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Research Papers

# Reconstruction of Electric Currents in a Fuel Cell by Magnetic Field Measurements

[+] Author and Article Information
H. Lustfeld

Forschungszentrum Juelich, IFF and IAS, 52425 Juelich, Germanyh.lustfeld@fz-juelich.de

M. Reißel

Fachhochschule Aachen, Campus Juelich, 52428 Juelich, Germanyreissel@fh-aachen.de

U. Schmidt

Fachhochschule Aachen, Campus Juelich, 52428 Juelich, Germanyu.schmidt@fh-aachen.de

B. Steffen

Forschungszentrum Jülich, JSC, 52425 Juelich, Germanyb.steffen@fz-juelich.de

Here we assume that (i) the conductivity $σ(r)$ does not depend on the current, and (ii) it is scalar and not a tensor. (i) 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. (ii) is completely justified outside the MEA. Moreover we did not find a process in the MEA that would require a tensor for the conductivity.

Appropriate boundary conditions are presumed.

Outside the MEA the conductivities are known and therefore fixed.

Note that this procedure reduces efficiently the number of degrees of freedom. A typical computation in Sec. 5 gives $NM=436$ and $3Nj≈16,000$.

Note that $3Nj>NM$.

In a typical case $Nc≈30–450$ and $3Nj≈30⋅Nc$.

The estimate of Eq. 20 is on the safe side. It does not require any assumptions about the kind of errors. Estimates can become more favorable if (i) the kind of systematic errors is known or if (ii) one can be sure that the errors are random, i.e., nonsystematic. This will be discussed in a forthcoming paper.

We numerate the indices $k=1,…,3NH$ in the following manner: $dHx(rj)=dȞk=3j+0$, $dHy(rj)=dȞk=3j+1$, and $dHz(rj)=dȞk=3j+2$, where $rj$ denotes the position of the measuring point $j$.

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.

Due to additional frames and equipment the experimental fuel cell considered in this paper fits into a cuboid of size $39×200×280mm3$. The $dG$ refer to distances measured from this cuboid.

J. Fuel Cell Sci. Technol 6(2), 021012 (Feb 26, 2009) (8 pages) doi:10.1115/1.2972171 History: Received August 10, 2007; Revised October 04, 2007; Published February 26, 2009

## Abstract

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 $j(r)$ 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.

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## Figures

Figure 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 0.6mm. 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 138mm. We use this model in the numerical calculations.

Figure 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.

Figure 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=3cm. Dashed line: the spacing dP between the measuring points on the planes is about 5cm: dP≈5cm. Dashed dotted line: dP≈2.5cm. Dotted line: dP≈1.2cm. Full line: dP≈0.6cm. Magenta and blue line nearly coincide in this case. Note that a spacing of 5cm is much too large.

Figure 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=1cm. Dashed line: dP≈5cm. Dashed dotted line: dP≈2.5cm. Dotted line: dP≈1.2cm. Full line: dP≈0.6cm. Note that a spacing of 5cm is much too large leading to meaningless results.

Figure 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=1cm. Spacing between the measuring points is about 0.6cm. At each measuring point the value of the ζ 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 ζ values. Note that the scales in depth, width, and height are not identical.

Figure 7

The remaining 868 measuring points after all measuring points with ζ(rj)<2 have been dropped. Nomenclature and other parameters as in Fig. 6.

Figure 8

Singular values (dG=1cm and dP=0.6cm) 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 ζ(rj)⩾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.

Figure 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 0.6cm between measuring points. Upper line: out of the 5536 measuring points only the 868 points with ζ(rj)⩾2 are taken into account. Note that reducing the number of measuring points leads to a better resolution.

Figure 2

Singular values of the matrix C in Eq. 11 for the following resolution: 0.35cm vertical to the MEA plane, otherwise 2.75cm. 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 σ by the same amount. This change can be detected via the cell voltage.

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