Patent Publication Number: US-11024862-B2

Title: Fuel cell arrangement

Description:
The invention relates to a novel fuel cell design, in particular for a polymer electrolyte membrane fuel cell (PEMFC), for ascertaining overvoltages. 
     BACKGROUND OF THE INVENTION 
     In an electrochemical cell, the electrochemical reactions at the involved electrodes are decisively influenced by the half-cell overvoltages. These overvoltages include both contributions from the activation and from transport losses of the corresponding half cells. It is therefore considered to be a primary objective to minimize these contributions within the scope of a novel cell concept. One of the most common previous methods for measuring half-cell overvoltages is a method using a reference electrode (RE). 
     A reference electrode is always aimed at measuring the potential Φ of the electrolyte membrane at an arbitrary point between two electrodes. Having knowledge of this potential makes it possible, in principle, to separate the cathode overvoltage and the anode overvoltage. However, it has been found that suitable positioning of the reference electrode is not readily apparent. 
     A typical option for positioning a reference electrode within an electrochemical cell is to dispose the reference electrode on one cell side, at a certain distance L gap  from the edges of the working electrode, which are aligned with one another. For this purpose, a design according to  FIG. 1  is proposed, in which the two electrodes or the lateral edges thereof are aligned with one another. 
     Within the scope of simulations, B. Adler et al., in  J. Electrochem. Soc.  149, E166 (2002), have shown that the distance L gap  must be greater than three times the membrane layer thickness l m . At this distance, the inhomogeneities that are regularly caused by the edge of a working electrode, and the losses of potential resulting from the hydrogen oxidation reaction (HOR), are negligible, and the potential measured at the reference electrode regularly corresponds to the membrane potential at any arbitrary point along the z-axis between the anode and the cathode in the working area. 
     The measurements of the membrane potential Φ and of the electrode potentials then allow the overvoltages at the anode and the cathode to be separated. 
     Furthermore, with the aid of reference electrodes it is possible to carry out electrochemical impedance spectroscopy between the reference electrode and any other cell electrode. This technique has already been applied in the field of solid oxide fuel cells (SOFC). 
     In electrochemical cells, in which protons act as charge carriers, the potential of a hydrogen-fed reference electrode generally corresponds to the membrane potential Φ at the location of the reference electrode, neglecting the voltage losses due to the hydrogen oxidation/generation reaction. 
     In the implementation for positioning a reference electrode in a fuel cell according to  FIG. 1 , however, two essential problems occur. Unless indicated otherwise, it is assumed hereafter that the anode is grounded, and that all potentials are measured with respect to the anode. 
     Initially, it shall be noted that even a very small shift (δ) in the alignment of the edges of the working electrodes with respect to the counter electrode nonetheless causes a large change in the potential at the reference electrode. This effect has already been sufficiently discussed in the literature on solid oxide fuel cells (SOFCs). Several suggestions on minimizing this effect have also already been described there. An analogous application of this solution to PEMFCs has likewise already been proposed in the literature, which reports on a system of working electrodes precisely aligned by way of laser ablation. 
     A further problem, which occurs even in electrochemical cells in which the edges of the working electrodes are precisely aligned (δ=0), is that, even though the value of the potential Φ measured by the reference electrode corresponds to a point on the z-axis between the working electrodes, the exact position of this point on the z-axis is not known. It is therefore not possible, in such a case, to unambiguously evaluate and interpret the measured DC voltage signal of the reference electrode. In this case as well, the separation of the half-cell overvoltages can only take place by way of impedance spectroscopy. 
     The latest developments by A. A. Kulikovsky and P. Berg (DE 102015001572.9), however, demonstrate that it is possible to determine the overvoltage of a working electrode in a fuel cell by measuring the potential of a reference electrode compared to the grounded counter electrode. A fuel cell according to  FIG. 2  is used for this purpose. This includes a large-surface-area (endless) working electrode, which in this instance is a cathode, and a counter electrode, which has at least one lateral edge. Within the scope of the invention, an electrode is referred to as having a large surface area (being endless) when the extension thereof corresponds at least to 10 times the parameter l * , and in particular when the extension of the electrode is greater than the membrane layer thickness by several orders of magnitude. As a result of the edge of the counter electrode, an electrode-free region is obtained, compared to the working electrode, on the electrolyte membrane surface adjoining the counter electrode, in which the reference electrode is disposed on the electrolyte membrane surface. Due to the fact that the working electrode no longer has an edge in the vicinity of the working area, there is likewise no longer a problem of the alignment of the electrode edges. 
     It was possible to demonstrate that the minimum distance L gap  between the reference electrode and the edge of the counter electrode must meet the following condition: 
                   L   gap     ≥     3   ·     l   *         =     3   ·           σ   m     ⁢     b     a   ⁢           ⁢   x       ⁢     l   m         2   ⁢     j     a   ⁢           ⁢   x       +   0                 ,         
where σ m =ionic conductivity of the electrolyte membrane (Ω −1  cm −1 ), b ox =Tafel slope of the half cell for the electrochemical reaction of the cathode, l m =membrane layer thickness (cm) and j ox   0 =exchange current density of the catalyst of the cathode per unit of electrode surface in (A cm −2 ). At distances between the reference electrode and the counter electrode which correspond at least to this minimum distance, it can be ensured that the measurement of the potential of the reference electrode substantially corresponds to the overvoltage of the working electrode, which is to say the cathode.
 
     Using typical cell parameters (see Table 1), this yields a minimum distance L gap  on an order of magnitude of several centimeters, in the case of the reference electrode being disposed on the anode side for ascertainment of the overvoltage on the cathode side. This distance, however, is a very large value for a test fuel cell of normally 10 cm*10 cm or smaller. 
     SUMMARY OF THE INVENTION 
     It is an object of the invention to provide a method for ascertaining the overvoltage of a working electrode of a fuel cell which overcomes the existing disadvantages of the prior art and, in particular, the aforementioned minimum distance from the reference electrode in a fuel cell arrangement. The method should, in particular, be usable for polymer electrolyte membrane fuel cells (PEMFC), direct methanol fuel cells (DMFC) and high-temperature solid oxide fuel cells (SOFC). 
     It is furthermore an object of the invention to provide an improved fuel cell arrangement, with the aid of which the aforementioned method according to the invention can be carried out. 
     The objects of the invention are achieved by a novel fuel cell arrangement according to the main claim and by a method according to the additional independent claim. Advantageous embodiments of the device and of the method can be found in the respective dependent claims. 
     Within the scope of the present invention, it was found, supported by numerical simulations for a polymer electrolyte membrane fuel cell (PEMFC) comprising a concentric small anode and a concentric large-surface-area (endless) cathode, that a novel arrangement overcomes the existing disadvantages of the prior art, based on, and in a continuation of, the arrangement of a reference electrode in a fuel cell as proposed in A. A. Kulikovsky and P. Berg (DE 102015001572.9). For this purpose, a fuel cell arrangement is proposed, in which it is possible to position a reference electrode at a considerably smaller distance than before from a counter electrode, and in particular an anode, and it is nonetheless possible to ascertain the overvoltage of the counter electrode, and in particular of a cathode, with sufficient accuracy. Moreover, this novel arrangement of the fuel cell regularly has electrode surface areas that are able to ensure sufficient electrochemical conversion. 
     A model was developed for a radial distribution of the cathode overvoltage η c  and of the membrane potential Φ in the anode-free region of the fuel cell. Mathematically, the problem results in an axially symmetric Poisson-Boltzmann equation for the cathode overvoltage η c . The solution to this problem demonstrates that |η c | shows a rapid drop to zero as a function of the radius, while |Φ| quickly rises to the value of |η c   0 | of the cathode overvoltage in the working area of the cell. The smaller the radius of the concentric anode R a , the faster the membrane potential Φ reaches the value of the cathode overvoltage in the working area η c   0 . 
     From this, it follows that, in measuring the cathodic overvoltage between the (working) electrodes, the smaller the local radius R a  of this region, the closer the reference cell (RE) can be positioned to the convexly curved edge region of the curved anode. 
     Hereafter, the electrode of the fuel cell on which the overvoltage is to be ascertained is referred to as the working electrode. In contrast, a counter electrode shall be understood to mean a further electrode of this fuel cell, which is required for the necessary conduction of current. Both the working electrode and the counter electrode are electrodes at which electrochemical processes take place in a controller manner. 
     In the literature, model notions of the fuel cells comprising a reference electrode are based on the 2D simulations in which the membrane potential Φ is present in the plane perpendicular to the working electrodes and the reference electrode (x-z plane in  FIG. 1 ). The numerical calculations, however, show neither a dependence nor a characteristic quantification during variations, for the membrane potential Φ in such systems. 
     The invention now achieves the object of being able to measure a meaningful overvoltage in a fuel cell by way of a reference electrode, by proposing a fuel cell arrangement comprising an electrolyte membrane, which comprises a counter electrode that is delimited by at least a single edge and includes at least one convexly curved edge region having a local small radius of curvature R a , and a continuous (endless) working electrode. 
     Within the scope of the invention, a continuous working electrode shall be understood to mean an electrode having edges disposed so far away from the working areas, which is to say from the edges of the counter electrode, that any influence on the working area by the counter electrode edges can generally be precluded. Again, the term ‘continuous working electrode’ shall mean that the extension thereof is greater than the parameter l *  by orders of magnitude. 
     Adjoining the counter electrode, the fuel cell according to the invention comprises an electrode-free region on the electrolyte membrane surface. Within this region, the reference electrode is disposed at a distance L gap  from the edge of the counter electrode. This distance is generally a multiple of the membrane thickness. The working electrode extends on the opposite side of the electrolyte membrane. This is located opposite both the counter electrode and the reference electrode, and in this regard shall be considered to be continuous. All common standard reference electrodes may be used as reference electrodes, for example a reverse hydrogen electrode (RHE), or a dynamic hydrogen electrode (DHE), in which an electrolyte potential is measured may be used. 
     In contrast to the arrangement of A. A. Kulikovsky and P. Berg (DE 102015001572.9), which is illustrated in  FIG. 3  in a cross-sectional view (a) and in a top view (b), the counter electrode in the inventive embodiment of the invention does not show only a straight edge, compared to the reference electrode, but in the top view has a convex area of curvature having a smaller radius of curvature R a , at least in the edge region disposed closest to the reference electrode. This is met, in a first approximation, when the anode itself is designed in the manner of a concentric disk, as is apparent from  FIG. 4 . 
     Compared to the previously described straight edges of a counter electrode according to  FIG. 3 , the convexly curved region of the counter electrode having a radius of curvature R a  has the advantage that the minimum distance L gap , at which the reference electrode is disposed from the counter electrode, is considerably smaller than the corresponding distance L gap  in the arrangement according to  FIG. 4 , while the overvoltage of the working electrode, and in particular of the cathode, can nonetheless be ascertained with sufficient accuracy. This is of crucial importance, in particular with small fuel cells. 
     The reference electrode can advantageously be designed to be very small with the arrangement according to the invention. In this way, it is possible to ensure that the tip of the convexly curved region of the counter anode is the closest region of the counter electrode. If this were not the case, the membrane potential could be impaired at the location of the reference electrode by contributions from remote regions of the anode. 
     In a first embodiment of the invention ( FIG. 5 ), the counter electrode has a substantially convex edge, which includes a tip having a radius R a , in the immediate vicinity of which the reference electrode is disposed at a minimum distance L gap . 
     In this arrangement of the counter electrode, the distance L gap  can be selected to be significantly smaller, which advantageously results in a much more compact design comprising the reference electrode. Nonetheless, such an arrangement advantageously makes it possible that the potential measured at the reference cell corresponds with sufficient accuracy to the overvoltage of the cathode. 
     During the actual measurement for ascertaining the overvoltage, the reference electrode is fed hydrogen. The counter and working electrodes are fed an operating agent and an oxidizing agent, depending on whether they are switched as an anode or cathode. During measurement, the potential of the reference electrode compared to the grounded counter electrode is measured. 
     A PEM fuel cell is usually operated with hydrogen and air or oxygen, a DMFC with methanol or a methanol/water mixture and air or oxygen, and a SOFC with hydrogen and air or oxygen. 
     In the particular in case where the counter electrode is switched as the anode and the working electrode as the cathode, the measured potential of the reference cell corresponds to the overvoltage of the cathode with sufficient accuracy. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       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: 
         FIG. 1  shows an exemplary schematic fuel cell arrangement comprising a reference electrode (from the prior art); 
         FIG. 2  shows an exemplary schematic fuel cell arrangement comprising a reference electrode (from the prior art); 
         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); 
         FIG. 4  shows a schematic fuel cell arrangement in a top view, comprising a concentric counter electrode; 
         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; 
         FIG. 6  shows an exemplary schematic fuel cell arrangement comprising a concentric anode having a radius R a  and an (infinite) concentric cathode having a radius R c  as a basis for the model calculations; 
         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 m  for the indicated standardized anode radius R a /l m  according to the model from  FIG. 7 ; 
         FIG. 8  shows the membrane potential Φ +  and the local overvoltage on the cathode side η c   +  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 
         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 ox  as a function of the standardized anode radius. 
     
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     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. 
     The following symbols are used: 
     ˜ denotes non-dimensional variables 
     b Tafel slope of the half cell for the anodic or cathodic reaction (V) 
     E eq  equilibrium potential of a half cell (V) 
     F Faraday constant 
     J average current density in the working area (A cm −2 ) 
     j a  local proton current density on the anode side (A cm −2 ) 
     j c  local proton current density on the cathode side (A cm −2 ) 
     j hy  hydrogen exchange current density (A cm −2 ) 
     j hy   0  hydrogen exchange current density in the working area (A cm −2 ) 
     j ox  oxygen exchange current density (A cm −2 ) 
     j ox   0  oxygen exchange current density in the working area (A cm −2 ) 
     L gap  minimum distance between the edge of the counter electrode and the edge of the reference electrode 
     L l,x  distance between the straight edge of the counter electrode and the point at which η c   + =b ox  (cm) 
     L l,r  radial distance between the edge of the counter electrode and the point at which x η c   + =b ox  (cm) 
     l m  membrane layer thickness (cm) 
     r radial position 
     R a  anode radius 
     R c  cathode radius 
     z coordinate perpendicular to the membrane surface (cm) 
     The following subscript indices are used: 
     a anode 
     c cathode 
     HOR hydrogen oxidation reaction 
     hy hydrogen 
     m membrane 
     ORR oxygen reduction reaction 
     ox oxygen 
     ref reference electrode 
     x system comprising a straight edge of the counter electrode 
     Furthermore, the following superscript indices are used: 
     + positive value 
     0 center of the concentric counter electrode (={circumflex over ({tilde over (r)})}) 
     ∞ for an infinite radius 
     Additionally, the following Greek symbols are used: 
     η local overvoltage (V) 
     η c   +,0  positive cathode overvoltage at r=0 (V) 
     σ m  ionic conductivity of the membrane (Ω −1  cm −1 ) 
     Φ membrane potential (V) 
     Φ +  positive membrane potential (V) 
     Φ +,∞  positive membrane potential for r→∞ (V) 
     ϕ potential of the carbon phase (V) 
                     TABLE 1               Selected physical parameters for the calculations                                                ORR exchange current density   j ox     −0       A cm −2     10 −6         ORR equilibrium potential   E ox   eq     V   1.23       ORR Tafel slope of the half cell   b ox     V   0.03       HOR exchange current density in   j hy   0     A cm −2     1.0       the working area       HOR Tafel slope of the half cell   b hy     V   0.015       Membrane layer thickness   l m     cm   0.005 (50 μm)       Proton conductivity of the   σ m     Ω −1  cm −1     0.1       membrane       Cell potential   ϕ c     V   0.82642       Average current density in the   J   A cm −2     1       working area                    
1. Cell Model and Basic Equations
 
     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 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. 
     The center of the concentric anode then corresponds to r=0, and r=R 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 c  where R c →∞. The region R a &lt;r≤R c  or ∞ is referred to as an anode-free region (recess), and the region for 0≤r≤R a  as the working area. This arrangement is represented schematically in  FIG. 6 . 
     A model was developed for distribution of the cathode overvoltage η c  and the membrane potential Φ in the anode-free region of the fuel cell. Mathematically, the problem results in the axially symmetric Poisson-Boltzmann equation for the cathode overvoltage η c . The solution to this problem demonstrates that the anode-free region |η c | shows a rapid drop to zero as a function of the radius, while |Φ| quickly rises to the value of |η c | of the cathode overvoltage in the working area of the cell. The smaller the radius of the concentric anode R a , the faster the membrane potential 1 reaches the value of the cathode overvoltage in the working area η c   0 . 
     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 a  of this region, the closer the reference cell (RE) can be positioned to the convexly curved edge region of the anode. 
     The case considered here is one in which the concentric cathode having a radius R 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 a , wherein R a &lt;R 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. 
     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. 
     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 Φ, which follows the Poisson equation 
     
       
         
           
             
               
                 
                   
                     
                       
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     Here, j a  and j c  correspond to the current densities at the anode side and the cathode side of the membrane. Moreover, it is assumed that j a  and j c  follow Butler-Volmer kinetics. 
     
       
         
           
             
               
                 
                   
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     Here, j hy  and j ox  each denote the corresponding surface exchange current densities of the anode catalyst layer (ACL) and of the cathode catalyst layer (CCL), η a  and η c  are the corresponding local electrode overvoltages at the anode and the cathode, and b hy  and b 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 hy  and j ox . 
     The respective half-cell overvoltages follow from
 
η a =ϕ a   −Φ−E   HOR   eq   (3)
 
η c =ϕ c   −Φ−E   ORR   eq   (4)
 
wherein ϕ a  and ϕ c  represent the electrode potentials and E HOR   eq =0 V and E ORR   eq =1.23 V represent the equilibrium potentials of the corresponding half-cell reactions. It is assumed that the anode is grounded (ϕ a )=0, and thus ϕ c  represents the cell potential.
 
     Inserting equation (3) to equation (6) into equation (2) and introducing the non-dimensional variables 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                               
                           
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     Here, {tilde over (j)} ox   o  including the superscript symbol  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. 
     Within the scope of the invention, the distribution of the membrane potential Φ 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 
     
       
         
           
             
               
                 
                   
                     
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                   9 
                   ) 
                 
               
             
           
         
       
     
     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 (η)} c ={tilde over (ϕ)} c   −{tilde over (Φ)}−{tilde over (E)}   ORR   eq   (10)
 
     Inserting this into equation (9) yields 
     
       
         
           
             
               
                 
                   
                     
                       1 
                       
                         r 
                         ~ 
                       
                     
                     ⁢ 
                     
                       d 
                       
                         d 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           r 
                           ~ 
                         
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           r 
                           ~ 
                         
                         ⁢ 
                         
                           
                             d 
                             ⁢ 
                             
                               
                                 η 
                                 ~ 
                               
                               c 
                             
                           
                           
                             d 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               r 
                               ~ 
                             
                           
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       k 
                       2 
                     
                     ⁢ 
                     sinh 
                     ⁢ 
                     
                       
                         η 
                         ~ 
                       
                       c 
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
             
               
                 
                   
                     where 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     k 
                   
                   = 
                   
                     
                       2 
                       ⁢ 
                       
                         
                           j 
                           ~ 
                         
                         
                           a 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           x 
                         
                         0 
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     By introducing the positive overvoltage,
 
{tilde over (η)} c   + =−{tilde over (η)} c &gt;0  (13)
 
the following equation is obtained for
 
                       1     r   ˜       ⁢     d     d   ⁢           ⁢     r   ~         ⁢     (       r   ˜     ⁢       d   ⁢       η   ˜     c   +         d   ⁢           ⁢     r   ~           )       =       k   2     ⁢   sinh   ⁢       η   ~     c   +               (   14   )               
2. General Conditions
 
     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 (η)} 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 ñ c   + ({tilde over (R)} a )≅η c   +0 , wherein {tilde over (η)} c   +0 ≅{tilde over (η)} c   + (0)=−{tilde over (η)} 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 (η)} c   + ({tilde over (R)} a )={tilde over (η)} c   +0  at the edge of the anode. In this way, the general conditions for equation 14 result as follows: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             η 
                             ˜ 
                           
                           c 
                           + 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               R 
                               ˜ 
                             
                             a 
                           
                           ) 
                         
                       
                       = 
                       
                         
                           η 
                           ˜ 
                         
                         c 
                         
                           + 
                           0 
                         
                       
                     
                     , 
                     
                       
                         
                           
                             d 
                             ⁢ 
                             
                               
                                 η 
                                 ~ 
                               
                               c 
                               + 
                             
                           
                           
                             d 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               r 
                               ~ 
                             
                           
                         
                         ⁢ 
                         
                           | 
                           
                             
                               r 
                               ~ 
                             
                             = 
                             ∞ 
                           
                         
                       
                       = 
                       0 
                     
                   
                   ⁢ 
                   
                       
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
     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: κ{tilde over (R)} c &gt;&gt;1. The condition, however, is redundant. A more detailed analysis shows that the approximation works well with an infinite cathode as long as
 
 R   c ≥3 L   l,r   (16)
 
wherein L l,r  is given by equation 18. Such a condition can be regularly regarded as given for fuel cells on a laboratory scale.
 
3. Different Anodic Radii R a  
 
       FIG. 7  shows the solutions for equation 14 for different anode radii R a . The particular characteristic of this problem is that the gradient of the potential of the oxygen reduction reaction (ORR) η c   +  increases very close to the anode edge as the anode radius decreases. Qualitatively, this resembles the behavior of Laplace&#39;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. 
     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. 
     4. Position of the Reference Electrode 
     It becomes apparent from  FIG. 8  that the positive membrane potential Φ +  approaches the limiting value η c   +,0  more quickly as the anode radius R a  decreases. Likewise, the radial width of the region {tilde over (L)} l,r  dominated by the oxygen reduction reaction is apparent from  FIG. 8 . This radial width increases with the increasing anode radius R a . 
     For further estimations, the assumption is made that the reference electrode (RE) is disposed at a distance {tilde over (L)} l,r  from the at least partially convexly curved anode edge having the local radius R a .  FIG. 8  shows that this assumption results in the determination of η c   +  with a 10% accuracy. It is advisable to compare this distance {tilde over (L)} l,r  ( FIG. 9 , solid line) to the analogous {tilde over (L)} l,x  distance for a straight anode edge according to  FIG. 3 . The dotted line in  FIG. 9  represents the value of {tilde over (L)} l,s . Physically, {tilde over (L)} l,r  is an asymptote, approached by the {tilde over (L)} l,x  curve for {tilde over (R)} a →∞ ( FIG. 9 ). For very small radii of the anode, which is to say for {tilde over (R)} a ≤10, the radial distance {tilde over (L)} l,r  between the curved anode edge and the reference electrode is only half as small as the distance 
                 L   ˜       l   ,   x       ≅       1   ,   4     κ           
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)} a ≈100 (right edge in  FIG. 9 ), the radial distance {tilde over (L)} l,r  between the curved anode edge and the reference electrode is still almost 30% smaller than the distance {tilde over (L)} l,x  that would result for the corresponding anode geometry having a straight edge.
 
     It must be noted that the distance {tilde over (L)} l,r  decreases drastically for very small anode radii. For the range of {tilde over (R)} 0 ≤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)} a  would advantageously be selected in the range of 2 to 3. 
     A possible expression that describes the dependencies of {tilde over (L)} l,r  as a function of {tilde over (R)} a  in the region of 0≤κ{tilde over (R)} a ≤1 is given by 
                       L   ˜       l   ,   r       ≅           π     2   ⁢   κ       ⁡     [     ln   ⁢     {       6   ⁢   7       1   ⁢   8   ⁢       (     κ   ⁢       R   ˜     a       )       7   /   45           )       ]       -     ⁢   for   ⁢           ⁢   0     ≤     κ   ⁢       R   ˜     a       ≤   1           (   17   )               
and is represented in  FIG. 9  by open circles. In the non-dimensional case, this yields the following equation:
 
     
       
         
           
             
               
                 
                   
                     
                       L 
                       
                         l 
                         , 
                         r 
                       
                     
                     ≅ 
                     
                       
                         
                           
                             π 
                             ⁢ 
                             
                               λ 
                               D 
                             
                           
                           2 
                         
                         ⁡ 
                         
                           [ 
                           
                             ln 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   6 
                                   ⁢ 
                                   7 
                                 
                                 
                                   18 
                                   ⁢ 
                                   
                                     
                                       ( 
                                       
                                         
                                           R 
                                           a 
                                         
                                         / 
                                         
                                           λ 
                                           D 
                                         
                                       
                                       ) 
                                     
                                     
                                       7 
                                       / 
                                       45 
                                     
                                   
                                 
                               
                               ) 
                             
                           
                           ] 
                         
                       
                       
                         - 
                         1 
                       
                     
                   
                   , 
                   
                     
                       where 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         λ 
                         D 
                       
                     
                     = 
                     
                       
                         
                           
                             σ 
                             m 
                           
                           ⁢ 
                           
                             b 
                             
                               a 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               x 
                             
                           
                           ⁢ 
                           
                             l 
                             m 
                           
                         
                         
                           2 
                           ⁢ 
                           
                             j 
                             
                               a 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               x 
                             
                             0 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
     This equation 18 can advantageously be used to estimate the required minimum distance L gap  between the anode tip (convexly curved edge region) having the local radius R a  and the reference electrode in an application in terms of development. If a certain the measurement accuracy is required, should L gap ≅3·L l,r  be used as the minimum distance. 
       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 
     
       
         
           
             
               
                 L 
                 gap 
               
               ≅ 
               
                 3 
                 · 
                 
                   λ 
                   D 
                 
               
             
             , 
             
               
                 where 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   λ 
                   D 
                 
               
               = 
               
                 
                   
                     
                       σ 
                       m 
                     
                     ⁢ 
                     
                       b 
                       
                         a 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         x 
                       
                     
                     ⁢ 
                     
                       l 
                       m 
                     
                   
                   
                     2 
                     ⁢ 
                     
                       j 
                       
                         a 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         x 
                       
                       0 
                     
                   
                 
               
             
             , 
           
         
       
     
       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 gap ≅3·L l,r  applies for the minimum distance, wherein L l,r  is given by equation 18. It should be noted that L 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 Φ +  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 gap  in  FIG. 5  can be easily determined by equation 18 by using the local radius of the anode tip for R a . 
     Using the parameters from Table 1 and an anode radius R a =0.01 cm, a distance L 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 l,x =3.83 cm. At smaller anode radii R a , the advantage would be even greater, due to the curved anode edge.