membrane. In view of eq.(28) the chemical potential can be also expressed by an effective Nernst potential via Ec Eco
RT c ln F co
(40)
Governing equation for this system is eq. (36) divided by F Ec 0 2
(41)
subject to the potential in the two electrodes: Ec (0 x x1 , y H ) U fc
(42)
Ec ( 0 x x3 , y 0) Uc 0
(43)
The high conductivity of the auxiliary electrode guarantees that it is in a uniform, but unknown scaled electrochemical potential. Imposing no overall flux through this electrode generates the condition needed to determine this potential: x3
x2
Ec y
dx 0
(44)
yH
This condition necessitates the existence of a zero-flux point on the auxiliary electrode, irrespective of its distance from the anode, i.e. a point at x xzfp , y H where
Ec y
0 . No proton flow yH
can take place through the side surface of the membrane, or its surface between the anode and the auxiliary electrode: Ec ( x x3 , y ) 0 x
(45)
Ec ( x1 x x2 , y H ) 0 y
(46)
Finally, the potential field is insulated along surface x 0 : Ec ( x 0, y ) 0 x
(47)
5.2 Triode fuel cell operation In this case U aux Uc 0 in the region x 2 x x3 inducing a proton current from the cathode to the auxiliary electrode through a second external circuit, while U fc which is the boundary value along the anode is computed from eq.(18) of the Kirchhoff model. To this end, eq.(36) is solved again, but with boundary conditions on the electrodes modified as follows: / F Ec (0 x x1, y H ) U fc
(48)
/ F (x 2 x x3 , y H ) Uo Uaux
(49)
/ F Ec (0 x x3 , y 0) 0
(50)
The rest of the boundary conditions are same as in section 5.1.
5.3 Solution method The method of finite elements is used to solve eq.(35) in Cartesian geometry (e.g. [25]) in a similar geometry, along with the two sets of boundary conditions, each one for the different operation of the cell. According to it, the part of the membrane depicted in fig. 15(c) is tessellated in orthogonal and equal elements. Accuracy tests demonstrated that 600 elements in both directions were sufficient, resulting in 361,201 unknowns. Bilinear basis functions, i , are used to locally approximate the chemical potential field, Ec (or the electrochemical potential field, ): Ec i i ( x, y )
(51)
i
Galerkin’s method is used to compute the unknown coefficients, i , as follows: This equation is multiplied by a basis function, the divergence theorem is used, and the relevant boundary conditions are introduced. The integrals are computed via a 3-point Gauss quadrature in each direction. The final set of linear algebraic equations is solved using the LAPACK library. The total currents in each electrode are computed in post processing the solution by computing the gradient of the potential on the respective interface and integrating via Gauss quadrature and the location of the zero-flux point.
6. Computed potential fields 6.1 Conventional operation First, we examined the proton distribution in the membrane. As representative values, we used for the length of the membrane element x3 1.5mm , for the potential at the anode U a 1V and kept in all cases the length of the auxiliary electrode at x3 x2 0.5mm . We first examined the importance of the distance between the anode and the auxiliary electrode, d x2 x1 , while maintaining
the
auxiliary
electrode
constant.
The
resulting
equipotential
lines
for
d 0.05mm,0.2 mm,0.5 mm, and 0.9 mm are shown in Fig. 16. Close inspection of these lines yields
several interesting and general observations. These contours are clearly normal to the membrane surface between the anode and the auxiliary electrode, attesting to the accuracy of our computations, which imposed no flux there. In the vicinity of the anode and the cathode they are nearly parallel to the respective electrodes. A proton current exists from the anode to the cathode, I1, from the anode to the auxiliary electrode, I3 and from the auxiliary electrode to the cathode, I2, all in agreement with Figs 4 and 15c. The fact that currents in opposite directions are related with the auxiliary electrode should have been anticipated given that eq. (39) has imposed on it zero total flux. The zero-flux point is always located closer to the anode. Although the auxiliary electrode is in a uniform potential U aux , which is computed to have a value in between those in the anode and the cathode, the contour lines in the immediate proximity of the auxiliary electrode are not parallel to it. The further away and the smaller the anode is, the more uniform and closer to the cathode value the potential in the membrane between the auxiliary electrode and the cathode becomes. Having computed the three current densities in A m-2 (through eq. (31) and eq. (37) using for the proton diffusivity, Deff, the calculated from eq. (32) value of 610-10 m2 s-1) and the equilibrium potential of the auxiliary electrode, Uaux, one can readily determine the values for the resistances between these electrodes. For the case of TC1 (Figure 6c), the three resistances can be computed from eq. (47) taking into account the given in Figure 16 current densities and the geometric surfaces of the anodic and auxiliary electrode. R1
U fc I1
, R2
U fc U aux U c U aux , R3 I2 I3
(52)
The calculations are shown in more detail in the Supplementary material. As shown in Figure 17, decreasing d, causes a pronounced decrease in R3 , while causing a small decrease in R1 and leaving R2 practically unaffected. This dependence on d could have been anticipated given that a resistance
is proportional to the distance between electrodes and inversely proportional to the cross section
through which a current pass. Both are constant for R2 , while the former increases and the later decreases for R3 .
6.2 Triode operation In triode operation the total potential can be assumed as the superposition of two potentials (alternative form of eq. (25)): the effective Nernst potential, Ec, (eq. 40) and the electrical potential, ( Uaux Uo ) . The membrane geometry remains the same as in 6.1.
Fig. 18a shows the contour lines of the Nernst potential Ec for conventional operation (diffusive process) for U fc 0.5V (calculated from Eq. 18 for U o 0.5V , Uaux 0 V and R3 Raux 2 ), while Fig. 18b depicts the corresponding contour lines during triode operation for
the diffusive operation alone and the corresponding directions of proton diffusion. Fig. 18c shows the corresponding variables generated only by the electrical potential, ( Uaux Uo ) . Now a much more intense flow of protons takes place in the opposite direction. The contour lines are parallel to the electrodes, except for a short area just to the left of the auxiliary electrode, where they turn sharply upwards to satisfy the no-flux condition for
x1 x x2 . Fig. 18d shows the
superposition of the previous two fields. The flux of protons now is from the anode to the cathode in the left portion and more intensive from the cathode to the entire auxiliary electrode in the right part of the membrane. These two fluxes are separated by a sharp front of contour lines. For the above set of parameters, the Nernst-Planck model predicts a value equal to 2.5 and a value equal to 1.1 which are both very close to the experimental values (Figure 9). The calculations are shown in more detail in the Supplementary material.
7. Conclusions The triode operation of PEM fuel cells was investigated and modeled using the laws of Kirchhoff and the Nernst-Planck equation. Both model and experiment showed the importance of minimizing the resistance between auxiliary electrode and anode or cathode. This led to a novel comb-type electrode geometry and significantly enhanced triode performance, with power enhancements ratios up to 6, i.e. enhanced power up to 500% and power gain ratios up to 1.2, i.e. 20% enhancement in thermodynamic efficiency. The model provides good semiquantitative estimates of the performance indicators and as a function of the external resistance of the fuel cell circuit and of the total imposed auxiliary potential. Overall the present results have shown that triode operation of PEM fuel cells can be quite advantageous and with proper fuel cell design may lead to practically useful results. The crucial
design problem is the minimization of the resistance between auxiliary and anode or cathode and this has been demonstrated both by model and by experiment utilizing novel comb-type electrode geometries which minimize distance and maximize three phase boundary length. This type of electrode geometries may be easily adapted for SOFCs and for monolithic electropromoted reactors [20,26]. Triode operation appears to be quite promising for poisoned anodes or cathodes and the explicit inclusion of activation and diffusion overpotentials to the model presented here will be necessary for obtaining closer agreement between model and experiment.
Acknowledgements This work was supported by the European Union’s Seventh Framework Programme (FP7/20072013) for the Fuel Cells and Hydrogen Joint Technology Initiative, under the project T-CELL (G.A. Number: 298300).
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Figure Captions Figure 1: Schematic of the triode fuel cell concept, showing the fuel cell circuit and the auxiliary circuit; P/G: potentiostat - galvanostat Figure 2: Schematic of the triode fuel cell concept, showing (a) the geometry of the triode PEM fuel cell and the electrical circuits and (b) the geometry of the membrane electrode assembly. The fuel cell cathode acts simultaneously as an electrode of the auxiliary circuit. P/G: potentiostat– galvanostat. (a) Graphite casing design, (b) Electrode geometry. Figure 3: (a) Layout of the new comb-type triode membrane electrode assembly (MEA) electrode design for decreasing the resistance between the anode and the auxiliary electrode. (b) Polytetrafluoroethylene (PTFE,Teflon) custom-made plate with two detachable graphite blocks acting as current collectors. Figure 4: Schematic of the electrical circuit of a triode PEM fuel cell operating in the conventional mode without imposition of auxiliary potential. Uo is the open-circuit potential. Figure 5: Typical dependence of fuel cell potential, Ufc, and of anode-auxiliary electrode potential difference, U3, on fuel cell current, Ifc, and current density, jfc, showing the definition of R1* and
R *3 ; pure H2 feed; TC1, R3=1.8 Ω, conventional operation (Iaux=0 A). Figure 6: Ifc-Ufc and Ifc-U3 curves under conventional mode (Iaux=0 mA) of fuel cell operation, and corresponding computed R1, R2 and R3 values for cells (a) GS1, (b) GC1 and (c) TC1. Figure 7: (a) Schematic of the electrical circuit of a triode PEM fuel cell operating in the triode mode via auxiliary potential (Uaux) application. (b) Plot of Eq. 18 showing the contribution of the applied Uaux, to the potential of the fuel cell, Ufc, as a function of the external resistance, Rex, for three different values of the resistance R3 (=5, 50 and 100 Ω) and Uo=1V. Figure 8: Comparison of experimentally measured and model predicted (equation 18) dependence of the fuel cell potential, Ufc, on the external resistance, Rex, and on the imposed auxiliary potential (Uo to Uaux=1 to 1.6 V) (a) for cell GS1 (R3=50 ) and (b) for case TC2 (R3=2 ); Anode feed: 100%H2. Figure 9: Comparison of experimentally measured and model predicted (equations (22) and (23)) dependence of the power enhancement ratio, (top), and of the power gain ratio, (bottom) on the imposed auxiliary potential Uaux, using the triode cell TC3. Experimental resistance values: R1=1.50 , R2=2.00 , R3=1.95 . Experimental Rex values shown in the Figures and also used in the model: Rex=1.75 , 4 , and 15 . Electrolyte resistance values used in the model: R1=1.50 , R2=2.00 , R3=1.95 (same as the experimental values, Fig. 9a) and R1=1.50 , R2=2.00 , R3=1.2 , Fig. 9b, see text for discussion. Figure 10: (a) Top left: Time variation of imposed auxiliary potential ΔUaux and corresponding auxiliary current Iaux and auxiliary power Paux. Lower left: Corresponding variation of Ufc, Ifc and power output Pfc. Top right: Corresponding variation of power enhancement ratio and power gain ratio. Anode feed: 100% H2 ; external resistance Rex=15 Ω; TC1, R3 =1.8 Ω. (b) Ifc - Ufc curves of the fuel cell circuit obtained during conventional operation (Iaux=0) and triode operation (Uo+ΔUaux=1.6V) by varying Rex; anode feed 100% H2; TC1; R3 =1.8 Ω. Figure 11: (a) Top left: Time variation of imposed auxiliary potential Uaux and of corresponding auxiliary current Iaux and auxiliary power Paux. Lower left: corresponding variation of Ufc, Ifc and power output Pfc. Top right: corresponding variation of power enhancement ratio ( Pfc Pfc0 ) and power gain ratio ( Pfc Paux ) . Anode feed: 90 ppm CO, 81.6% H2. (b) Ifc–Ufc curves of the fuel cell circuit obtained during conventional operation (Iaux = 0 mA) for the case of 100% H2 feed for
the anode (filled circles) and for the case of 90 ppm CO/81.6% H2 feed for the anode (open diamonds) and Ifc–Ufc curve obtained during triode operation (Uo+Uaux=1.6V) for the case of 90 ppm CO/81.6% H2 feed for the anode (filled diamonds) by varying the external resistive load, R ex; TC1, R3=1.8 Ω. Figure 12: Experimental values of ratios ρ and Λ, as a function of Ifc, under various CO concentrations (30, 60, 120, 160 ppm) and applied potential Uaux of 200 mV (a), 500 mV (b) and 900 mV (c); TC2, R3=2.0 . Figure 13: (a) Time variation of fuel cell power output, Pfc, (dense line), fuel cell current, Ifc, (lighter line) and fuel cell potential, Ufc, (dashed line) on the lower left, sacrificed power and current in the auxiliary circuit, Paux and I aux respectively on the top left, power enhancement ratio, ρ, on the top right and power gain ratio, Λ, on the lower right and time-averaged values for Pfc, ρ and Λ for triode operation during imposition of a constant auxiliary potential, U o+Uaux =1.9V, under a fixed external resistive load (Rex = 3 Ω). Anode feed: 50 ppm CO/16% H2. (b) Ifc–Ufc curves of the fuel cell circuit obtained during conventional operation (Iaux = 0 mA) for the case of 16% H2 feed for the anode (filled circles) and for the case of 50 ppm CO/16% H2 feed for the anode (open diamonds) and Ifc–Ufc curve obtained during triode operation mode (Uo+Uaux=1.9V) for the case of 50 ppm CO/16% H2 feed for the anode (filled diamonds) by varying the external resistive load, R ex; GS1, R3=50 Ω. Figure 14: (a) Time variation of fuel cell power output, Pfc, (dense line), fuel cell current, Ifc, (lighter line) and fuel cell potential, Ufc, (dashed line) on the lower left, sacrificed power and current in the auxiliary circuit, Paux and I aux respectively on the top left, power enhancement ratio, ρ, on the top right and power gain ratio, Λ, on the lower right and time-averaged values for Pfc, ρ and Λ for triode operation during imposition of a constant auxiliary potential, U o+Uaux =1.6 V, under a fixed external resistive load (Rex = 1.2 Ω). Anode feed: 90 ppm CO/81.6% H2. (b) Ifc–Ufc curves of the fuel cell circuit obtained during conventional operation (Iaux = 0) for the case of 100% H2 feed for the anode (filled circles) and for the case of 90 ppm CO/81.6% H2 feed for the anode (open diamonds) and Ifc–Ufc curve obtained during triode operation mode (Uo+Uaux=1.6V) for the case of 90 ppm CO/81.6% H2 feed for the anode (filled diamonds) by varying the external resistive load, Rex; GC1, R3=10.1 Ω. Figure 15: Schematic of (a) the comb-type structured anode and auxiliary electrodes and (b) the spatial unit of the electrolyte (membrane) used for the Nernst-Planck mode solution. Current fluxes due to (c) diffusion (conventional operation-chemical potential gradient driven) and (d) diffusion and electrolysis process (triode operation-chemical potential and electric potential gradients driven). Figure 16: Potential distribution in the electrolyte for different anode-auxiliary electrode distances (d=0.9, 0.5, 0.2, 0.05 mm). Rex =2 . Figure 17: Effect of the anode - auxiliary distance, d on the resistances R1, R2, R3 computed via the Nernst-Planck model during conventional operation. Figure 18: Potential distribution inside the electrolyte taking into account (a) the diffusion process during conventional operation (b) the diffusion process during triode operation, (c) the electrolytic process during triode operation and (d) the two processes together (for the triode operation) according to the Nernst-Planck Equation, R3=Rex=2 Ω, d=0.2 mm , Uaux=0.6 V, S2 is the auxiliary electrode surface area.
Table 1: Geometric characteristics of the anodic and auxiliary electrodes used in the graphite and Teflon casing designs (G: Graphite, T: Teflon, S: Square, C: Comb, #: number of sample)
Name
GS1 GC1 TC1 TC2 TC3
number of comb teeth 4 6 6 6
Anode Surface area/ cm2 3.85 2.35 8.5 9.42 8.90
number of comb teeth 3 6 5 5
Auxiliary Surface area/ cm2
R3/
0.49 1.96 5.88 6.71 6.50
50 10.1 1.8 2.0 1.95
Figure 1
(a)
(b) Figure 2
(a)
(b) Figure 3
Figure 4
Figure 5
R1=1.8Ω R2=14.2Ω R3=49.9Ω
Figure 6 (a)
R1=1.1Ω R2=1.3Ω R3=10.1Ω
Figure 6 (b)
R1=0.7Ω R2=1.1Ω R3=1.8Ω
Figure 6 (c)
(a)
(b)
Figure 7
(a)
(b) Figure 8
Figure 9 (a)
Figure 9 (b)
(a)
(b) Figure 10
(a)
(b) Figure 11
(a)
Figure 12
(b)
Figure 12
(c) Figure 12
(a)
(b) Figure 13
(a)
(b) Figure 14
(a)
(b)
(c)
(d)
Figure 15
Figure 16
Figure 17
(a)
(b)
(c)
(d)
Figure 18