Through-plane

Symmetry operators leave the eleetronie Hamiltonian H invariant beeause the potential and kinetie energies are not ehanged if one applies sueh an operator R to the eoordinates and momenta of all the eleetrons in the system. Beeause symmetry operations involve refleetions through planes, rotations about axes, or inversions through points, the applieation of sueh an operation to a produet sueh as H / gives the produet of the operation applied to eaeh term in the original produet. Henee, one ean write ... 

The total rate of transfer of momentum through plane 1 2 ... 

A mass flow of fluid equal to the difference between the flows at planes 3 -4 and l -2 (equation 11.3) must therefore occur through plane 2-4, as it is assumed that there is uniformity over the width of the element. 

Since plane 2-4 lies outside the boundary layer, the fluid crossing this plane must have a velocity us in the X-direction. Because the fluid in the boundary layer is being retarded, there will be a smaller flow at plane 3-4 than at 1-2, and hence the flow through plane 2-4 is outwards, and fluid leaves the element of volume. 

The net rate of change of momentum in the X-direction on the element must be equal to the momentum added from outside, through plane 2-4, together with the net force acting on it. 

The heat transferred by thermal conduction into the element through plane, 1-3... 

Schematic diagram of the catalyst loading gradient through plane (left) and in plane (right).

Product Name Thickness (mm) Weight (g/m ) Bulk Density (g/cm ) Porosity (%) Tensile Strength (MPa) Stiffness (Taber) Through-Plane Air Permeability (sec/100 cc) In-Plane Air Permeability (sec/100 cc) Compressibility (%) Through- Plane Resistivity (mohm cm ) In-Plane Resistivity (mohm cm) Comments ... 

Product Thickness Weight Bulk Density Porosity Tensile Strength Stiffness Through-Plane Air Permeability In-Plane Air Permeability Compressibility Through- plane Resistivity In-Plane Resistivity... 

One of the main parameters that would improve the overall performance of a fuel cell is better mass transport of reactants through the diffusion layer toward the active catalyst zones. In order to quantify and characterize how well the gas mass transport is in a specific DL material and design, it is important to measure the in-plane and through-plane permeabilities. Most of the published permeability results report the viscous permeability... 

Although in-plane permeability is critical in order to understand in detail the transport mechanisms of fluids inside diffusion layers, it has not been as commonly used (and measured) as through-plane permeability. The following are a few examples of how in-plane permeability can be determined... 

Through-plane permeability is usually one of the most common parameters given by manufacturers for carbon fiber papers and carbon cloths, even though it is often not specified as through-plane permeability. It is important to note that commercial instruments, such as permeameters and Gurley method instruments, are used in the fuel cell industry to measure this permeability [197,218]. 

In order to determine the viscous and inert through-plane gas permeabilities of diffusion layers at varied compression pressures, Gostick et al. [212] designed a simple method in which a circular specimen was sandwiched between two plates that have orifices in the middle, aligned with the location of the material. Pressurized air entered the upper plate, flowed through the DL, and exited the lower plate. The pressure drop between the inlet and the outlet was recorded for at least ten different flow rates for each sample. The inert and viscous permeabilities were then determined by fitting the Forchheimer equation to the pressure drop versus flow rate data as explained earlier. 

Experimental apparatus to measure through-plane permeability. (Reprinted from V. Gurau... 

The pressure difference between the lower cylindrical compartment and the lower annular compartment was measured. To make sure that the flow through the sample material had only a through-plane component (z-direc-tion), the back-pressure valve was adjusted until the lower annular pressure was equal to the pressure experienced by the lower cylindrical compartment. The temperatures of the inlet and outlet lines, as well as the flow rate of the gas leaving the lower cylindrical compartment, were monitored. A wide... 

Other methods to study the through-plane permeabilities were presented by Chang et al. [183] and Williams et al. [90]. However, these methods only determined the viscous permeability coefficient with Darcy s law and did not take into account the inertial component of the permeability. 

The following subsection will briefly discuss the main methods used to measure in-plane and through-plane electrical conductivity for diffusion layer materials. This parameter is critical for optimal fuel cell performance. 

Similarly to in-plane conductivity, through-plane conductivity seems to be a linear function of the compressed thickness of the DL that is, conductivity increases linearly with thickness decrease. Nitta et al. [216] also observed that in-plane conductivity was larger than through-plane conductivity however, the difference was not as large as that found in previous studies [9]. 

M. Khandelwal and M. M. Mench. Direct measurement of through-plane thermal conductivity and contact resistance in fuel cell materials. Journal of Power Sources 161 (2006) 1106-1115. 

J. T. Gostick, M. W. Fowler, M. D. Pritzker, M. A. loannidis, and L. M. Behra. In-plane and through-plane gas permeability of carbon fiber electrode backing layers. Journal of Power Sources 162 (2006) 228-238. 

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