Download Reconstruction of Electric Currents in a Fuel Cell by Magnetic Field

H. Lustfeld
Forschungszentrum Juelich,
IFF and IAS,
52425 Juelich, Germany
e-mail: [email protected]
M. Reißel
e-mail: [email protected]
U. Schmidt
e-mail: [email protected]
Fachhochschule Aachen,
Campus Juelich,
52428 Juelich, Germany
B. Steffen
Forschungszentrum Jülich,
JSC,
52425 Juelich, Germany
e-mail: [email protected]
1
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 discussed. This is concerned with how well the electric current density jsrd be 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 measurable 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
Introduction
Fuel cells have been known for a long time f1–4g. Their principal 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 energy f5g. 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 life f6g h of
fuel cells is far away from h < 1 f7,8g. To give an example consider the direct methanol fuel cell sDMFCd. The theoretical limit
for the efficiency factor h = 0.95 has by far not been reached. The
mean value for DMFCs in operation is about h = 0.35 f9,10g.
Hence a lot of further development and ideas are required to
increase the efficiency considerably f7,8,11g. 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
noncatalyticd processes in a fuel cell are very complex f11g. Any
direct measurement represents an interference that can influence
these processes that to an extent make these measurements questionable 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 cell f12g. The method being used is to be distinguished
from NMR techniques in which the resonant magnetic field is
measured f13g.
Then two problems arise: sad When knowing the exterior magnetic field is it possible to uniquely reconstruct the internal current
distribution f14g? 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
Manuscript received August 10, 2007; final manuscript received October 4, 2007;
published online February 26, 2009. Review conducted by Nigel M. Sammes.
important one and having solved it, problem sbd is of minor importance. But in fact, the reverse is true: Proving merely the existence 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 quality 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 reconstructions 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, the x-direction
any current density, j, not depending on x, j = jsy , zd can at each
position sy 0 , z0d be replaced by one having cylindrical symmetry
around sy 0 , z0d without 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 problem sad. 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
nearly equal with regard to their external magnetic fields. In principle 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 successful reconstruction all properties of j and the constraints on j have
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8.4 mm
16 mm
layer sand possibly other layers except the MEAd. All the complicated chemical and physical processes taking place in the MEA
can be expressed by the sin this regime unknownd conductivity
distribution1 ssrd, which has to be determined.
Within this framework we can formulate the properties of and
the constraints on jsrd.
For properties,
14 mm
0.6mm
Hextern depends uniquely on jsrd
s1d
and can be computed from f15g
nonmagnetic steel
graphite
178 mm
MEA layer
nonmagnetic steel
¹ 3 H = 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
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 between 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.
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
density j. 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 assembly sMEAd. 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. siid The 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 without losing 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.
¹3
1
jsrd = 0
ssrd
s6d
Equations s3d, s4d, and s6d can be combined to
− ¹ · sssrd ¹ Fd = 0
s7d
Here, F is a scalar potential with the property
− ¹F = E
s8d
Since ssrd . 0 Eq. s7d represents an elliptic differential equation,
which leads to a unique solution2 of jsrd for each given ssrd.
Then Eq. s2d leads to a unique solution of the magnetic field. And
therefore we can write
Hsrd = Hsr,hsjd
s9d
Note that ssrd can vary freely whereas jsrd cannot. Note also that
there is no guarantee yet that jsrd is determined by H in 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 Distribution
2 The Current Density in the Fuel Cell Properties and
Constraints
Solving the partial differential equations s2d and s7d numerically amounts to discretizing the problem. Let ǰ be the vector
representing the discretized jsrd. If there are NJ discretized vectors then ǰ has 3NJ components. In an analogous manner we define Ȟ as the magnetic field vector representing the external magnetic field at all positions where H is measured. If NH is the
number of measuring points then the dimension of this vector is
3NH. Furthermore we denote that vector as š, whose NM components contain all the discretized conductivities in the MEA3 layer.
It follows from Eq. s2d that a given current distribution j leads
to a uniquely determined external H field and therefore Ȟ is a
unique function of ǰ.
The heart of a fuel cell is the MEA containing the electrolytic
layers and the layers where the catalytic reactions and recombinations 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 conductivity ssrd is
known in the regime of the metallic layers and of the graphite
1
Here we assume that sid the conductivity ssrd does not depend on the current,
and siid it is scalar and not a tensor. sid is 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. siid is completely justified outside the
MEA. Moreover we did not find a process in the MEA that would require a tensor for
the conductivity.
2
Appropriate boundary conditions are presumed.
3
Outside the MEA the conductivities are known and therefore fixed.
021012-2 / Vol. 6, MAY 2009
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s10d
Ȟ = Ȟsǰd
On the other hand the information contained in Eq. s2d is insufficient to construct the inverse map of Eq. s10d. We will now show
that the constraint equations s3d, s4d, and s6d strongly reduce the
degrees of freedom leading to criteria from which one is able to
decide whether or not ǰ can uniquely be determined for a given
resolution.
Because of the constraints not all variations dǰ are allowed.
Rather all possible variations are exhausted by4 dš,
dȞ = Sda
This equation shows how drastically the constraints sEqs. s3d, s4d,
and s6dd reduce the number of independent variables. Without
these constraints we would have S = S0 and S0 would be a 3NH
3 3Nj matrix. In contrast to that S is a 3Nj 3 Nc matrix and6
Nc ≪ 3Nj.
We can extract practical results from S by applying the singular
value decomposition again, this time to S
dǰ = Cdš
S = US · WS · VST
]ǰ
]š
C=
US = 3Nj 3 Nc
s11d
WS = Nc 3 Nc
C can be calculated by using first Eqs. s4d and s8d
jsrd = − ssrd · ¹Fsrd
s12d
Differentiating this equation with respect to s yields C provided
we can get ]F / ]s. However, this derivative can be computed by
differentiating Eq. s7d with respect to š and solving it in quite the
same way as Eq. s7d.
The properties of the matrix C can be detected by applying the
singular value decomposition:5
WC = NM 3 NM
unitary matrix
the resolution chosen in the MEA. The finer the discretization the larger the dimension Nc, i.e., the number of
independent dai.
siid the relative error f rel of the measuring devices. Due to
these errors small matrix elements in the matrix WS become irrelevant. In other words there is an integer NR beyond that we may set
WSk,k → 0
e is a parameter. It is not very critical and in this paper we have set
e = 0.01. Then we may write
Nc
dǰ =
o u da
k
k
s15d
k=1
The dai are independent of each other, but nevertheless guarantee
the constraints sEqs. s3d, s4d, and s6dd. Therefore the relevant
variations are the dai and the relevant matrix, connecting them
with the dǰ, is the 3NJ 3 Nc matrix Uc consisting of the column
vectors ui, i = 1 , . . . , Nc.
What we have achieved so far is the replacement of restricted
variations dǰ by unrestricted ones dai. It remains to determine the
matrix connecting dȞ and the dan. However, analogous to the
computation of C it is a straightforward procedure to compute
from Eq. s2d the derivative
S0 =
]Ȟ
]ǰ
s16d
for k . NR
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
NR given by7
s13d
s14d
s18d
sid
The smaller the WCi,i the larger dš has to be for a perceptible
variation in the direction of ui, ui being the ith column vector of
UC. This justifies a cutoff for avoiding unphysically large variations of s, i.e., we take into account only indices i with the property
thus i # NCsed
unitary matrix
The rank of S is determined by
diagonal matrix and WCi,i $ WCi+1,i+1 $ 0
WCi,i
. e,
WC11
diagonal matrix and WSi,i $ WSi+1,i+1 $ 0
orthogonal matrix
VC = NM 3 NM
orthogonal matrix
VS = Nc 3 Nc
C = UC · WC · VCT
UC = 3Nj 3 NM
s17d
S = S 0U c
f rel ,
WSNR,NR
WS11
,
f rel .
WSNR+1,NR+1
WS11
s20d
Now Eq. s20d contains the criterion we were looking for
Ȟsǰd is locally invertible if NR = Nc
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.
s21d holds 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. Equation s21d does
not guarantee the global invertibility but the local one only. Furthermore Eq. s21d is 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 of jsrd.
4
Positioning the Measuring Points
We suppose that either external magnetic fields are shielded by
plates with a high permeability m and the exterior boundaries have
been moved to these plates, or that external magnetic fields can be
and applying the chain rule we get
6
In a typical case Nc < 30– 450 and 3Nj < 30· Nc.
The estimate of Eq. s20d is on the safe side. It does not require any assumptions
about the kind of errors. Estimates can become more favorable if sid the kind of
systematic errors is known or if siid one can be sure that the errors are random, i.e.,
nonsystematic. This will be discussed in a forthcoming paper.
7
4
Note that this procedure reduces efficiently the number of degrees of freedom. A
typical computation in Sec. 5 gives NM = 436 and 3Nj < 16,000.
5
Note that 3Nj . NM.
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neglected altogether. In both srealisticd cases 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
NH measuring points. To see how it works consider a perturbation
da that causes a deviation dHsr jd at a measuring point with position r j. dHsr jd contains contributions from all orthogonal column vectors of the matrix US times the corresponding singular
value of the matrix WS. But for getting high resolution all column
vectors of US, 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 function8 z
2
zsr jd =
NH
sUSUST d3j+m,3j+m
Nc m=0
o
s22d
44
2
00
−2
−4
−4
−6
−8
00
5
10
10
20
20
15
25
30
30
35
Fig. 2 Singular values of the matrix C in Eq. „11… for the following 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.
z has the property
NH
1
zsrid = 1
NH i=1
o
s23d
z can be used as a criterion for finding the relevant set from NH
measuring points. Let dz be a constant we have fixed in the beginning. Then we select the relevant measuring positions r j according to
if zsr jd $ dz
then the measuring point at r j is kept
s24d
dz need 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.
s24d this 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 DMFC f16g. The MEA of this cell consists of three very
thin layers containing two catalytic layers splatinum and platinumrutheniumd 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 V s0.7 V
is the voltage for zero current in this cell f17gd.
8
We numerate the indices k = 1 , . . . , 3NH in the following manner: dHxsr jd
= dȞk=3j+0, dHysr jd = dȞk=3j+1, and dHzsr jd = dȞk=3j+2, where r j denotes the position
of the measuring point j.
021012-4 / Vol. 6, MAY 2009
To get the current distribution in the cell we solve Eq. s7d by
applying the finite difference method with one grid for the electric
potential F and another grid for both the current and the conductivity, with grid points located between the gridpoints of the first
grid. In this way edVj · ndo = 0 is guaranteed for any inner surface
dV. The resolution of j 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 / ]s can be obtained by differentiating Eq. s7d and by solving
the corresponding equation. In this way we get the matrix C defined 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 corresponds to a variation of all s values by the same amount. This and
the corresponding u vector are not taken into account. Components of some other ui, located in the MEA, are plotted in Fig. 3.
One can easily verify that with increasing the index i the 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.9 Then we need not solve Eq. s2d but can
use Biot–Savarts law f18,19g directly
Hsrd = ¹ 3
E
jsr8d 3
d r8
ur − r8u
s25d
In our case the fuel cell is a rectangular solid. The measuring
points are located on six rectangular planes, a distance dG apart
from the fuel cell’s surface.10 On 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 to j is easily computed using the same formula. We get the
matrix S0 scf. Eq. s16dd and from that the important matrix S scf.
Eq. s17dd.
9
This approximation is too crude when calculating currents j and fields H. But
when estimating variations dj and dH for the calculations in this paper this approximation is quite appropriate.
10
Due to additional frames and equipment the experimental fuel cell considered in
this paper fits into a cuboid of size 393 2003 280 mm3. The dG refer to distances
measured from this cuboid.
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Fig. 3 For the parameters given in Fig. 2 the components of various ui are shown „cf. Eq. „15……. One
can easily recognize the increasing resolution with increasing index i.
We have calculated the 3NH 3 Nc 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
density jsrd is resolved in the MEA. The required relative preci-
sion guaranteeing a rank Nc of the S matrix can be read off by
doing a singular value decomposition of S. 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-
−2
−2
−4−4
−2
−2
−6−6
−4
−4
−8−8
−6
−6
−10
−10
−8
−8
−12
−12
−10
−10
−14
−14
−12
−12
−14
−14
−16
−16
0.5
0.5
1
1.0
1.5
1.5
[cm]
2.02
2.5
2.5
[cm]
3
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 magnetic field is measured is dG = 3 cm. Dashed line: the spacing dP
between the measuring points on the planes is about 5 cm:
dP É 5 cm. Dashed dotted line: dP É 2.5 cm. Dotted line: dP
É 1.2 cm. Full line: dP É 0.6 cm. Magenta and blue line nearly
coincide in this case. Note that a spacing of 5 cm is much too
large.
Journal of Fuel Cell Science and Technology
−16
−16
0.5
0.5
1
1.0
1.5
1.5
[cm]
2
2.0
2.5
2.5
[cm]
3
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 magnetic field is measured is dG = 1 cm. Dashed line: dP É 5 cm.
Dashed dotted line: dP É 2.5 cm. Dotted line: dP É 1.2 cm. Full
line: dP É 0.6 cm. Note that a spacing of 5 cm is much too large
leading to meaningless results.
MAY 2009, Vol. 6 / 021012-5
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Fig. 6 All 5536 measuring points located in the six planes surrounding the fuel cell are
marked. Distance between the fuel cell and the planes dG = 1 cm. Spacing between the
measuring points is about 0.6 cm. At each measuring point the value of the z function 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 highest z values. Note that the
scales in depth, width, and height are not identical.
lution. One recognizes from Fig. 4 that the resolution is approximately 2 cm in the MEA for a distance dG = 3 cm. In contrast Fig.
5 shows that the resolution is approximately 1 cm for the same
relative precision if dG = 1 cm. Furthermore only for this value of
dG the 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 the z function is shown for a distance between
measuring planes dG = 1 cm. We have chosen a spacing between
Fig. 7 The remaining 868 measuring points after all measuring points with z„rj… < 2 have
been dropped. Nomenclature and other parameters as in Fig. 6.
021012-6 / Vol. 6, MAY 2009
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Fig. 8 Singular values „dG = 1 cm and dP = 0.6 cm… of the full S matrix built from all 5536
measuring points „upper line… and singular values of the strongly reduced S matrix built
from the remaining 868 measuring points with z„rj… Ð 2 „lower line…. Plotted are the index
numbers versus the logarithm of the corresponding values. One can verify by inspection
that the condition of the reduced S matrix is better than the condition of the full S matrix.
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 large z values are
found on these planes because there the true diameters of cylin-
drical currents can be detected best. Thus measurements are necessary 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 full S matrix and of the reduced one are compared.
In Fig. 9 the plots corresponding to Fig. 5 are shown. It is remark-
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 with z„rj… Ð 2 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
less points. 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 valued becomes 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 errors se.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 experiment 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
experimental sDMFCd fuel cell.
Acknowledgment
We would like to thank H. Dohle and M. Wannert for stimulating discussions.
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