Method for ascertaining overvoltages in fuel cells

Abstract

The invention relates to a method for ascertaining the overvoltage of a working electrode in a fuel cell, in which the potential of a reference electrode compared to the grounded counter electrode is measured. For the measurement, a fuel cell comprising a polymer electrolyte membrane is used, in which the counter electrode comprises a lateral edge having at least one convexly curved region, and the electrolyte membrane surface, adjoining the counter electrode, comprises an electrode-free region in which the reference electrode is disposed on the electrolyte membrane surface. In contrast, the working electrode is continuous, which is to say has a large surface. The minimum distance L.sub.gap between the reference electrode and the edge of the counter electrode L.sub.gap=3L.sub.l,r with (a) and (b), where m=ionic conductivity of the electrolyte membrane (.sup.1 cm-.sup.1), b.sub.ox=Tafel slope of the half cell for the electrochemical reaction of the working electrode l.sub.m=membrane layer thickness (cm) and j.sub.ox.sup.0=exchange current density of the catalyst of the working electrode per unit of electrode surface in (A cm.sup.2). This arrangement can advantageously be used to ensure that the potential measured at the hydrogen-fed reference electrode corresponds to the overvoltage of the working electrode with sufficient accuracy. The method can be applied to polymer electrolyte membrane fuel cells (PEMFC), to direct methanol fuel cells (DMFC) or to high-temperature fuel cells (SOFC).

Claims

1. A method for ascertaining the overvoltage of a working electrode in a fuel cell, comprising a polymer electrolyte membrane, the working electrode, a counter electrode, and a reference electrode, and in which the working electrode is a continuous electrode and is disposed on one side of the polymer electrolyte membrane and the counter electrode is grounded and is disposed on the other side of the polymer electrolyte membrane, the method comprising: measuring and comparing a potential of the reference electrode and a potential of the counter electrode in the fuel cell configured in which the counter electrode has an outer edge spanning an entire circumference of the counter electrode along a surface of the polymer electrolyte membrane, said outer edge comprising a first portion forming a lateral edge having a convex curvature with a local radius R.sub.a and said outer edge comprising a second portion with a local radius greater than R.sub.a, so that said lateral edge forms a convexly curved tip which is more convexly curved than said second portion, the polymer electrolyte membrane surface comprises an electrode-free region adjoining the counter electrode and opposite the working electrode, and the reference electrode is disposed on the polymer electrolyte membrane surface in the region of the electrode-free region and in the immediate vicinity of the convex edge region of the counter electrode at a distance L.sub.gap, wherein the minimum distance for L.sub.gap between the reference electrode and the convexly curved edge region of the counter electrode in the region is given by $L_{\mathrm{gap}}=3.Math.L_{l,r}\mathrm{where}$ $L_{1,r}{\frac{{}_D}{2}\left[\ln \left(\frac{67}{18{\left(R_a\text{/}{}_D\right)}^{7/45}}\right)\right]}^{-1}\mathrm{and}$ ${}_D=\sqrt{\frac{{}_m{b}_{\mathrm{ox}}{l}_m}{2j_{\mathrm{ox}}^0}}$ where .sub.m=ionic conductivity of the polymer electrolyte membrane (.sup.1 cm.sup.1), b.sub.ox=Tafel slope of the half cell for the electrochemical reaction of the working electrode (V), l.sub.m=polymer electrolyte membrane layer thickness (cm), j.sub.ox.sup.0=exchange current density of the catalyst of the working electrode per unit of electrode surface in (A cm.sup.2), and R.sub.a=radius of the convexly curved edge region of the counter electrode (cm).

2. The method according to claim 1, wherein the distance between the reference electrode and the convexly curved edge region of the counter electrode is selected in the range between L.sub.gap and 3 L.sub.gap, from the convexly curved edge region of the counter electrode.

3. The method according to claim 1, wherein the reference electrode is supplied with hydrogen during the method.

4. A method according to claim 1, wherein an anode is used as the counter electrode, and a cathode is used as the working electrode.

5. A method according to claim 1, wherein a value between 0.01 and 1 cm is selected for the local radius of curvature of the curved edge region of the counter electrode.

6. A method according to claim 1, wherein the fuel cell is one of a polymer electrolyte membrane fuel cell (PEMFC), a direct methanol fuel cell (DMFC), a solid oxide fuel cell (SOFC), and a high-temperature polymer electrolyte membrane fuel cell (HT-PEMFC).

7. The method according to claim 1, wherein the counter electrode disposed on said other side of the polymer electrolyte membrane has an outer edge spanning an entire outer periphery along a surface of polymer electrolyte membrane, and wherein a shortest distance from a lateral edge of the working electrode to a nearest edge along said outer edge of the counter electrode is prescribed so that said nearest edge of the counter electrode does not influence the overvoltage of the working electrode in the working area of the fuel cell.

8. The method according to claim 1, wherein the distance between the reference electrode and the convexly curved edge region of the counter electrode is selected in the range of Lgap to 100 Lgap from the convexly curved edge region of the counter electrode.

9. The method according to claim 1, wherein the distance between the reference electrode and the convexly curved edge region of the counter electrode is selected in the range between Lgap and 10 Lgap from the convexly curved edge region of the counter electrode.

10. A method according to claim 1, wherein a value of less than 0.1 cm is selected for the local radius of curvature of the curved edge region of the counter electrode.

11. A method for ascertaining the overvoltage of a working electrode in a fuel cell comprising a polymer electrolyte membrane, the working electrode, a counter electrode, and a reference electrode, and in which the working electrode is a continuous electrode and is disposed on one side of the polymer electrolyte membrane and the counter electrode is grounded and is disposed on the other side of the polymer electrolyte membrane, the method comprising: determining a minimum distance to be configured between the reference electrode and the counter electrode; configuring the fuel cell to have said polymer electrolyte membrane with said working electrode disposed as a continuous electrode on one face of the polymer electrolyte membrane and to have the counter electrode and the reference electrode disposed on an opposite face of the polymer electrolyte membrane, the counter electrode and the reference electrode being spaced by at least said minimum distance, the counter electrode being grounded, the polymer electrolyte membrane surface comprising an electrode-free region adjoining the counter electrode and opposite the working electrode; and measuring and comparing a potential of the reference electrode and a potential of the counter electrode in the configured fuel cell; and wherein the counter electrode has an outer edge spanning an entire outer circumference of the counter electrode along a surface of the polymer electrolyte membrane, said outer edge comprising a first portion forming a lateral edge having a convex curvature with a local radius R.sub.a and said outer edge comprising a second portion with a local radius greater than R.sub.a, so that said lateral edge forms a convexly curved tip which is more convexly curved than said second portion, said minimum distance being from a distal tip of said convexly curved tip through said electrode-free region to a nearest edge of said reference electrode; wherein said minimum distance is determined as being $L_{\mathrm{gap}}=3.Math.L_{l,r}\mathrm{where}$ $L_{1,r}{\frac{{}_D}{2}\left[\ln \left(\frac{67}{18{\left(R_a\text{/}{}_D\right)}^{7/45}}\right)\right]}^{-1}\mathrm{and}$ ${}_D=\sqrt{\frac{{}_m{b}_{\mathrm{ox}}{l}_m}{2j_{\mathrm{ox}}^0}}$ where .sub.m=ionic conductivity of the polymer electrolyte membrane (.sup.1 cm.sup.1), b.sub.ox=Tafel slope of the half cell for the electrochemical reaction of the working electrode (V), l.sub.m=polymer electrolyte membrane layer thickness (cm), f.sub.ox.sup.0=exchange current density of the catalyst of the working electrode per unit of electrode surface in (A cm.sup.2), and R.sub.a=radius of the convexly curved edge region of the counter electrode (cm).

12. A method according to claim 11, wherein an anode is used as the counter electrode, and a cathode is used as the working electrode.

13. A method according to claim 11, wherein the fuel cell is one of a polymer electrolyte membrane fuel cell (PEMFC), a direct methanol fuel cell (DMFC), a solid oxide fuel cell (SOFC), and a high-temperature polymer electrolyte membrane fuel cell (HT-PEMFC).

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) Additionally, several select figures are used to further illustrate the invention; however, these shall not be construed to limit the subject matter of the invention. In the drawings:

(2) FIG. 1 shows an exemplary schematic fuel cell arrangement comprising a reference electrode (from the prior art);

(3) FIG. 2 shows an exemplary schematic fuel cell arrangement comprising a reference electrode (from the prior art);

(4) FIG. 3 shows an exemplary schematic fuel cell arrangement comprising a reference electrode (from the prior art) in a side view a) and in a top view b);

(5) FIG. 4 shows a schematic fuel cell arrangement in a top view, comprising a concentric counter electrode;

(6) FIG. 5 shows a schematic top view onto an embodiment according to the invention of a fuel cell arrangement comprising an anode having a convexly curved edge region in close proximity to a reference electrode;

(7) FIG. 6 shows an exemplary schematic fuel cell arrangement comprising a concentric anode having a radius R.sub.a and an (infinite) concentric cathode having a radius R.sub.c as a basis for the model calculations;

(8) FIG. 7 shows the curve of the potential of the oxygen reduction reaction as a function of the radial position, standardized for the membrane layer thickness l.sub.m for the indicated standardized anode radius R.sub.a/l.sub.m according to the model from FIG. 7;

(9) FIG. 8 shows the membrane potential .sup.+ and the local overvoltage on the cathode side .sub.c.sup.+ in the region around the anode edge, in each case plotted against the standardized radial distance from the edge of the anode (concavely curved edge region), based on the example of a PEMFC; and

(10) FIG. 9 shows the normalized distance from the anode edge to the point at which the overvoltages of the oxygen reduction reaction drop to the value of the ORR Tafel slope b.sub.ox as a function of the standardized anode radius.

DESCRIPTION OF THE PREFERRED EMBODIMENTS

(11) Hereafter, the fuel cell model and fundamental equations will be described, which serve as the basis for the numerical calculation for the fuel cell design according to the invention.

(12) The following symbols are used:

(13) denotes non-dimensional variables

(14) b Tafel slope of the half cell for the anodic or cathodic reaction (V)

(15) E.sup.eq equilibrium potential of a half cell (V)

(16) F Faraday constant

(17) J average current density in the working area (A cm.sup.2)

(18) j.sub.a local proton current density on the anode side (A cm.sup.2)

(19) j.sub.c local proton current density on the cathode side (A cm.sup.2)

(20) j.sub.hy hydrogen exchange current density (A cm.sup.2)

(21) j.sub.hy.sup.0 hydrogen exchange current density in the working area (A cm.sup.2)

(22) j.sub.ox oxygen exchange current density (A cm.sup.2)

(23) j.sub.ox.sup.0 oxygen exchange current density in the working area (A cm.sup.2)

(24) L.sub.gap minimum distance between the edge of the counter electrode and the edge of the reference electrode

(25) L.sub.l,x distance between the straight edge of the counter electrode and the point at which .sub.c.sup.+=b.sub.ox (cm)

(26) L.sub.l,r radial distance between the edge of the counter electrode and the point at which .sub.x.sub.c.sup.+=b.sub.ox (cm)

(27) l.sub.m membrane layer thickness (cm)

(28) r radial position

(29) R.sub.a anode radius

(30) R.sub.c cathode radius

(31) z coordinate perpendicular to the membrane surface (cm)

(32) The following subscript indices are used:

(33) a anode

(34) c cathode

(35) HOR hydrogen oxidation reaction

(36) hy hydrogen

(37) m membrane

(38) ORR oxygen reduction reaction

(39) ox oxygen

(40) ref reference electrode

(41) x system comprising a straight edge of the counter electrode

(42) Furthermore, the following superscript indices are used:

(43) + positive value

(44) 0 center of the concentric counter electrode (=0) {tilde over (r)}

(45) for an infinite radius

(46) Additionally, the following Greek symbols are used:

(47) local overvoltage (V)

(48) .sub.c.sup.+,0 positive cathode overvoltage at r=0 (V)

(49) .sub.m ionic conductivity of the membrane (.sup.1 cm.sup.1)

(50) membrane potential (V)

(51) .sup.+ positive membrane potential (V)

(52) .sup.+, positive membrane potential for r.fwdarw. (V)

(53) potential of the carbon phase (V)

(54) TABLE-US-00001 TABLE 1 Selected physical parameters for the calculations ORR exchange current density j.sub.ox.sup.0 A cm.sup.2 10.sup.6 ORR equilibrium potential E.sub.ox.sup.eq V 1.23 ORR Tafel slope of the half cell b.sub.ox V 0.03 HOR exchange current density in j.sub.hy.sup.0 A cm.sup.2 1.0 the working area HOR Tafel slope of the half cell b.sub.hy V 0.015 Membrane layer thickness l.sub.m cm 0.005 (50 m) Proton conductivity of the .sub.m .sup.1 cm.sup.1 0.1 membrane Cell potential .sub.c V 0.82642 Average current density in the J A cm.sup.2 1 working area

(55) 1. Cell Model and Basic Equations

(56) A PEMFC comprising concentric electrodes is considered, having a geometry and a coordinate system as illustrated in FIG. 6. The electrodes, which are a large-surface-area cathode and a small anode, are disposed on the two sides of the polymer electrolyte membrane having the layer thickness l.sub.m. A corresponding catalyst layer, and in particular the anode catalyst layer (ACL) and the cathode catalyst layer (CCL), is generally (not shown in FIG. 7) present between the electrodes and the membrane.

(57) The center of the concentric anode then corresponds to r=0, and r=R.sub.a applies for the convex edge of the anode toward the recess. In this model, the cathode can likewise be regarded as a concentric electrode having a radius R.sub.c where R.sub.c.fwdarw.>. The region R.sub.a<rR.sub.c or is referred to as an anode-free region (recess), and the region for 0rR.sub.a as the working area. This arrangement is represented schematically in FIG. 6.

(58) A model was developed for distribution of the cathode overvoltage .sub.c and the membrane potential in the anode-free region of the fuel cell. Mathematically, the problem results in the axially symmetric Poisson-Boltzmann equation for the cathode overvoltage .sub.c. The solution to this problem demonstrates that the anode-free region |.sub.c| shows a rapid drop to zero as a function of the radius, while || quickly rises to the value of |.sub.c.sup.0| of the cathode overvoltage in the working area of the cell. The smaller the radius of the concentric anode R.sub.a, the faster the membrane potential reaches the value of the cathode overvoltage in the working area .sub.c.sup.0.

(59) From this, it follows that, for measuring the cathodic overvoltage between the working electrodes (or the working and counter electrodes), the smaller the local radius R.sub.a of this region, the closer the reference cell (RE) can be positioned to the convexly curved edge region of the anode.

(60) The case considered here is one in which the concentric cathode having a radius R.sub.c, serving as the working electrode, extends across the entire region of the membrane (endlessly), the anode however, serving as the counter electrode, makes contact with only a portion of the surface of the membrane. The center of the anode corresponds to r=0. The anode has a concentric geometry having the radius R.sub.a, wherein R.sub.a<R.sub.c. This yields a kind of recess or anode-free region on the anode side on the surface area of the electrolyte membrane with which the anode does not make contact.

(61) It is also, of course, analogously possible to consider the case in which the anode represents the working electrode, and a concentric cathode is present as the counter electrode.

(62) The goal initially is to understand the distribution of current and the potentials in this system. The main variable in this problem is the membrane potential , which follows the Poisson equation

(63) $\begin{array}{cc}\frac{1}{r}\frac{}{r}\left(r\frac{}{r}\right)+\frac{{}^2}{z^2}=0& \left(1\right)\end{array}$

(64) An infinitely large cathode shall mean that the cathode radius is larger than the membrane layer thickness l.sub.m by several orders of magnitude. This assumption allows the second derivative along the z-axis in equation 3 to be replaced by the difference of the proton current density into the membrane, and out of the same, resulting in the following equation:

(65) $\begin{array}{cc}\frac{1}{r}\frac{}{r}\left(r\frac{}{r}\right)=\frac{j_c-j_a}{{}_m{l}_m}& \left(2\right)\end{array}$

(66) Here, j.sub.a and j.sub.c correspond to the current densities at the anode side and the cathode side of the membrane. Moreover, it is assumed that j.sub.a and j.sub.c follow Butler-Volmer kinetics.

(67) $\begin{array}{cc}{j}_a=2j_{\mathrm{hy}}\sinh \left(\frac{{}_o}{b_{\mathrm{hy}}}\right)\mathrm{and}& \left(1\right)\\ j_c=2j_{\mathrm{ox}}\sinh \left(-\frac{{}_c}{b_{\mathrm{ox}}}\right).& \left(2\right)\end{array}$

(68) Here, j.sub.hy and j.sub.ox each denote the corresponding surface exchange current densities of the anode catalyst layer (ACL) and of the cathode catalyst layer (CCL), .sub.a and .sub.c are the corresponding local electrode overvoltages at the anode and the cathode, and b.sub.hy and b.sub.ox are the Tafel slopes of the corresponding half-cell reaction corresponding thereto. Since it is assumed that the transport losses are small, the dependence on the reactant concentration is already included in j.sub.hy and j.sub.ox.

(69) The respective half-cell overvoltages follow from
.sub.=.sub.E.sub.HOR.sup.eq(3)
.sub.c=.sub.cE.sub.ORR.sup.eq(4)
wherein .sub. and .sub.c represent the electrode potentials and E.sub.HOR.sup.eq=0 V and E.sub.ORR.sup.eq=1.23 V represent the equilibrium potentials of the corresponding half-cell reactions. It is assumed that the anode is grounded (.sub.)=0, and thus .sub.c represents the cell potential.

(70) Inserting equation (3) to equation (6) into equation (2) and introducing the non-dimensional variables

(71) $\begin{array}{cc}\tilde{r}=\frac{r}{l_m},\tilde{j}=\frac{{\mathrm{jl}}_m}{{}_m{b}_{\mathrm{ox}}},\stackrel{~}{}=\frac{}{b_{\mathrm{ox}}},\stackrel{~}{}=\frac{}{b_{\mathrm{ox}}},{\tilde{b}}_{\mathrm{hy}}=\frac{b_{\mathrm{hy}}}{b_{\mathrm{ox}}}& \left(5\right)\end{array}$
yields

(72) $\begin{array}{cc}\frac{1}{\tilde{r}}\frac{d}{d\tilde{r}}\left(\tilde{r}\frac{d\stackrel{~}{}}{d\tilde{r}}\right)=2{\tilde{j}}_{\mathrm{ox}}^0\sinh \left(-{\stackrel{~}{}}_c+\stackrel{~}{}+{\tilde{E}}_{\mathrm{ORR}}^{\mathrm{eq}}\right)-2{\tilde{j}}_{\mathrm{hy}}^{0}H\left({\tilde{R}}_a-\tilde{r}\right)\sinh \left(-\stackrel{~}{}\text{/}{\tilde{b}}_{\mathrm{hy}}\right)& \left(6\right)\end{array}$

(73) Here, j.sub.ox.sup.0 including the superscript symbol .sup.0 corresponds to the value at the center of the concentric working electrode (cathode). H denotes the Heaviside function, which in the working area assumes the value 1, and in the anode-free region assumes the value 0. The absence of anodic catalyst outside the working area is taken into consideration by setting the exchange current density of the hydrogen oxidation reaction (HOR) to zero.

(74) Within the scope of the invention, the distribution of the membrane potential in the anode-free region is now examined. In this region, the current production at the anode side vanishes, and equation (8) is simplified to

(75) $\begin{array}{cc}\frac{1}{\tilde{r}}\frac{d}{d\tilde{r}}\left(\tilde{r}\frac{d\stackrel{~}{}}{d\tilde{r}}\right)=2{\tilde{j}}_{\mathrm{ox}}^0\sinh \left(-{\stackrel{~}{}}_c+\stackrel{~}{}+{\tilde{E}}_{\mathrm{ORR}}^{\mathrm{eq}}\right)& \left(9\right)\end{array}$

(76) It is now possible to represent this equation as a function of the cathode overvoltage according to equation (6). As a non-dimensional equation, this results in
{tilde over ()}.sub.c={tilde over ()}.sub.c{tilde over ()}{tilde over (E)}.sub.ORR.sup.eq(10)

(77) Inserting this into equation (9) yields

(78) $\begin{array}{cc}\frac{1}{\tilde{r}}\frac{d}{d\tilde{r}}\left(\tilde{r}\frac{d{\stackrel{~}{}}_c}{d\tilde{r}}\right)={}^2\sinh {\stackrel{~}{}}_c& \left(11\right)\end{array}$
where ={square root over (2{tilde over (j)}.sub.ox.sup.0)}(12)

(79) By introducing the positive overvoltage,
{tilde over ()}.sub.c.sup.+={tilde over ()}.sub.c>0(13)
the following equation is obtained for {tilde over ()}.sub.c.sup.+:

(80) $\begin{array}{cc}\frac{1}{\tilde{r}}\frac{d}{d\tilde{r}}\left(\tilde{r}\frac{d{\stackrel{~}{}}_c^{+}}{d\tilde{r}}\right)={}^2\sinh {\stackrel{~}{}}_c^{+}& \left(14\right)\end{array}$

(81) 2. General Conditions

(82) It was possible to demonstrate that, due to the very high exchange current density of the hydrogen oxidation reaction (HOR), the cathode potential {tilde over ()}.sub.c at the anode edge in polymer electrolyte fuel cells is very close to the bulk potential value in the working area according to equation 7. For a system having axial symmetry, it is therefore to be expected that {tilde over ()}.sub.c.sup.+({tilde over (R)}.sub.).sub.c.sup.+0, wherein {tilde over ()}.sub.c.sup.+0{tilde over ()}.sub.c.sup.+(0)={tilde over ()}.sub.c(0) is the overvoltage of the oxygen reduction reaction (ORR) at the center of the axis of symmetry. Numerical tests confirmed this assumption. As a result, however, the working area can advantageously be eliminated from the consideration by placing the potential {tilde over ()}.sub.c.sup.+({tilde over (R)}.sub.)=.sub.c.sup.+0 at the edge of the anode. In this way, the general conditions for equation 14 result as follows:

(83) 0$\begin{array}{cc}{{\stackrel{~}{}}_c^{+}\left({\tilde{R}}_a\right)={\stackrel{~}{}}_c^{+0},\frac{d{\stackrel{~}{}}_c^{+}}{d\tilde{r}}.Math.}_{\tilde{r}=}=0& \left(15\right)\end{array}$

(84) The second equation means that the proton current density along {tilde over (r)} moves toward zero for . The fact of an infinite cathode exists approximately when the following condition is met: {tilde over (R)}.sub.c>>1. The condition, however, is redundant. A more detailed analysis shows that the approximation works well with an infinite cathode as long as
R.sub.c3L.sub.l,r(16)
wherein L.sub.l,r is given by equation 18. Such a condition can be regularly regarded as given for fuel cells on a laboratory scale.

(85) 3. Different Anodic Radii R.sub.a

(86) FIG. 7 shows the solutions for equation 14 for different anode radii R.sub.a. The particular characteristic of this problem is that the gradient of the potential of the oxygen reduction reaction (ORR) .sub.c.sup.+ increases very close to the anode edge as the anode radius decreases. Qualitatively, this resembles the behavior of Laplace's potential between a charged metal tip and a plane: the narrower the radius of the metallic tip, the more strongly the potential drops in the vicinity of the tip as a function of the axial symmetry.

(87) The practical significance that can be derived from FIG. 7 is that, in a fuel cell arrangement comprising a large-surface-area cathode and a concentric anode, the smaller the radius of the anode, the closer the reference electrode (RE) can be disposed to the anode. This applies under the prerequisite that this arrangement is to be used, and can be used, to ascertain the cathodic overvoltage with sufficient accuracy.

(88) 4. Position of the Reference Electrode

(89) It becomes apparent from FIG. 8 that the positive membrane potential .sup.+ approaches the limiting value .sub.c.sup.+,0 more quickly as the anode radius R.sub.a decreases. Likewise, the radial width of the region {tilde over (L)}.sub.l,r dominated by the oxygen reduction reaction is apparent from FIG. 8. This radial width increases with the increasing anode radius R.sub.a.

(90) For further estimations, the assumption is made that the reference electrode (RE) is disposed at a distance {tilde over (L)}.sub.l,r from the at least partially convexly curved anode edge having the local radius R.sub.a. FIG. 8 shows that this assumption results in the determination of .sub.c.sup.+ with a 10% accuracy. It is advisable to compare this distance {tilde over (L)}.sub.l,r (FIG. 9, solid line) to the analogous distance {tilde over (L)}.sub.l,r for a straight anode edge according to FIG. 3. The dotted line in FIG. 9 represents the value of {tilde over (L)}.sub.l,r. Physically, {tilde over (L)}.sub.l,r is an asymptote, approached by the {tilde over (L)}.sub.l,r curve for {tilde over (R)}.sub..fwdarw. (FIG. 9). For very small radii of the anode, which is to say for {tilde over (R)}.sub.10, the radial distance {tilde over (L)}.sub.l,r between the curved anode edge and the reference electrode is only half as small as the distance

(91) ${\tilde{L}}_{1,x}\frac{1,4}{}$
for the corresponding anode geometry having a straight edge, as shown in FIG. 4. But even at larger radii, for example for {tilde over (R)}.sub.100 (right edge in FIG. 9), the radial distance {tilde over (L)}.sub.l,r between the curved anode edge and the reference electrode is still almost 30% smaller than the distance {tilde over (L)}.sub.l,r that would result for the corresponding anode geometry having a straight edge.

(92) It must be noted that the distance {tilde over (L)}.sub.l,r decreases drastically for very small anode radii. For the range of {tilde over (R)}.sub.1, it is questionable whether the underlying model can still be applied, since possible strong two-dimensional effects in close proximity to the concavely curved anode edge are ignored. Thus, for the optimal minimum distance, {tilde over (R)}.sub. would advantageously be selected in the range of 2 to 3.

(93) A possible expression that describes the dependencies of {tilde over (L)}.sub.l,r as a function of {tilde over (R)}.sub. in the region of 0{tilde over (R)}.sub.1 is given by

(94) $\begin{array}{cc}{\tilde{L}}_{1,r}{\frac{}{2}\left[\ln \left(\frac{67}{18{\left({\tilde{R}}_a\right)}^{7/45}}\right)\right]}^{-1}\mathrm{for}0{\tilde{R}}_{a}1& \left(17\right)\end{array}$
and is represented in FIG. 9 by open circles. In the non-dimensional case, this yields the following equation:

(95) $\begin{array}{cc}{L}_{1,r}{\frac{{}_D}{2}\left[\ln \left(\frac{67}{18{(R_a\text{/}{}_D)}^{7/45}}\right)\right]}^{-1},\mathrm{where}{}_D=\sqrt{\frac{{}_m{b}_{\mathrm{ox}}{l}_m}{2j_{\mathrm{ox}}^0}}& \left(18\right)\end{array}$

(96) This equation 18 can advantageously be used to estimate the required minimum distance L.sub.gap between the anode tip (convexly curved edge region) having the local radius R.sub.a and the reference electrode in an application in terms of development. If a certain the measurement accuracy is required, L.sub.gap3.Math.L.sub.l,r should be used as the minimum distance.

(97) FIGS. 4 and 5 show possible arrangements for a fuel cell, each comprising a reference electrode. FIG. 3 shows the arrangement of a reference cell in a fuel cell having a straight anode edge according to the prior art. In this case, the minimum distance results as

(98) $L_{\mathrm{gap}}3.Math.{}_D,\mathrm{where}{}_D=\sqrt{\frac{{}_m{b}_{\mathrm{ox}}{l}_m}{2j_{\mathrm{ox}}^0}}.$

(99) FIG. 5, in contrast, shows the case for an embodiment of the arrangement according to the invention in which the anode comprises a convexly curved edge region in close proximity to the reference electrode. Here, L.sub.gap3.Math.L.sub.l,r applies for the minimum distance, wherein L.sub.l,r is given by equation 18. It should be noted that L.sub.gap is smaller in FIG. 5 than in FIG. 3. FIG. 5 shows a fuel cell arrangement comprising an anode edge having a convexly curved tip. This tip causes the membrane potential .sup.+ to increase drastically and quickly as the distance away from the tip increases. In this case, the reference electrode can thus be disposed very close to the tip, without interfering with the accuracy of the measurement. The value of the distance L.sub.gap in FIG. 5 can be easily determined by equation 18 by using the local radius of the anode tip for R.sub.a.

(100) Using the parameters from Table 1 and an anode radius R.sub.a=0.01 cm, a distance L.sub.l,r=1.97 cm would be obtained from equation 18. For an arrangement comprising an appropriately straight anode edge, the same membrane potential would not be achieved until a distance of L.sub.l,x=3.83 cm. At smaller anode radii R.sub.a, the advantage would be even greater, due to the curved anode edge.