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Electrostatic broadening of current-free spots in a fuel cell stack: The mechanism of stack aging?
Electrochemistry Communications 8 (2006) 1225–1228 www.elsevier.com/locate/elecom
Electrostatic broadening of current-free spots in a fuel cell stack: The mechanism of stack aging? A.A. Kulikovsky
*
Institute for Materials and Processes in Energy Systems (IWV-3), Research Center ‘‘Ju¨lich’’, D-52425 Ju¨lich, Germany Received 18 April 2006; accepted 19 May 2006 Available online 3 July 2006
Abstract We develop a model of a fuel cell stack focused on effects due to current-free spots in the individual cells. The spots disturb potential of the bipolar plate over the distance, which exceeds the physical size of the spot. The variation of plate potential translates into the growth of cell local current around the spot. Analytical solution to a model problem shows that the amplitude of voltage perturbation grows quadratically with the spot radius and linearly with the resistivity of the bipolar plate and with the mean current in the stack. Possible scenario of stack aging due to this electrostatic broadening of spots is discussed. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Fuel cell stack; Current-free spots; Aging; Modeling
1. Introduction Dramatically increasing prices of oil and gas heat up the search for alternative power sources worldwide. In a very short list of environmental friendly and efficient energy converters, fuel cells are probably the most promising candidates to replace old-fashioned internal combustion engines. To achieve high power density individual cells are assembled into a stack, typically consisting of ten to hundred cells. The cells in a stack are separated by the bipolar plates (BPs), which distribute reactants over the cells surface and transport current from one cell to another. In addition to kinetic and transport voltage losses in the individual cells, transport of current through the BPs leads to stack-specific voltage losses [1,2]. Voltage loss in the BP is negligible if the distributions of local current over the surface of adjacent cells are the same [1]. In practice, however, due to various factors these distributions may strongly differ. One of the largest problems in stacks is formation of local current-free (CF) spots in the *
Tel.: +49 2461 61 5396; fax: +49 2461 61 6695. E-mail address: [email protected].
1388-2481/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.elecom.2006.05.010
cells. These spots arise due to a large contact resistance between the functional layers in the cell (or between the cell and the flow field), or due to a local degradation of one of the layers. The degradation may be caused by numerous factors, like catalyst agglomeration, degradation of membrane conductivity etc. Regardless of the physical nature, the spot disturbs BP potential. Local voltage of the cell depends on the BP potential; therefore CF spot induces redistribution of local current over the cell surface. How large is this effect? What is the characteristic size of potential disturbance due to the spot and how these disturbance depends on parameters? This Letter aims at answering this questions. 2. Model 2.1. Bipolar plate potential Consider the stack consisting of 4 square cells separated by 3 bipolar plates with the side L (Fig. 1). Potential Va of the bipolar plate a satisfies Poisson equation [2] o 2 V a o 2 V a jA jB ; þ ¼ ox2 oy 2 hp r
ð1Þ
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A.A. Kulikovsky / Electrochemistry Communications 8 (2006) 1225–1228
each spot has Gaussian shape. To avoid unphysical doubling of resistivity of the overlapping spots, we will calculate the local resistivity of the cell A due to the spots as ! ð~x ~xk Þ2 þ ð~y ~y k Þ2 e e ; ð4Þ R A ðx; yÞ ¼ R 0 exp s2 where (xk, yk) are coordinates of the spot nearest to the point (x, y), s is the characteristic radius of the spots. Our goal is to rationalize the electrostatic effect of spots on stack performance; to simplify calculations we will assume that local polarization curve of each individual cell is linear. For cell A we write V A ¼ V oc Rg jA RA jA ;
Fig. 1. Sketch of the 4-cell stack.
where Voc is cell open circuit voltage, Rg is internal cell resistivity (the same for all cells). In dimensionless variables Eq. (5) reads e A Þ~jA : Ve A ¼ 1 ð1 þ R
where jA and jB are the local current densities in the cells A and B, respectively (Fig. 1), hp is the BP thickness and r is the conductivity of BP material. Quite analogous equations can be written for the other BPs. Introducing dimensionless variables jRg x y V e¼ R; ~x ¼ ; ~y ¼ ; ~j ¼ ; Ve ¼ ; R ð2Þ L L V oc Rg V oc we get o2 Ve a o2 Ve a e ~ þ ¼ R bp ðjA ~jB Þ: ð3Þ o~x2 o~y 2 Here Voc is cell open circuit voltage, Rg is internal cell resistivity (X cm2, see below) and Rbp = L2/(hpr) is the BP resistivity. Normal current through the end faces of the plate is zero; we thus have the following boundary conditions for Eq. (3): o Ve a o Ve a ¼ ¼ 0: o~x o~y ~x¼0;1
~y ¼0;1
These conditions determine Ve within the additive constant; the latter can be defined assigning arbitrary potential to any point of the plate, e.g., taking Ve a ð0; 0Þ ¼ 1. Eq. (3) is well known in plasma physics and in the theory of electrolytes. If we replace ~jA and ~jB with, e.g., the number densities of ions and electrons, respectively, Eq. (3) will describe potential of electric field in the ion–electron plasma. Currents in Eq. (3) are thus quite analogous to charges in the plasma or in the electrolyte. We, therefore, may expect that the effect of spot on the BP potential has a long-range Coulomb character. In the following section we will show that this, indeed, is the case. 2.2. Local polarization curve Suppose that each cell in the stack has N randomly located CF spots. We will assume that the resistivity of
ð5Þ
ð6Þ
Local potential of cell A is equal to the voltage drop between the bipolar plates c and a (Fig. 1) Ve A ¼ Ve 0ca þ Ve c Ve a :
ð7Þ
Here Ve 0ca Ve c ð0; 0Þ Ve a ð0; 0Þ is voltage drop between the BPs c and a at the origin of coordinates ( Ve c and Ve a are calculated with respect to the plate potential at (0, 0)). Equating (7) and (6), and solving for ~jA we get e0 e e ~jA ¼ E ca ð V c V a Þ ; eA 1þR
ð8Þ
where e 0 ¼ 1 Ve 0 : E ca ca
ð9Þ
Quite analogous, for ~jB we write e0 e e ~jB ¼ E ab ð V a V b Þ ð10Þ eB 1þR e 0 is related to the mean current density in the cell e E J. ca e ¼ 1 we get Integrating (8) over the cell surface A Z e0 ¼ e e A þ ð Ve c Ve a ÞÞd e E J þ ð~jA R S: ð11Þ ca 3. Results and discussion The procedure of stack simulation is as following. Potential of each BP is governed by Eq. (3). The right side of this equation contains local current densities in the adjacent cells, which, in turn depend on potentials of the BPs seizing these cells. Model of stack thus leads to a selfconsistent problem for distribution of local current densities in the cells and potentials in the BPs. This problem is solved iteratively. Calculation of BP potential requires solution of 2D Poisson equation, which is rather lengthy procedure. Due to that the problem can be efficiently parallelized: each BP can be ‘‘solved’’ on a separate processor.
A.A. Kulikovsky / Electrochemistry Communications 8 (2006) 1225–1228
Each cell in the stack has N randomly located spots. The resistivity of spot has Gaussian shape (4); the characteristic radius s is the same for all spots. Parameters for calculations are listed in Table 1. Fig. 2 shows the maps of voltage of the bipolar plate a and local current in the cell A when the number of spots is N = 4. Current entering the BP a is produced in the cell A; the spots in A thus generate the largest disturbance of potential Ve a . The spots in the cell A and their physical radius are clearly seen: these are the circular ‘‘holes’’ on the map of current (Fig. 2b). These spots produce local peaks on the map of potential Ve a . The spots in the cell B also contribute Table 1 Parameters for numerical calculations e bp R
e0 R
s
e J
100
100
0.02
0.5
Fig. 2. Maps of potential of the bipolar plate a and local current density in the cell A ð~jA e J Þ (Fig. 1). Location of spots in the cell A and their physical size are seen on the map of current (the holes).
1227
to non-uniformity of Ve a : they generate negative cavities on the map of Ve a (Fig. 2a). Due to the coupling between cell current and BP voltage these cavities translate into the cavities on the map of current (Fig. 2b). Map of potential clearly shows that the size of potential disturbance exceeds the size of the spot. The spots manifest themselves as surface ‘‘charges’’, which induce a long-range disturbance of BP voltage. Analytical solution to a model problem shows that potential disturbance is the largest near the spot, but it extends over the whole cell active area (see Appendix). Due to Eq. (8), the disturbance in BP voltage Ve a translates into the disturbance of cell current. The largest growth of current occurs in the vicinity of spots (Fig. 2b). This effect leads to formation of the ‘‘bridge’’ of enhanced local current between two closely located spots (the two leftmost spots in Fig. 2b). Qualitatively the same effect depicts Fig. 3, which shows the maps in the case of 8 spots per cell. The spots located
Fig. 3. The same as in Fig. 2 for 8 spots per cell.
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A.A. Kulikovsky / Electrochemistry Communications 8 (2006) 1225–1228
near each other are connected with the ‘‘bridges’’ of enhanced cell current (Fig. 3b). This effect may lead to the following scenario of stack degradation. We may expect that the rate of aging is larger in domains where local current is higher. Thus, the highcurrent ‘‘bridges’’ between the spots may transform into new spots. This mechanism provides ‘‘agglomeration’’ of initially isolated spots into a large current-free cluster. The cluster will further grow with time, thus increasing local current in the rest part of the cell. This process will dramatically reduce the stack performance. Appendix. CF spot in the center of circular plate To understand the character of potential disturbance due to a spot and to rationalize the parametric dependencies consider the circular bipolar plate of a unit radius. Suppose that current enters the BP from the cell on top of the plate. Let current density in this cell has the following shape (Fig. 4): 2 ~r ~jA ¼ a e J 1 exp 2 ; s
ð12Þ
where r is the distance from the BP center and the characteristic spot radius s is small: s 1. Constant a can be found from the normalization requirement Z
1
2p~r~jA d~r ¼ p e J;
ð13Þ
which yields 1 1 ’ : 2 2 2 1 s þ s expðs Þ 1 s2
ð14Þ
field
6 1
4
Current
Potential and field
current
0.5
2 potential
0
0
~jB ¼ e J:
ð15Þ
With (12) and (15), Poisson equation (3) for BP potential is ! 2 2 1 o o Ve e bp Je 1 exp ð~r =s Þ 1 ; Ve ð1Þ ~r ¼R ~r o~r o~r 1 s2 o Ve ¼ 0; ð16Þ ¼ 0: o~r ~r¼1
At small s the general solution to this equation simplifies to 2 e bp e R J s2 1 r ð~r2 1Þ e ln ~r Ei 1; 2 þ V ’ ; ð17Þ 2 2 2ð1 s2 Þ s R1 where Eið1; xÞ ¼ 1 t1 expðtxÞdt is the exponential integral. The function in square brackets in (17) is shown in Fig. 4 for s = 0.1. We see, that though physically the spot is localized within the radius .2s = 0.2, potential disturbance induced by the spot extends over the whole BP surface (Fig. 4). Formally, due to ln ~r in Eq. (17) the disturbance of potential has no characteristic size. The peak BP voltage Ve 0 is achieved at ~r ¼ 0. Expanding Eq. (17), for Ve 0 we find e bp e R J s2 ðc 2 ln s 1Þ Ve 0 ¼ 4ð1 s2 Þ
0
a¼
Note that hereinafter we retain the first non-vanishing, quadratic in s terms and neglect vanishingly small terms with exp(s2). Let the cell under the bipolar plate has no spots and current in this cell is distributed uniformly over its surface:
0.2
0.4
0.6
0.8
1
0
ð18Þ
where c = 0.57721. . . is Euler’s constant. Neglecting weak logarithmic dependence of Ve 0 on s, we see that Ve 0 scales e bp and e linearly with R J and quadratically with the characteristic spot radius s. Differentiating (17) over ~r we get the electric field in the e ¼ o Ve =o~r: plate E 2 e e 2 e ¼ R bp J s 1 1 exp ~r ~ E r : ð19Þ 2ð1 s2 Þ ~r s2 The field (19) is zero at ~r ¼ 0; it reaches maximum at ~r ’ s and then decreases with ~r (Fig. 4). At ~r J 3s the function in the square brackets in (19) is 1=~r ~r (Fig. 4). Thus, outside the spot the shapes of the field and potential do not depend on spot radius s. This behavior is similar to the field of the charged ball, which outside the ball does not depend on ball radius and decreases as a field of a point charge 1=~r. In the electrostatics of current-free spots on a plane bipolar plate the field decreases faster, as 1=~r ~r. References
Fig. 4. Analytical solution to the problem of potential disturbance due to the single spot in the center of coordinates (Appendix). Local current in the cell A (dash-dotted line); BP potential (thick solid line) and electric field (thin solid line) induced by the spot. Dashed line: the function 1=~r ~r.
[1] A.A. Kulikovsky, Model of DMFC stack: resistive ‘‘spots’’ and stack performance, J. Electrochem. Soc. 153 (2006), in press. [2] A.A. Kulikovsky, Voltage loss in bipolar plates in a fuel cell stack, J. Power Sources (2006), in press, doi:10.1016/j.jpowsour.2006.01.058.
Electrostatic broadening of current-free spots in a fuel cell stack: The mechanism of stack aging? A.A. Kulikovsky
*
Institute for Materials and Processes in Energy Systems (IWV-3), Research Center ‘‘Ju¨lich’’, D-52425 Ju¨lich, Germany Received 18 April 2006; accepted 19 May 2006 Available online 3 July 2006
Abstract We develop a model of a fuel cell stack focused on effects due to current-free spots in the individual cells. The spots disturb potential of the bipolar plate over the distance, which exceeds the physical size of the spot. The variation of plate potential translates into the growth of cell local current around the spot. Analytical solution to a model problem shows that the amplitude of voltage perturbation grows quadratically with the spot radius and linearly with the resistivity of the bipolar plate and with the mean current in the stack. Possible scenario of stack aging due to this electrostatic broadening of spots is discussed. Ó 2006 Elsevier B.V. All rights reserved. Keywords: Fuel cell stack; Current-free spots; Aging; Modeling
1. Introduction Dramatically increasing prices of oil and gas heat up the search for alternative power sources worldwide. In a very short list of environmental friendly and efficient energy converters, fuel cells are probably the most promising candidates to replace old-fashioned internal combustion engines. To achieve high power density individual cells are assembled into a stack, typically consisting of ten to hundred cells. The cells in a stack are separated by the bipolar plates (BPs), which distribute reactants over the cells surface and transport current from one cell to another. In addition to kinetic and transport voltage losses in the individual cells, transport of current through the BPs leads to stack-specific voltage losses [1,2]. Voltage loss in the BP is negligible if the distributions of local current over the surface of adjacent cells are the same [1]. In practice, however, due to various factors these distributions may strongly differ. One of the largest problems in stacks is formation of local current-free (CF) spots in the *
Tel.: +49 2461 61 5396; fax: +49 2461 61 6695. E-mail address: [email protected].
1388-2481/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.elecom.2006.05.010
cells. These spots arise due to a large contact resistance between the functional layers in the cell (or between the cell and the flow field), or due to a local degradation of one of the layers. The degradation may be caused by numerous factors, like catalyst agglomeration, degradation of membrane conductivity etc. Regardless of the physical nature, the spot disturbs BP potential. Local voltage of the cell depends on the BP potential; therefore CF spot induces redistribution of local current over the cell surface. How large is this effect? What is the characteristic size of potential disturbance due to the spot and how these disturbance depends on parameters? This Letter aims at answering this questions. 2. Model 2.1. Bipolar plate potential Consider the stack consisting of 4 square cells separated by 3 bipolar plates with the side L (Fig. 1). Potential Va of the bipolar plate a satisfies Poisson equation [2] o 2 V a o 2 V a jA jB ; þ ¼ ox2 oy 2 hp r
ð1Þ
1226
A.A. Kulikovsky / Electrochemistry Communications 8 (2006) 1225–1228
each spot has Gaussian shape. To avoid unphysical doubling of resistivity of the overlapping spots, we will calculate the local resistivity of the cell A due to the spots as ! ð~x ~xk Þ2 þ ð~y ~y k Þ2 e e ; ð4Þ R A ðx; yÞ ¼ R 0 exp s2 where (xk, yk) are coordinates of the spot nearest to the point (x, y), s is the characteristic radius of the spots. Our goal is to rationalize the electrostatic effect of spots on stack performance; to simplify calculations we will assume that local polarization curve of each individual cell is linear. For cell A we write V A ¼ V oc Rg jA RA jA ;
Fig. 1. Sketch of the 4-cell stack.
where Voc is cell open circuit voltage, Rg is internal cell resistivity (the same for all cells). In dimensionless variables Eq. (5) reads e A Þ~jA : Ve A ¼ 1 ð1 þ R
where jA and jB are the local current densities in the cells A and B, respectively (Fig. 1), hp is the BP thickness and r is the conductivity of BP material. Quite analogous equations can be written for the other BPs. Introducing dimensionless variables jRg x y V e¼ R; ~x ¼ ; ~y ¼ ; ~j ¼ ; Ve ¼ ; R ð2Þ L L V oc Rg V oc we get o2 Ve a o2 Ve a e ~ þ ¼ R bp ðjA ~jB Þ: ð3Þ o~x2 o~y 2 Here Voc is cell open circuit voltage, Rg is internal cell resistivity (X cm2, see below) and Rbp = L2/(hpr) is the BP resistivity. Normal current through the end faces of the plate is zero; we thus have the following boundary conditions for Eq. (3): o Ve a o Ve a ¼ ¼ 0: o~x o~y ~x¼0;1
~y ¼0;1
These conditions determine Ve within the additive constant; the latter can be defined assigning arbitrary potential to any point of the plate, e.g., taking Ve a ð0; 0Þ ¼ 1. Eq. (3) is well known in plasma physics and in the theory of electrolytes. If we replace ~jA and ~jB with, e.g., the number densities of ions and electrons, respectively, Eq. (3) will describe potential of electric field in the ion–electron plasma. Currents in Eq. (3) are thus quite analogous to charges in the plasma or in the electrolyte. We, therefore, may expect that the effect of spot on the BP potential has a long-range Coulomb character. In the following section we will show that this, indeed, is the case. 2.2. Local polarization curve Suppose that each cell in the stack has N randomly located CF spots. We will assume that the resistivity of
ð5Þ
ð6Þ
Local potential of cell A is equal to the voltage drop between the bipolar plates c and a (Fig. 1) Ve A ¼ Ve 0ca þ Ve c Ve a :
ð7Þ
Here Ve 0ca Ve c ð0; 0Þ Ve a ð0; 0Þ is voltage drop between the BPs c and a at the origin of coordinates ( Ve c and Ve a are calculated with respect to the plate potential at (0, 0)). Equating (7) and (6), and solving for ~jA we get e0 e e ~jA ¼ E ca ð V c V a Þ ; eA 1þR
ð8Þ
where e 0 ¼ 1 Ve 0 : E ca ca
ð9Þ
Quite analogous, for ~jB we write e0 e e ~jB ¼ E ab ð V a V b Þ ð10Þ eB 1þR e 0 is related to the mean current density in the cell e E J. ca e ¼ 1 we get Integrating (8) over the cell surface A Z e0 ¼ e e A þ ð Ve c Ve a ÞÞd e E J þ ð~jA R S: ð11Þ ca 3. Results and discussion The procedure of stack simulation is as following. Potential of each BP is governed by Eq. (3). The right side of this equation contains local current densities in the adjacent cells, which, in turn depend on potentials of the BPs seizing these cells. Model of stack thus leads to a selfconsistent problem for distribution of local current densities in the cells and potentials in the BPs. This problem is solved iteratively. Calculation of BP potential requires solution of 2D Poisson equation, which is rather lengthy procedure. Due to that the problem can be efficiently parallelized: each BP can be ‘‘solved’’ on a separate processor.
A.A. Kulikovsky / Electrochemistry Communications 8 (2006) 1225–1228
Each cell in the stack has N randomly located spots. The resistivity of spot has Gaussian shape (4); the characteristic radius s is the same for all spots. Parameters for calculations are listed in Table 1. Fig. 2 shows the maps of voltage of the bipolar plate a and local current in the cell A when the number of spots is N = 4. Current entering the BP a is produced in the cell A; the spots in A thus generate the largest disturbance of potential Ve a . The spots in the cell A and their physical radius are clearly seen: these are the circular ‘‘holes’’ on the map of current (Fig. 2b). These spots produce local peaks on the map of potential Ve a . The spots in the cell B also contribute Table 1 Parameters for numerical calculations e bp R
e0 R
s
e J
100
100
0.02
0.5
Fig. 2. Maps of potential of the bipolar plate a and local current density in the cell A ð~jA e J Þ (Fig. 1). Location of spots in the cell A and their physical size are seen on the map of current (the holes).
1227
to non-uniformity of Ve a : they generate negative cavities on the map of Ve a (Fig. 2a). Due to the coupling between cell current and BP voltage these cavities translate into the cavities on the map of current (Fig. 2b). Map of potential clearly shows that the size of potential disturbance exceeds the size of the spot. The spots manifest themselves as surface ‘‘charges’’, which induce a long-range disturbance of BP voltage. Analytical solution to a model problem shows that potential disturbance is the largest near the spot, but it extends over the whole cell active area (see Appendix). Due to Eq. (8), the disturbance in BP voltage Ve a translates into the disturbance of cell current. The largest growth of current occurs in the vicinity of spots (Fig. 2b). This effect leads to formation of the ‘‘bridge’’ of enhanced local current between two closely located spots (the two leftmost spots in Fig. 2b). Qualitatively the same effect depicts Fig. 3, which shows the maps in the case of 8 spots per cell. The spots located
Fig. 3. The same as in Fig. 2 for 8 spots per cell.
1228
A.A. Kulikovsky / Electrochemistry Communications 8 (2006) 1225–1228
near each other are connected with the ‘‘bridges’’ of enhanced cell current (Fig. 3b). This effect may lead to the following scenario of stack degradation. We may expect that the rate of aging is larger in domains where local current is higher. Thus, the highcurrent ‘‘bridges’’ between the spots may transform into new spots. This mechanism provides ‘‘agglomeration’’ of initially isolated spots into a large current-free cluster. The cluster will further grow with time, thus increasing local current in the rest part of the cell. This process will dramatically reduce the stack performance. Appendix. CF spot in the center of circular plate To understand the character of potential disturbance due to a spot and to rationalize the parametric dependencies consider the circular bipolar plate of a unit radius. Suppose that current enters the BP from the cell on top of the plate. Let current density in this cell has the following shape (Fig. 4): 2 ~r ~jA ¼ a e J 1 exp 2 ; s
ð12Þ
where r is the distance from the BP center and the characteristic spot radius s is small: s 1. Constant a can be found from the normalization requirement Z
1
2p~r~jA d~r ¼ p e J;
ð13Þ
which yields 1 1 ’ : 2 2 2 1 s þ s expðs Þ 1 s2
ð14Þ
field
6 1
4
Current
Potential and field
current
0.5
2 potential
0
0
~jB ¼ e J:
ð15Þ
With (12) and (15), Poisson equation (3) for BP potential is ! 2 2 1 o o Ve e bp Je 1 exp ð~r =s Þ 1 ; Ve ð1Þ ~r ¼R ~r o~r o~r 1 s2 o Ve ¼ 0; ð16Þ ¼ 0: o~r ~r¼1
At small s the general solution to this equation simplifies to 2 e bp e R J s2 1 r ð~r2 1Þ e ln ~r Ei 1; 2 þ V ’ ; ð17Þ 2 2 2ð1 s2 Þ s R1 where Eið1; xÞ ¼ 1 t1 expðtxÞdt is the exponential integral. The function in square brackets in (17) is shown in Fig. 4 for s = 0.1. We see, that though physically the spot is localized within the radius .2s = 0.2, potential disturbance induced by the spot extends over the whole BP surface (Fig. 4). Formally, due to ln ~r in Eq. (17) the disturbance of potential has no characteristic size. The peak BP voltage Ve 0 is achieved at ~r ¼ 0. Expanding Eq. (17), for Ve 0 we find e bp e R J s2 ðc 2 ln s 1Þ Ve 0 ¼ 4ð1 s2 Þ
0
a¼
Note that hereinafter we retain the first non-vanishing, quadratic in s terms and neglect vanishingly small terms with exp(s2). Let the cell under the bipolar plate has no spots and current in this cell is distributed uniformly over its surface:
0.2
0.4
0.6
0.8
1
0
ð18Þ
where c = 0.57721. . . is Euler’s constant. Neglecting weak logarithmic dependence of Ve 0 on s, we see that Ve 0 scales e bp and e linearly with R J and quadratically with the characteristic spot radius s. Differentiating (17) over ~r we get the electric field in the e ¼ o Ve =o~r: plate E 2 e e 2 e ¼ R bp J s 1 1 exp ~r ~ E r : ð19Þ 2ð1 s2 Þ ~r s2 The field (19) is zero at ~r ¼ 0; it reaches maximum at ~r ’ s and then decreases with ~r (Fig. 4). At ~r J 3s the function in the square brackets in (19) is 1=~r ~r (Fig. 4). Thus, outside the spot the shapes of the field and potential do not depend on spot radius s. This behavior is similar to the field of the charged ball, which outside the ball does not depend on ball radius and decreases as a field of a point charge 1=~r. In the electrostatics of current-free spots on a plane bipolar plate the field decreases faster, as 1=~r ~r. References
Fig. 4. Analytical solution to the problem of potential disturbance due to the single spot in the center of coordinates (Appendix). Local current in the cell A (dash-dotted line); BP potential (thick solid line) and electric field (thin solid line) induced by the spot. Dashed line: the function 1=~r ~r.
[1] A.A. Kulikovsky, Model of DMFC stack: resistive ‘‘spots’’ and stack performance, J. Electrochem. Soc. 153 (2006), in press. [2] A.A. Kulikovsky, Voltage loss in bipolar plates in a fuel cell stack, J. Power Sources (2006), in press, doi:10.1016/j.jpowsour.2006.01.058.