As **fuel cell** size decreases, the transport phenomena of the fuels and water changes in the fuel cell. In standard fuel cell designs, the movement of fuel and water is governed by volumetric effects, but surface effects become critical as dimensions shrink. A good rule of thumb is that millimeter-scale devices are small enough for surface effects to be significant, and surface effects will be dominant in the micron regimen. Some of the differences between macroscopic and microscopic systems include the following:

• Surface effects matter more than bulk effects

• Very small dead volumes

• Issues with bubbles

• No unwanted turbulent flow

The classification of microchannels varies in the literature, but a good guideline is shown in Table 1.

Classification |
Hydraulic Diameter Range |

Convectional | Dh > 3 mm |

Minichannel | 3 mm > Dh > 200 µm |

Microchannel | 200 µm > Dh > 10 µm |

**Table 1. Classification of Microchannels.**

Therefore, some of the parameters that can be ignored when modeling macro-scale fuel cells need to be included when modeling **micro fuel cells**. Some of the performance considerations with microdevices are minimal dead volume, low leakage, good flow control, and rapid diffusion. The branch of science that dictates fluid flow on a microscopic level is called microfluidics.

**Microfluidics**

The science of microfluidics primarily uses the Navier-Stokes equations for liquids and gases. The Navier-Stokes equations are used when gases are compressible, and liquids are not. The Navier-Stokes equations need to be modified if the fluid is assumed to be incompressible, the fluid is inviscid, or if the flow assumes a characteristic velocity. These assumptions are important because as the dimensions become smaller, the differences between gases and liquids become more apparent. The first difference is that liquids have interfaces, and there may be a definite boundary for liquids flowing in a channel – depending upon the miscibility of the liquids. In contrast, gases readily mix. The second difference becomes apparent when Navier-Stokes equations are analyzed for MEMS systems. A Knudsen number that is less than 0.01 indicates that the equations of the continuum theory should provide a good approximation, while a Knudsen number approaching unity means that the gas must be treated as a collection of particles rather than a continuum.

**Navier-Stokes Equation**

The Navier-Stokes equations describe the behavior of a fluid in terms of stress and strain. In fluids, the addition to conservation of momentum, there is also an equation derived from the principle of conservation of mass:

where ρ denotes the density of the fluid and u_{i} is a vector of fluid velocities whose i^{th} component is fluid velocity in direction i. The strain rate tensor:

The stress tensor can be thought of as a 3 x 3 matrix; therefore, it is written as:

The ij^{th} element of this matrix is the force per unit area in the direction i exerted on a surface element with normal in the j direction. The stress tensor is related to the strain tensor through:

In equation 4, the dot implies differentiation with respect to time, p is the pressure in the fluid, μ is the dynamic viscosity, and λ is a second viscosity coefficient. The equation of conservation can now be written as:

where F_{i}, represents body forces, while the stress tensor captures the internal stresses. Equation 5 is a statement of Newton’s second law, F = ma. Using Equations 4 in Equation 5:

Equation 6 is usually called the Navier-Stokes equation of motion and Equations 1 and 6 are called Navier-Stokes equations. These can be rewritten in vector form:

The Navier-Stokes equations for a viscous, compressible fluid are a system of four nonlinear partial differential equations. However, the system contains five unknown functions: pressure, density, and the three components of the velocity vector. To solve for the unknowns, the conservation of energy equation is usually added to the Navier-Stokes equations, which introduces one more equation and one more unknown function, the temperature, T. A final equation relating the ρ, p, and T then needs to be added to solve for the six unknowns.

**Bubbles and Particles**

Bubbles can influence the performance of micro and MEMS systems. Since the channel size is very small, bubbles can sometimes block entire channels, inhibiting flow, creating large void fractions, or introducing many other issues in microsystems. Small volumes of one fluid in another fluid have spherical shapes due to surface tension. If the liquids have different densities, the droplets will move upward or downward due to the buoyant forces acting upon them.

The buoyant force, F_{B}, on a spherical air bubble of radius r in a liquid density, is given by:

The force acting to hold a bubble in place on a surface is the interfacial force, F is:

where γ is the interfacial tension, and d is the diameter of the contact area of the bubble.

For bubbles in flow channels, the pressure drop across a liquid-gas interface and the pressure difference needed to move the bubbles are given by:

where r is the channel radius and γ is a frictional surface tension parameter.

Depending upon how a bubble is positioned in a microchannel, the pressure drop and pressure required to move the bubble can vary. If the bubble impedes the flow in a capillary, the pressure may be low. If the bubble ends up in a region with different curvatures, the pressure drop may be significant, and considerable pressures may be required to remove the bubble.

When considering the movement of a particle in a fluid, the friction coefficient is given by Stokes’ law:

Like bubbles, particles in microfluidic systems are important because they are of comparable size to flow channels. To prevent issues with particles, careful filtration of fluids and gases is required before introducing them into the fuel cell.

**Capillary Effects**

The surface tension force that draws liquid into a small flow channel or capillary is:

where Θ is the contact angle between the liquid and surface. For a vertical capillary, the gravitational force on the rising column of liquid is given by:

When these forces are made equal, the maximum rise in the height of a fluid in a capillary against gravity is:

Therefore, the height of the fluid column will significantly increase as the size of the channel is decreased. Capillary forces are very useful in microfluidics because fuel cells can be designed to take advantage of these forces. Very long channel lengths can be filled with fluid using this force alone -- as long as the capillary force is not opposed by gravity.

**Velocity in Microchannels**

There are two distinct regions of flow in a microchannel: the entrance and regular flow region. When the fluid or gas enters the channel, the flow (velocity) profile changes from flat to a more rounded and eventually to the characteristic parabolic shape. Once this occurs, it is in the fully developed region of flow, as shown in Figure 1. Figure 2 shows a 3D model of velocity in a micro flow field.

**Figure 1. Developing velocity profiles from the entrance region to the fully developed region in a microchannel.**

**Figure 2. 3-D Velocity field in microchannels.**

The parabolic profile is typical of laminar flow in channels and is caused by the existence of the boundary layer. When the fluid first enters the channels, the velocity profile will not yet be parabolic. Instead, this profile will develop over a distance called the entrance length. The length of the entrance region for a circular duct is given by:

If the entrance is well rounded, the velocity profile is nearly uniform. Boundary layers form at the entrance as the fluid enters. The fluid acts according to the continuity law, which says the fluid will slow down at the walls of the channel, while the fluid in the center of the wall will accelerate. There is an excess pressure drop across the entrance length due to the increased shear forces in the entrance boundary layers and the acceleration of the core.

In a circular pipe, the velocity distribution across the diameter of the pipe is given by:

**Conclusion**

When designing and modeling MEMS fuel cells, many properties need to be considered that are neglected in the macroscale fuel cell system models. Important micro fuel cell properties are surface effects, dead volumes, bubbles and the consideration of both gas and liquid phases. Also, the properties can differ significantly between 1 mm and 1 micron; therefore, the system must consider the necessary parameters. Although micro fuel cells have been investigated for several years, it appears that this science is still in its infancy based upon the current micro fuel cell designs in the literature. To make progress in micro fuel cells, mathematical modeling needs to be an integral part of the design process since most of the system variables cannot be measured. This is due to the small dimensions of the fuel cell and lack of available sensors that can be incorporated into the small dimensions without interfering with the fuel cell performance.

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