Reconstruction of Electric Currents in a Fuel Cell by

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H. Lustfeld

Forschungszentrum Juelich, IFF and IAS, 52425 Juelich, Germany e-mail: h.lustfeld@fz-juelich.de

M. Reißel

e-mail: reissel@fh-aachen.de

U. Schmidt

e-mail: u.schmidt@fh-aachen.de Fachhochschule Aachen, Campus Juelich, 52428 Juelich, Germany

B. Steffen

Forschungszentrum Jülich, JSC, 52425 Juelich, Germany e-mail: b.steffen@fz-juelich.de

Reconstruction of Electric Currents in a Fuel Cell by

Magnetic Field Measurements

In this paper the tomographic problem arising in the diagnostics of a fuel cell is dis- cussed. This is concerned with how well the electric current densityjsrdbe reconstructed by measuring its external magnetic field. We show that (i) exploiting the fact that the current density has to comply with Maxwell’s equations it can, in fact, be reconstructed at least up to a certain resolution, (ii) the functional connection between the resolution of the current density and the relative precision of the measurement devices can be obtained, and (iii) a procedure can be applied to determine the optimum measuring positions, essentially decreasing the number of measuring points and thus the time scale of mea- surable dynamical perturbations—without a loss of fine resolution. We present explicit results for (i)–(iii) by applying our formulas to a realistic case of an experimental direct methanol fuel cell. fDOI: 10.1115/1.2972171g

Keywords: fuel cell, DMFC, PEFC, MEA, tomography, current density distribution, magnetic field measurement

1 Introduction

Fuel cells have been known for a long time f1–4g. Their prin- cipal advantage is the direct conversion of chemical into electric energy thus avoiding the by Carnot’s law the limited efficiency of machines that first convert chemical energy into heat and then heat into electric energyf5g. For this reason the theoretical efficiency factor of fuel cells, h, is distinctively larger than that of thermal engines and is close to 1. However, in real lifef6g hof fuel cells is far away fromh_<1f7,8g. To give an example consider the direct methanol fuel cellsDMFCd. The theoretical limit for the efficiency factorh= 0.95 has by far not been reached. The mean value for DMFCs in operation is abouth= 0.35f9,10g.

Hence a lot of further development and ideas are required to increase the efficiency considerablyf7,8,11g. One prerequisite for that is a good diagnostics, in particular a reasonable insight into the distribution of electric currents in the fuel cell. But here another problem is lurking: The physical chemical scatalytic and noncatalyticdprocesses in a fuel cell are very complexf11g. Any direct measurement represents an interference that can influence these processes that to an extent make these measurements ques- tionable at best. Therefore indirect measurements are the tools that one has to resort to.

Typical for every fuel cell are the high electric currents in the range of 10– 100 A. This suggests measurements of the exterior magnetic fields for reconstructing the internal current distribution of the fuel cellf12g. The method being used is to be distinguished from NMR techniques in which the resonant magnetic field is measuredf13g.

Then two problems arise:sadWhen knowing the exterior magnetic field is it possible to uniquely reconstruct the internal current distributionf14g?sbd Which relative precision of magnetic field measurements is required to obtain a reasonable resolution of the internal currents?

At first sight it may seem that problem sad is the much more

important one and having solved it, problemsbd is of minor importance. But in fact, the reverse is true: Proving merely the ex- istence of a unique reconstruction does not tell anything about its practical feasibility. The requirements on the precision of measurements, as well as on the suppression of noise, may be completely unrealistic. On the other hand if we can show which qual- ity of a reconstruction is possible for a given precision of measurements and a realistic level of noise, then it is of minor interest whether or not an exact reconstruction would be possible if infinitely precise measurements were available.

In any tomographic problem the question of possible recon- structions comes up right in the beginning and there are cases where the impossibility of a reasonable reconstruction is easy to recognize, e.g., the external static electric field is known yet we are still unable to find from that information the internal charge distribution. Indeed, at any internal position the electric charge can be replaced by a rotational symmetric charge distribution without changing the external electric field. This gives rise to an infinite number of charge distributions all having the same external electric field.

Analogous phenomena arise when trying to determine internal currents from the external magnetic field, e.g., internal circular currents need not change external magnetic fields at all. And in the inside of a material with infinite extension in, say, thex-direction any current density,j, not depending onx, j=jsy,zd can at each positionsy0,z₀d be replaced by one having cylindrical symmetry aroundsy₀,z₀dwithout modifying the external magnetic field. Of course there is no material with infinite extension in one direction.

But this demonstrates again that problem sbd is more important than problemsad. At least in situations where the extension of the current in one direction is much larger than in the other two there are solutions clearly distinct in the current distribution but are nearlyequal with regard to their external magnetic fields. In prin- ciple the correct current distribution could still be determined. But due to noise and the finite precision of measurements the distributions in question may easily become indistinguishable.

The considerations above show very clearly that for a success- ful reconstruction all properties ofjand the constraints onjhave

Manuscript received August 10, 2007; final manuscript received October 4, 2007;

published online February 26, 2009. Review conducted by Nigel M. Sammes.

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to be exploited. This and the consequences thereof are discussed in Sec. 2. In Sec. 3 we treat the problem of how to find the allowed variations under the constraints imposed on the current densityj. We show in particular the connection between the relative precision of the measuring devices and the resolution of the currents in the layer of the fuel cell we are interested in, namely, in the membrane electrode assemblysMEAd. In Sec. 4 we discuss where to position the measuring points for obtaining the ptimum information out of the magnetic field measurements. The measuring points are located on six planes surrounding the fuel cell.

There are two rules of the thumb: sid The smaller the distance between the fuel cell and the measuring plane the better the resolution.siidThe smaller the spacing between the measuring points in a given plane the better the resolution. We show that this second rule has to be modified because the number of measuring points on a given plane can be reduced in a systematic manner drastically and withoutlosing resolution. This is very important because the timescale of the measuring apparatus depending on the number of measuring points is thus essentially reduced as well, enabling the reconstruction of faster dynamical perturbations. In Sec. 5 we apply our results to an experimental DMFC to make all the results of the previous sections quantitative. The conclusion ends this paper.

2 The Current Density in the Fuel Cell Properties and Constraints

The heart of a fuel cell is the MEA containing the electrolytic layers and the layers where the catalytic reactions and recombina- tions take place. If the cell is a single experimental cell—and that is what we are interested in here—the MEA together with a graphite layer is embedded into two metallic layers, cf. Fig. 1. The current and its properties can be represented by an equivalent circuit diagram that is relatively simple: The conductivityssrdis known in the regime of the metallic layers and of the graphite

layersand possibly other layers except the MEAd. All the compli- cated chemical and physical processes taking place in the MEA can be expressed by the sin this regime unknownd conductivity distribution¹ssrd, which has to be determined.

Within this framework we can formulate the properties of and the constraints onjsrd.

For properties,

Hexterndepends uniquely onjsrd s1d

and can be computed fromf15g

¹3H=jsrd, ¹ ·B= 0 s2d sOf course it is assumed here that the boundary conditions can be determined.d

For constraints,

¹·j= 0 s3d

jsrd=ssrd·Esrd s4d

which includes the condition

¹3E= 0 s5d

or

¹3 1

ssrdjsrd= 0 s6d Equationss3d,s4d, ands6dcan be combined to

−¹·sssrd¹Fd= 0 s7d

Here,Fis a scalar potential with the property

−¹F=E s8d

Sincessrd.0 Eq.s7drepresents an elliptic differential equation, which leads to a unique solution² of jsrd for each given ssrd.

Then Eq.s2dleads to a unique solution of the magnetic field. And therefore we can write

Hsrd=Hsr,hsjd s9d

Note thatssrdcan vary freely whereasjsrdcannot. Note also that there is no guarantee yet thatjsrdis determined byHin a unique way—in spite of the constraints of Eqs. s3d, s4d, and s6d. This problem will be dealt with in the next chapter.

3 The Inverse Problem, the Problem of Independent Variations, and the Resolution of the Current Distribu- tion

Solving the partial differential equations s2d and s7d numeri- cally amounts to discretizing the problem. Let jˇ be the vector representing the discretizedjsrd. If there areN_Jdiscretized vectors thenjˇ has 3N_Jcomponents. In an analogous manner we de- fineHˇ as the magnetic field vector representing the external magnetic field at all positions where H is measured. If N_H is the number of measuring points then the dimension of this vector is 3N_H. Furthermore we denote that vector assˇ , whoseN_Mcompo- nents contain all the discretized conductivities in the MEA³layer.

It follows from Eq.s2dthat a given current distributionjleads to a uniquely determined externalH field and thereforeHˇ is a unique function ofjˇ.

1Here we assume thatsidthe conductivityssrddoes not depend on the current, andsiidit is scalar and not a tensor.sidis well justified for estimating the connection between variations of the internal current and the external magnetic field not too far away from the operating point of the fuel cell.siidis completely justified outside the MEA. Moreover we did not find a process in the MEA that would require a tensor for the conductivity.

2Appropriate boundary conditions are presumed.

3Outside the MEA the conductivities are known and therefore fixed.

MEAlayer nonmagneticsteel

14 mm

178mm

0.6mm

graphite

nonmagneticsteel

8.4 mm 16 mm

+

Fig. 1 Scheme of an experimental fuel cell of the DMFC type.

The heart of the cell is the MEA layer consisting of a porous layer, a catalytic layer, an electrolytic layer, a second catalytic layer, and a second porous layer. Note that in spite of these five layers the thickness of the MEA is only 0.6 mm. The positive pole of the fuel cell is indicated on the bottom of the front end, the negative pole on the bottom of the back end. The line be- tween them is the unscreened part of the cable connecting the fuel cell with external electric equipment. The effective width of the cell is 138 mm. We use this model in the numerical calculations.

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Hˇ =Hˇsjˇd s10d On the other hand the information contained in Eq.s2dis insufficient to construct the inverse map of Eq.s10d. We will now show that the constraint equationss3d,s4d, ands6dstrongly reduce the degrees of freedom leading to criteria from which one is able to decide whether or not jˇ can uniquely be determined for a given resolution.

Because of the constraints not all variations djˇ are allowed.

Rather all possible variations are exhausted by⁴dsˇ , djˇ =Cdsˇ

C= ]jˇ

]sˇ s11d

Ccan be calculated by using first Eqs.s4dands8d

jsrd= −ssrd·¹Fsrd s12d Differentiating this equation with respect tos yieldsCprovided we can get]F/]s. However, this derivative can be computed by differentiating Eq.s7dwith respect tosˇ and solving it in quite the same way as Eq.s7d.

The properties of the matrixCcan be detected by applying the singular value decomposition:⁵

C=U_C·W_C·V_C^T

UC= 3Nj3NM orthogonal matrix

WC=N_M3N_M diagonal matrix andW_Ci,i$W_Ci+1,i+1$0 VC=NM3NM unitary matrix s13d The smaller theW_Ci,ithe larger dsˇ has to be for a perceptible variation in the direction ofu_i,u_ibeing theith column vector of U_C. This justifies a cutoff for avoiding unphysically large variations ofs, i.e., we take into account only indicesiwith the property

WCi,i

W_C11.e, thusi#NCsed s14d eis a parameter. It is not very critical and in this paper we have set e= 0.01. Then we may write

djˇ =

o

k=1 N_c

u_kda_k s15d

Theda_iare independent of each other, but nevertheless guarantee the constraints sEqs. s3d, s4d, and s6dd. Therefore the relevant variations are the da_i and the relevant matrix, connecting them with thedjˇ, is the 3NJ3N_c matrixU_c consisting of the column vectorsu_i,i= 1 , . . . ,N_c.

What we have achieved so far is the replacement of restricted variationsdjˇ by unrestricted onesda_i. It remains to determine the matrix connecting dHˇ and the dan. However, analogous to the computation of C it is a straightforward procedure to compute from Eq.s2dthe derivative

S₀= ]Hˇ

]jˇ s16d

and applying the chain rule we get

dHˇ =Sda

S=S₀U_c s17d This equation shows how drastically the constraintssEqs.s3d,s4d, and s6dd reduce the number of independent variables. Without these constraints we would have S=S₀and S₀would be a 3N_H 33N_j matrix. In contrast to that S is a 3N_j3N_c matrix and⁶ N_c≪3N_j.

We can extract practical results fromSby applying the singular value decomposition again, this time toS

S=U_S·W_S·V_S^T

U_S= 3Nj3Nc orthogonal matrix

W_S=N_c3N_c diagonal matrix andW_Si,i$W_Si+1,i+1$0 V_S=N_c3N_c unitary matrix s18d The rank ofSis determined by

sid the resolution chosen in the MEA. The finer the discreti- zation the larger the dimension N_c, i.e., the number of independentda_i.

siid the relative error f_rel of the measuring devices. Due to these errors small matrix elements in the matrixW_Sbe- come irrelevant. In other words there is an integerN_Rbe- yond that we may set

WSk,k→0 fork.N_R s19d In this paper we restrict ourselves to relative errors due to systematic, i.e., nonrandom, deviations of the true values.

This is the worst-case scenario leading to a lower limit for N_Rgiven by⁷

frel,WSN_R,N_R

W_S11 , frel.WSN_R+1,N_R+1

W_S11 s20d

Now Eq.s20dcontains the criterion we were looking for Hˇsjˇdis locally invertible ifN_R=N_c s21d In other words, for any given relative error of the measuring devices we vary the resolution in the MEA of the fuel cell until Eq.

s21dholds true. Then we have found the available resolution for jsrd. Examples for obtaining the resolution as a function of the relative error are presented in Sec. 5.

Two objections to this procedure may arise. Equations21ddoes not guarantee the global invertibility but the local one only. Fur- thermore Eq.s21dis rather a numerical recipe than a mathematical proof. Both objections are correct but—as we think—not very relevant. The fuel cell works best at a certain operating point and detecting changes in the current are mainly of interest in the neighborhood of this operating point. And—as stated above—a general mathematical proof of invertibility remains rather useless as long as there is no connection visible between the relative error of the measuring devices plus noise and the resolution ofjsrd.

4 Positioning the Measuring Points

We suppose that either external magnetic fields are shielded by plates with a high permeabilitymand the exterior boundaries have been moved to these plates, or that external magnetic fields can be

4Note that this procedure reduces efficiently the number of degrees of freedom. A typical computation in Sec. 5 givesN_M= 436 and 3N_j<16,000.

5Note that 3N_j.N_M.

6In a typical caseN_c<30– 450 and 3N_j<30·N_c.

7The estimate of Eq.s20dis on the safe side. It does not require any assumptions about the kind of errors. Estimates can become more favorable ifsidthe kind of systematic errors is known or ifsiidone can be sure that the errors are random, i.e., nonsystematic. This will be discussed in a forthcoming paper.

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neglected altogether. In bothsrealisticdcases there are no exterior boundary conditions depending on perturbations of the currents in the MEA. Consequently measuring points need not be placed at these exterior boundaries, on the contrary they can be placed at will.

Here we investigate the optimum positions of measuring points under the assumption that perturbations of the current distribution in the MEA are well described using the linear approximation.

This approximation, certainly valid for not too large perturbations, remains reasonable in general, we think, for determining the optimum positions of measuring points. With this approximation a rather natural scheme can be given extracting the relevant set from N_Hmeasuring points. To see how it works consider a perturbation dathat causes a deviationdHsr_jd at a measuring point with po- sitionr_j. dHsrjd contains contributions from all orthogonal column vectors of the matrix U_S times the corresponding singular value of the matrixW_S. But for getting high resolution all column vectors ofU_S, the vectors with large as well as those with small singular values, are of equal importance. Therefore the relevance of the measuring point depends on the information it contains about all the column vectors. A measure for that is the function⁸z

zsrjd=NH

Nc

o

m=0 2

sU_SU_S^Td3j+m,3j+m s22d zhas the property

1 NH

o

i=1 N_H

zsrid= 1 s23d zcan be used as a criterion for finding the relevant set fromN_H measuring points. Letd_zbe a constant we have fixed in the beginning. Then we select the relevant measuring positionsr_j according to

ifzsrjd$d_z then the measuring point atr_jis kept s24d d_zneed not be small. A typical value is 2, i.e., twice the average of z. This will be demonstrated in the next section.

The perturbations in the MEA are dynamical ones. Such perturbations can be reconstructed only if their timescale is larger than that of the measuring apparatus. Since the latter is proportional to the number of measuring points it is of vital interest to reduce this number essentially as long as this does not result in a loss of precision. We will show in the next section that by applying Eq.

s24dthis can be achieved.

5 Numerical Results for an Experimental DMFC In this section we present numerical results of our theory by applying the formulas of the previous sections to an experimental single DMFCf16g. The MEA of this cell consists of three very thin layers containing two catalytic layerssplatinum and platinum- rutheniumd embedded in porous graphite and the electrolyte sNafiond. The MEA is only 0.6 mm thick and embedded in a graphite layer, a teflon frame, two nonmagnetic steel plates, and a thin titanium layer. A schematic view is given in Fig. 1. The positive and negative contacts are at the bottom edges of the steel plates, the positive cable passes the negative contact close by and is then—together with the negative cable—completely screened.

The conductivity is unknown in the MEA only. We select the operating point such that the conductivity is constant throughout the MEA. Its value is chosen according to the condition that the total electric current is 60 A and the voltage drop is 0.4 Vs0.7 V is the voltage for zero current in this cellf17gd.

To get the current distribution in the cell we solve Eq.s7d by applying the finite difference method with one grid for the electric potentialFand another grid for both the current and the conductivity, with grid points located between the gridpoints of the first grid. In this waye_d_Vj·ndo= 0 is guaranteed for any inner surface dV. The resolution ofj in the MEA is determined by the spacing between the lattice points in the MEA.

Next we differentiate Eq.s12d with respect to s. The matrix ]F/]scan be obtained by differentiating Eq.s7d and by solving the corresponding equation. In this way we get the matrixCde- fined in Eq.s11d.

The result of the singular value decomposition is shown in Fig.

2. Only the last value is much smaller than the others and corre- sponds to a variation of allsvalues by the same amount. This and the corresponding u vector are not taken into account. Compo- nents of some otheru_i, located in the MEA, are plotted in Fig. 3.

One can easily verify that with increasing the index ithe resolution becomes finer.

Having computed for the current distribution in the fuel cell, we can now determine the magnetic field. In this paper we neglect all disturbances that might arise due to external additional currents and magnetic materials.⁹Then we need not solve Eq.s2dbut can use Biot–Savarts lawf18,19gdirectly

Hsrd=¹3

E

ur^j^s−^r⁸r^d8ud³r8 ^s25d In our case the fuel cell is a rectangular solid. The measuring points are located on six rectangular planes, a distanced_Gapart from the fuel cell’s surface.¹⁰On these planes the spacing of measurement positions is chosen, e.g., to be a constant. Now inserting the j values obtained from Eq. s7d we can compute H at any position by applying Eq. s25d. Furthermore, the derivative with respect tojis easily computed using the same formula. We get the matrixS₀scf. Eq.s16ddand from that the important matrixSscf.

Eq.s17dd.

8We numerate the indices k= 1 , . . . , 3N_H in the following manner: dH_xsr_jd

=dHˇ

k=3j+0,dH_ysrjd=dHˇ

k=3j+1, anddH_zsrjd=dHˇ

k=3j+2, wherer_jdenotes the position of the measuring pointj.

9This approximation is too crude when calculating currentsjand fieldsH. But when estimating variationsdjanddHfor the calculations in this paper this approximation is quite appropriate.

10Due to additional frames and equipment the experimental fuel cell considered in this paper fits into a cuboid of size 3932003280 mm³. Thed_Grefer to distances measured from this cuboid.

0 5 10 15 20 25 30 35

−8

−6

−4

−2 0 2 4

0

10 20

4

30 0

−4

Fig. 2 Singular values of the matrixCin Eq.„11…for the fol- lowing resolution: 0.35 cm vertical to the MEA plane, otherwise 2.75 cm. Plotted are the index numbers versus the logarithm of the corresponding values. The last singular value is much smaller corresponding to a change of all s by the same amount. This change can be detected via the cell voltage.

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We have calculated the 3N_H3N_c matrix S for various grid distances in the fuel cell. In particular the grid spacing in the MEA is crucial being a direct measure of how well the current densityjsrd is resolved in the MEA. The required relative preci-

sion guaranteeing a rankN_c of the Smatrix can be read off by doing a singular value decomposition ofS. The results are shown in Figs. 4 and 5.

The relative precision of a standard measuring device is about 10⁻⁴– 10⁻⁵. It turns out—not very surprisingly—that the closer to the fuel cell the measuring planes are chosen, the better the reso- Fig. 3 For the parameters given in Fig. 2 the components of various u_iare shown„cf. Eq.„15……. One

can easily recognize the increasing resolution with increasing indexi.

0.5 1 1.5 2 2.5 3

−16

−14

−12

−10

−8

−6

−4

−2

[cm]

−2

−12

−4

−10

−14

−16

−6

−8

0.5 1.0 1.5 2.0 2.5 [cm]

Fig. 4 Resolution of j„r… in the MEA versus logarithm of the required relative precision when measuring the magnetic field.

Distance between the fuel cell and the planes in which the mag- netic field is measured isd_G= 3 cm. Dashed line: the spacingd_P between the measuring points on the planes is about 5 cm:

d_PÉ5 cm. Dashed dotted line: d_PÉ2.5 cm. Dotted line: d_P É1.2 cm. Full line: d_PÉ0.6 cm. Magenta and blue line nearly coincide in this case. Note that a spacing of 5 cm is much too large.

0.5 1 1.5 2 2.5 3

−16

−14

−12

−10

−8

−6

−4

−2

0.5 1.0 1.5 [cm] 2.0 2.5 [cm]

−16

−14

−12

−10

−8

−6

−4

−2

Fig. 5 Resolution of j„r…in the MEA versus logarithm of the required relative precision when measuring the magnetic field.

Distance between the fuel cell and the planes in which the mag- netic field is measured is d_G= 1 cm. Dashed line: d_PÉ5 cm.

Dashed dotted line: d_PÉ2.5 cm. Dotted line: d_PÉ1.2 cm. Full line:d_PÉ0.6 cm. Note that a spacing of 5 cm is much too large leading to meaningless results.

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lution. One recognizes from Fig. 4 that the resolution is approximately 2 cm in the MEA for a distanced_G= 3 cm. In contrast Fig.

5 shows that the resolution is approximately 1 cm for the same relative precision ifd_G= 1 cm. Furthermore only for this value of d_Gthe spacing of about 0.6 cm between the measuring points is adequate.

Figure 5 demonstrates also that expecting a resolution distinctively better than 0.5 cm is not realistic.

Next we show that the efficiency can be dramatically increased by reducing the number of measuring points—without loss of precision. In Fig. 6 thez function is shown for a distance between measuring planesd_G= 1 cm. We have chosen a spacing between Fig. 6 All 5536 measuring points located in the six planes surrounding the fuel cell are

marked. Distance between the fuel cell and the planesd_G= 1 cm. Spacing between the measuring points is about 0.6 cm. At each measuring point the value of thezfunction is indicated by the color and thickness of the corresponding marker. The two planes close to the front end and back end of the fuel cell have the highestz values. Note that the scales in depth, width, and height are not identical.

Fig. 7 The remaining 868 measuring points after all measuring points withz„r_j…<2 have been dropped. Nomenclature and other parameters as in Fig. 6.

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the measuring points of about 0.6 cm, which amounts to 5536 measuring points. In Fig. 7 all points with zsr_jd,2 have been discarded, resulting in a remaining number of 868 points only.

Note that these points are located at the front end and at the back end planes. It is perhaps not surprising that the largezvalues are found on these planes because there the true diameters of cylin-

drical currents can be detected best. Thus measurements are nec- essary only on two out of six planes. Also the resolution does not decrease. This is demonstrated in Fig. 8 in which the singular spectra of the fullSmatrix and of the reduced one are compared.

In Fig. 9 the plots corresponding to Fig. 5 are shown. It is remark- Fig. 8 Singular values„d_G= 1 cm andd_P= 0.6 cm…of the fullSmatrix built from all 5536

measuring points„upper line…and singular values of the strongly reducedSmatrix built from the remaining 868 measuring points withz„r_j…Ð2„lower line…. Plotted are the index numbers versus the logarithm of the corresponding values. One can verify by inspection that the condition of the reducedSmatrix is better than the condition of the fullSmatrix.

Fig. 9 Resolution of j„r…in the MEA versus logarithm of the required relative precision when measuring the magnetic field. Lower line: all 5536 measuring points are taken into account with a spacing of 0.6 cm between measuring points. Upper line: out of the 5536 measuring points only the 868 points withz„r_j…Ð2 are taken into account. Note that reducing the number of measuring points leads to a better resolution.

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able in this figure that given a relative error of the measuring devices a better resolution is predicted for the measurments with lesspoints. At first sight this might appear as a paradoxical and counterintuitive phenomenon but it has a very simple explanation:

The condition of a matrix sratio of the largest to the smallest singular valuedbecomes better after superflous information have been dropped. Thus this phenomenon demonstrates that the z function, introduced in Sec. 4 and used here, is a very good tool for deciding on which points should be kept.

6 Conclusion

We have demonstrated in this paper that the following tomographic problem is solvable: Determining the current density in a fuel cell by measuring external magnetic fields. Furthermore we have computed the resolution of the current density in the MEA of the fuel cell as a function of the relative errors of the measurement. This resolution can be guaranteed independently of the kind of errorsse.g., systematic or random errorsd. Moreover we have introduced a function z evaluating and discarding measuring points with insufficient or superfluous information. In a real ex- periment this means that the number of measuring points can be reduced dramatically without loss of resolution. Thus the timescale of the measuring apparatus is reduced essentially too, enabling the reconstruction of faster dynamical perturbations. All these results are demonstrated by explicit computations for an experimentalsDMFCdfuel cell.

Acknowledgment

We would like to thank H. Dohle and M. Wannert for stimulat- ing discussions.

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f15gSteffen, B., 1989, “Some Remarks on the Calculation of Static Magnetic Fields,” Z. Angew. Math. Mech., 69, pp. T174–T176.

f16gMore detailed results are presented in Schmidt, U., 2007, “Numerische Sensi- tivitätsanalyse für ein Tomographieproblem bei Brennstoffzellen,” diploma thesis, Fachhochschule Aachen, Abteilung Jülich, Jülich.

f17gDohle, H., private communication.

f18gJackson, D., 1998,Classical Electrodynamics, 3rd ed., Wiley, New York.

f19gPotthast, R., and Kühn, L., 2003, “On the Convergence of the Finite Integra- tion Technique for the Anisotropic Boundary Value Problem of Magnetic To- mography,” Math. Methods Appl. Sci., 26, pp. 739–757.