In a microfluidic channel, the relationship between the fluid velocity and the absolute pressure for an incompressible viscous liquid is given by the classical fluid dynamics theory and the well-known Navier-Stokes equation:

9V — — (P’] —

—— НІ»-V) v = — VI — I +u Av

9t { ’ (1)

Where V stands for the fluid velocity vector with components (u, v, w), each expressed for a set of Euler components (x, y, z, t), P is the absolute pressure, p is the relative density, and |i is the kinematic viscosity.

In the case of a microfluidic horizontal straight channel (x-direction), the flow is always laminar under low pressure drop (typically a few bar), leading thus to a unidirectional flow and a uniform absolute pressure in the cross-section. For a fixed pressure drop AP between the inlet and the outlet of the channel, Eq. 1 simplifies to:

9u 1 AP (9 2u 92u

91 p L ^ 9y 2 9z 2

Where L is the length of the microchannel. When the permanent flow is reached, the time derivative term becomes zero and Eq. 2 simplifies to:

-AP 92u 92u pL 9y2 9z2

Where p is the dynamic viscosity (10-3 Pa. s for water at 20°C), defined as the product of the dynamic viscosity and the relative density p. Due to the very large aspect ratio of the rectangular cross-section of the microchannel, a 2D approach is usually considered that leads to a pseudo infinite-plate flow (except in the borders). The directions along the length and height of the microchannel are indicated as x and y coordinates, respectively (see fig. 2). A typical parabolic rate profile is obtained for pressure driven flow:

AP h2

— у 2)

2.p. L 4

Where h is the height of the microchannel and y is defined as y=0 at the middle of the microchannel and y= ± h/2 at the upper and under walls. By considering a rectangular microchannel (with l the width of the microchannel) in Eq. 4, the flow rate, Q, in laminar regime, is deduced and is proportional to the applied pressure (Eq. 5):

AP. l.h3

12. p. L

In electrochemical laminar flow systems, the mass transport is achieved by both diffusion and convection transport. In the case of Y-shaped microchannel, the mixing between the two laminar streams occurs by transverse diffusion. Microscale devices are generally characterized by high Peclet number, Pe, (Pe = Uavh/ D, with Uav the average velocity of the flow, h the height of the microchannel and D the diffusion coefficient of the molecule). In

this condition, the transverse diffusion is much lower than the convection, and the diffusive mixing of the co-laminar streams is restricted to a thin interfacial width, Smix, in the center of the channel (Fig. 3) that grows as a function of the downstream position (x) and the flow rate, determined from Eq. 6 (Ismagilov et al., 2000):

<6>

Where D is the diffusion coefficient for ions of type i and Uav is the average flow velocity defined as:

AP. h2

12iyL

Fig. 3. Schematic of a laminar flow in a microchannel with the formation of the diffusion region during operation of a microfluidic BFC.

For fast electron transfer and in excess of supporting electrolyte, the kinetics of a simple electrochemical redox reaction is controlled by diffusion and convection. The concentration profiles of the chemical species involved in the reaction are determined by solving the convective diffusion equation (Eq. 8):

-c — + V(-DVc) + vVc + R = 0 dt ‘ ‘

Where ci is the concentration of species i, Di its diffusion coefficient, t the time, v the fluid velocity vector (given by Eq. 1) and Ri a term describing the rate of net generation or consumption of species i formed by homogeneous chemical reaction.

In the case of a microfluidic biofuel cell as described in this work, Eq. 8 can be simplified into a 2-dimensionnal cartesian steady state (Eq. 9):

I C2c d2c і dc

-D I + Cc — 1 + u(y)^ = 0

‘ y dx dy2 ) dx

The boundary conditions associated are usually: (i) c = c° (bulk concentration) at the inlet of the microchannel, (ii) c = 0 at the electrode surface and (iii) no flux at the other walls (no electrochemical reaction).

Those simulations were exploited in order to calculate the diffusive flux at the electrode, defined as:

(x) = D

electrode

And, therefore, the total current is expressed as:

Where n is the number of electron exchanged, and F is the Faraday constant.

The pumping power Wpumpmg, required to sustain steady laminar flow in the microchannel by the syringe pump, is estimated on laminar flow theory (Bazylak et al., 2005) as the pressure drop multiplied by the flow rate:

One can note that the contributions from inlet and outlet feed tubes are not included, because they are negligible.

The fuel utilization (FU) is estimated by the following Eq. 13, defined as the current output divided by the flux of reactant entering the channel (Bazylak et al., 2005):

I

’.F. C.Q

The fuel utilization is maximized for the lowest flow rate and decreases with flow rate. Typical fuel utilization for microfluidic fuel cell is ~ 1% (Hayes et al., 2008).