ABSTRACT In this paper we present how advection in the electric double layer (EDL) affects the kinetic performance of electrochemical cells. To accomplish this we use a laminar flow fuel cell model based on the Poisson-NernstPlanck and Frumkin-Butler-Volmer equations. The model contains nonlinear physics with very disparate length scales due to the complex 3-dimensional nature of the nano-porous device. To account for these difficulties, the full mathematical model is solved numerically using a novel numerical algorithm developed based on domain decomposition method. Numerical results show that the presence of an advection flux through nano-pores on the order of the EDL width yields some novel physics that affect the structure of electrode-electrolyte interface. We also show that electrolyte advection within the EDL can be used to enhance the kinetic performance of electrodes in electrochemical cells. In the device presented the peak power density can be increased significantly with flow velocity

Isaac B Sprague

Prashanta Dutta

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 1 Copyright © 2012 by ASME 
Flow Through Nanoporous Electrodes in a Microfluidic Fuel Cell 
Isaac B. Sprague and Prashanta Dutta 
School of Mechanical and Materials Engineering 
Washington State University, Pullman, WA 99164-2920 
 
 
ABSTRACT 
In this paper we present how advection in the electric 
double layer (EDL) affects the kinetic performance of 
electrochemical cells. To accomplish this we use a 
laminar flow fuel cell model based on the Poisson-Nernst-
Planck and Frumkin-Butler-Volmer equations. The model 
contains nonlinear physics with very disparate length 
scales due to the complex 3-dimensional nature of the 
nano-porous device. To account for these difficulties, the 
full mathematical model is solved numerically using a 
novel numerical algorithm developed based on domain 
decomposition method. Numerical results show that the 
presence of an advection flux through nano-pores on the 
order of the EDL width yields some novel physics that 
affect the structure of electrode-electrolyte interface. We 
also show that electrolyte advection within the EDL can 
be used to enhance the kinetic performance of electrodes 
in electrochemical cells. In the device presented the peak 
power density can be increased significantly with flow 
velocity. 
 
Nomenclature 
A anion 
C cation 
c concentration 
D diffusion coefficient 
e
-
 electron 
F Faraday’s constant 
H hydrogen 
I total current density 
j current density 
k  forward rate constant 
k  backward rate constant 
N flux density of species 
n electron number per mole of fuel 
O oxygen 
P power desnity 
p pressure 
R universal gas constant 
T temperature 
v

 velocity field 
z ionic charge 
 
Greek Letters 
β  kinetic symmetry factors 
sΔ  potential drop across Stern layer 
  Debye length 
ε  permittivity 
μ  viscosity 
  density 
  electrolyte potential 
cell  cell potential 
El  electrode potential 
PR  potential at reaction plane 
ω  mobility 
 
INTRODUCTION 
In the past decade, the paradigm of using micro fuel cells 
for portable power applications has inspired novel 
innovations in fuel cell technology. One such example is 
the laminar flow fuel cell (LFFC) which utilizes colaminar 
flow to maintain the separation between the anode and 
cathode instead of a solid electrolyte such as the 
membrane used in polymer electrolyte membrane (PEM) 
fuel cells. The original concept of a LFFC was first 
presented as a vanadium redox cell in 2002 [1].  LFFCs 
were then investigated experimentally by addressing 
parameters such as flow rates and fuel concentrations [2-
4]. Further adaptations of the original LFFC concept have 
also been proposed, usually focusing on the electrode 
structure such as an air breathing cathode [5-6], 
electrodes comprised of graphite rods aligned in the 
streamwise direction [7], and reactants fed through 
porous electrodes [8], among others.  
As microfabrication and nano-patterning techniques 
improve, new possibilities for improving the effective 
Proceedings of the ASME 2012 International Mechanical Engineering Congress & Exposition 
IMECE2012 
November 9-15, 2012, Houston, Texas, USA 
IMECE2012-85769
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Figure 1.  Device configuration used for electrolyte 
advection through nano-porous electrodes. The 
device is comprised of two inlets on opposing sides of 
a microchannel that feed to porous electrodes which 
are separated by a central region. The nano-pores 
(inset) are considered to be slots extending the full 
height of the planar channel. For the bulk electrolyte 
advection through the pores to influence the electrode 
performance, the physical values of the pore width 
     must be on the order of the Debye length 
                                . Hydrogen 
and oxygen (H-O) are considered for the fuel and 
oxidant using the following stoichiometric reaction: 
OH2OH2
222
 . 
electrode performance are available through electrode 
microstructure design with features down to the order of 
the EDL width. Cylindrical electrodes with diameters down 
to the nano-scale are found to greatly alter electrode 
kinetics [10]. The EDL structure has been found to be 
altered through the use of blind nano-pores in a planar 
electrode [11]. Here, we introduce an electrode design 
with nano-scale features to improve laminar flow fuel cell 
performance. This is accomplished by flowing an 
electrolyte and reactant mixture through nano-pores in 
the electrode, providing an advection flux in regions of 
nonzero charge density. Figure 1 presents the proposed 
counter flow type microfluidic fuel cell where fuel and 
oxidant reactant streams are fed through opposing 
porous electrodes before they are then evacuated to the 
waste. The two streams are in contact in the middle of the 
device allowing the conduction of ionic current across the 
electrolyte.  
To limit the complexity of the device analysis, the nano-
pores are considered to be slots (i.e. the pore width << 
pore height) allowing for a 2-dimensional simulation in the 
electrode regions. The anode and cathode electrodes are 
separated by a distance    in the planar microchannel of 
height    and width   . Each electrode has a stream-wise 
dimension of   . The nano-pores are assumed to extend 
the full height of the electrode and are regularly spaced 
along the electrode. The nano-pores have a width of    
and a spacing of    as shown in the inset of Figure 1.  
For the bulk electrolyte advection through the pores to 
influence the electrode performance, the physical values 
of the dimensions    and    must be on the order of the 
Debye length. The remaining dimensions                 
are consistent with typical planar microchannel length 
scales. Table 1 summarizes the intended order of 
magnitude for the typical values of the dimensions. 
 
Table 1. Intended order of magnitude for key 
dimensions in porous electrode device. 
                  
                                    
 
To focus the study on the nano-porous electrode, generic 
fuel cell chemistry is used. The general overall device 
reaction can be written as: 
                                           (1) 
 
The overall reaction is split into two separate half 
reactions so that the exchange of electrons can be 
harnessed as electrical work. The two half reactions are 
therefore: 
    
        
   
 
     
                        (2.a) 
for fuel oxidation and 
     
              
       
  
  
               (2.b) 
for oxidant reduction. Here   is the number of electrons 
     liberated per mole of fuel. The term     is the 
stoichiometric coefficient for the cation     . Each half 
reaction is allowed to proceed in either direction as 
determined by the forward and reverse rate constants 
        , respectively. 
 
MATHEMATICAL MODEL 
Governing Equations 
The flux density of the reactant species and the 
electrolyte components is given by 
  iiiiiii FczcDcvN

   (3) 
where   indicates the specific species and          . 
The flux density is comprised of three components: 
advection, diffusion, and electromigration. The first term 
on the right hand side of equation (3) is advection; this 
term is important to consider for LFFCs which have a 
flowing liquid electrolyte. The second term is diffusion and 
the third term is electromigration, which is critical to the 
function of fuel cells because it is the primary motive force 
for the charged ions to migrate from one electrode to the 
other. It is obvious from equation (3) that the flux density 
of a charged ion is dependent on the electrolyte potential 
gradient.  
The concentration distributions are governed by the 
steady state Nernst-Plank equation which reduces to 
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Figure 2. (a) V-I plot and (b)  power density 
performance for hydrogen-oxygen fuel cell with 
nano-pore widths are equal to that of the Debye 
length         for various average electrolyte 
velocities.  
0 iN    (4) 
The potential distribution in the electrolyte is governed by 
Poisson’s equation 
 iiE czF)(     (5) 
Here, the right hand side of the equation is the charge 
density, which is the sum of the total charge for all of the 
species present. For neutral species such as fuel and 
oxidant, the ionic charge is zero so they do not contribute 
to the total charge density. 
The flow field is governed by the steady state Navier-
Stokes and continuity equations as [11]: 
pvvv  )()(

   (6a) 
0)(  v

    (6b) 
 
Finally, the rate at which the half reactions proceed is 
determined by the generalized-Frumkin-Butler-Volmer 
equation: 
           
        
  
      
        
    
  
         (7) 
where     is the change in potential between the 
electrode and reaction plane (i.e. across the Stern layer) 
            , and   and   are the ideal gas constant 
and absolute temperature.   is the kinetic symmetry 
factor and is assumed to be 0.5.  
 
In this study, the primary interest is the effect of the bulk 
electrolyte advection on ion transport and how this affects 
the electric double layer and electrode kinetics. 
Therefore, we assume that the neutral reactants of fuel 
and oxidant are present in excess and we do not consider 
their influence on electrode kinetics. Also, to maintain 
simplicity in this investigation we consider a simple acidic 
fuel cell with a binary electrolyte consisting of a cation 
     and an anion      of unit charge,         . 
Additionally, since the electrolyte is acidic, the cation is 
the working ion, being produced by fuel oxidation and 
consumed by oxidant reduction. The anion is chemically 
inert and does not participate in the electrode reactions. 
The full device is simulated numerically using custom in-
house software, the details of which are presented 
elsewhere [12]. 
 
RESULTS AND DISCUSSION 
Hydrogen and oxygen (H-O) are considered for the fuel 
and oxidant using the following stoichiometric reaction: 
      
    
   
    
 
                                         (8) 
where          are the oxidation and reduction rate 
constants, respectively. The H-O device operates using 
reactant streams mixed with a liquid electrolyte. Due to 
the solubility limits, the maximum concentration of 
hydrogen and oxygen in the electrolyte are around 10 mM 
and 2 mM, respectively. 
Figure 2 shows the V-I and power density performance 
data for a device with nano-pore widths that are equal to 
that of the Debye length        .  As the electrolyte flow 
rate increases from            to             the 
overall device performance is approximately doubled. The 
different electrolyte flow rates introduce different levels of 
an advection flux in the charged regions of the EDL. 
Since the EDL is a near wall phenomenon the fluid 
velocity near the wall is essentially zero and the advection 
flux within the EDL is typically negligible. However, by 
flowing the electrolyte through a nano-pore these 
advection fluxes generate a novel performance 
improvement by altering the balance between the 
diffusion and migration fluxes within the EDL. This change 
in EDL structure along the nano-porous electrode yields a 
higher total electrode current for an equivalent potential. 
As shown in Figure 2, the greater the electrolyte velocity 
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 4 Copyright © 2012 by ASME 
 
 
Figure 3. (a) V-I plot and (b)  power density 
performance for hydrogen-oxygen fuel cell with 
nano-pore widths 10 times that of Debye length 
          for various average electrolyte 
velocities.  
(i.e. the greater the advection flux) the greater the 
increase in device performance caused by the change in 
the EDL structure. Further study (not shown) revealed 
that the change in EDL structure is a shift in the 
distribution of positive and negative ion concentrations 
reducing the electrostatic restriction on the electrode 
kinetics, thus allowing them to proceed at rates nearer 
their spontaneous thermodynamic rates. These results 
suggest that a device capitalizing on this novel EDL 
advection effect, such as the nano-porous counter flow 
device presented here, could yield significantly greater 
performance with device efficiencies much closer to the 
thermodynamic maximum.  
The performance data for a device where the nano-pore 
widths are 10 times that of the Debye length           
is shown in Figure 3 in the form of a V-I and power 
density plot.  In this configuration the performance is no 
longer significantly altered by electrolyte velocity. There is 
a noticeable slight improvement with greater electrolyte 
advection but this is merely due to improved reactant 
transport to the porous electrodes. The device in figure 3 
has nano-pores that are significantly wider than the EDLs 
reducing/eliminating the overlap between the charged 
regions of the EDL and the electrolyte advection. 
Therefore, the electrolyte advection flux within the EDL 
has become negligible and no longer alters the EDL 
structure. It is also worth noting that the wider pore 
devices have significantly lower performance than the 
narrower pore devices. The wider pores have a lower 
effective surface area when viewed at the device level. 
This, in conjunction with the lack of the novel flowing 
electrolyte performance enhancement, yields reduced 
device performance. 
 
REFERENCES 
[1] Ferrigno, R., Stroock, A. D., Clark, T. D., Mayer, M., 
Whitesides, G. M., J. American Chemical Society 
2002, Vol. 124, 12930-12931. 
[2] Choban, E. R., Markoski, L. J., Wieckowski, A., Kenis, 
P. J. A., J. Power Sources 2004, Vol. 128, 54-60. 
[3] Sprague, I., Dutta, P., and Ha, S., J. New Materials 
for Electrochemical Systems 2010, Vol. 13(4), 305-
313. 
[4] Sprague, I. B., Dutta, P., Ha, S., Proc. IMechE Part A: 
J. Power and Energy 2009, Vol. 223, 799-808. 
[5] Jayashree, R. S., Gancs, L., Choban, E. R., Primak 
A., Natarajan, D., Markoski, L. J., Kenis, P. J. A., J. 
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[6] Jayashree, R. S., Egas, D., Spendelow, J. S., 
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[7] Kjeang, E., McKechnie, J., Sinton, D., Djilali, N., J. 
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[8] Kjeang, E. Michel, R., Harrington, D. A., Djilali, N. 
Sinton, D., J. American Chemical Society 2008, Vol. 
130, 4000-4006. 
[9]  M.-H. Chang, F. Chen, and N.-S. Fang, J. of Power 
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[10]  He, R., Chen, S. L., Yang, F., and Wu, B. L., Journal 
of Physical Chemistry B 2006, Vol. 110(7), 3262-3270. 
[11]  Lim, J., Whitcomb, J. D., Boyd, J. G., and Varghese, 
J., Computational Mechanics 2009, Vol. 43(4), 461-475. 
[12]  Sprague, I. B., and Dutta, P., Numerical Heat 
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Flow Through Nanoporous Electrodes in a Microfluidic Fuel Cell

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Flow Through Nanoporous Electrodes in a Microfluidic Fuel Cell

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