Diffusion medium

The side-by-side diffusion cell has also been calibrated for drug delivery mass transport studies using polymeric membranes [12], The mass transport coefficient, D/h, was evaluated with diffusion data for benzoic acid in aqueous solutions of polyethylene glycol 400 at 37°C. By varying the polyethylene glycol 400 content incrementally from 0 to 40%, the kinematic viscosity of the diffusion medium, saturation solubility for benzoic acid, and diffusivity of benzoic acid could be varied. The resulting mass transport coefficients, D/h, were correlated with the Sherwood number (Sh), Reynolds number (Re), and Schmidt number (Sc) according to the relationships... 

Since we don t usually know enough about pore structure and other matters to assess the relative importance of these modes, we fall back on the phenomenological description of the rate of diffusion in terms of Fick s (first) law. According to this, for steady-state diffusion in one dimension (coordinate x) of species A, the molar flux, NA, in, say, mol m-2 (cross-sectional area of diffusion medium) s-1, through a particle is... 

For example, if fhe DL is used on the side of fhe cell where fhe fuel or oxidant is in gas phase, then this part can be referred to as gas diffusion layer (GDL). When bofh fhe CL and the DL are mentioned as one component, then the name "diffusion electrode" is commonly used. Because the DL is of a porous nature, it has also been called "diffusion medium" (DM) or "porous transporf layer" (PTL). Sometimes the DL is also referred to as fhe component formed by an MPL and a backing layer. The MPL has also been called the "water management layer" (WML) because one of its main purposes is to improve the water removal inside the fuel cell. In this chapter, we will refer to these components as MPL and DL because these names are widely used in the fuel cell indusfry. 

C. Ji and V. Kumar. Fuel cells with hydrophobic diffusion medium. US Patent 2007087260 (2007). 

J. H. Nam and M. Kaviany. Effective diffusivity and water-saturation distribution in single- and two-layer PEMEC diffusion medium. International Journal of Heat Mass Transfer 46 (2003) 4595-4611. 

The last part of the polarization curve is dominated by mass-transfer limitations (i.e., concentration overpotential). These limitations arise from conditions wherein the necessary reactants (products) cannot reach (leave) the electrocatalytic site. Thus, for fuel cells, these limitations arise either from diffusive resistances that do not allow hydrogen and oxygen to reach the sites or from conductive resistances that do not allow protons or electrons to reach or leave the sites. For general models, a limiting current density can be used to describe the mass-transport limitations. For this review, the limiting current density is defined as the current density at which a reactant concentration becomes zero at the diffusion medium/catalyst layer interface. 

The fuel is fed into the anode flow field, moves through the diffusion medium, and reacts electro-... 

To determine the saturation for any of the models, the capillary pressure must be known at every position within a diffusion medium. Hence, the two-phase models must determine the gas and liquid pressure profiles. In typical two-phase flow in porous media, the movement of both liquid and gas is determined by Darcy s law for each phase and eq 47 relates the two pressures to each other. Many models utilize the capillary pressure functionality as the driving force for the liquid-water flow... 

Figure 9. Idealized schematic of the cathode cataiyst iayer (going from z = 0 to z = L) between the membrane and cathode diffusion medium showing the two main iength scales the agglomerate and the entire porous electrode. Gray, white, and black indicate membrane, gas, and electrocatalyst, respectively, and the gray region outside of the dotted line in the agglomerate represents an external film of membrane or water on top of the agglomerate.

A more sophisticated and more common treatment of the catalyst layers still models them as interfaces but incorporates kinetic expressions at the interfaces. Hence, it differs from the above approach in not using an overall polarization equation with the results, but using kinetic expressions directly in the simulations at the membrane/diffusion medium interfaces. This allows for the models to account for multidimensional effects, where the current density or potential changes 16,24,46-48,51,52,54,56,60-62,66,80,82,87,107,125 although... 

Figure 11. Tafel plot of flooded porous-electrode simulation results for the cathode at three different values of xp = 2.2nFIfQ 2 02, z=dbK. The z coordinate ranges from 0 (catalyst layer/membrane interface) to L (catalyst layer/diffusion medium interface), the dimensionless overpotential is defined as // = —o FIRT r]oRR, - ), and the ORR rate constant is defined as A = hFFq 2 (Reproduced with permission from ref 36. Copyright 1998 The Electrochemical Society, Inc.)...

These are typically cathode models that include the diffusion medium and perhaps a membrane water flux. Next are the models that treat all of the layers of the sandwich and are only... 

Figure 15. Simulation results showing membrane dehydration (a) and cathode flooding (b). (a) 1 as a function of membrane position (cathode on the left) for different current densities. (Reproduced with permission from ref 14. Copyright 1991 The Electrochemical Society, Inc.) (b) Dimensionless oxygen mole fraction as a function of cathode-diffusion-medium position and cathode overpotential. (Reproduced with permission from ref 120. Copyright 2000 The Electrochemical Society, Inc.)...

Figure 21. Temperature distribution in kelvin inside the cathode diffusion medium at a current density of 1.2 A/cm and with saturated feeds. (Reproduced with permission from ref 80. Copyright 2003 The Electrochemical Society, Inc.)...

Diffusion medium properties for the PEFC system were most recently reviewed by Mathias et al. The primary purpose of a diffusion medium or gas diffusion layer (GDL) is to provide lateral current collection from the catalyst layer to the current collecting lands as well as uniform gas distribution to the catalyst layer through diffusion. It must also facilitate the transport of water out of the catalyst layer. The latter function is usually fulfilled by adding a coating of hydrophobic polymer such as poly(tet-rafluoroethylene) (PTFE) to the GDL. The hydrophobic polymer allows the excess water in the cathode catalyst layer to be expelled from the cell by gas flow in the channels, thereby alleviating flooding. It is known that the electric conductivity of GDL is... 

The density of the water employed as diffusion medium was raised by the addition of urea in order that the gold suspension could be run under a column of water without undue disturbance so as to obtain a uniform level surface of separation. 

The mass conservation equation only relates concentration variation with flux, and hence cannot be used to solve for the concentration. To describe how the concentrations evolve with time in a nonuniform system, in addition to the mass balance equations, another equation describing how the flux is related to concentration is necessary. This equation is called the constitutive equation. In a binary system, if the phase (diffusion medium) is stable and isotropic, the diffusion equation is based on the constitutive equation of Pick s law ... 

If the diffusion medium is isotropic in terms of diffusion, meaning that diffusion coefficient does not depend on direction in the medium, it is called diffusion in an isotropic medium. Otherwise, it is referred to as diffusion in an anisotropic medium. Isotropic diffusion medium includes gas, liquid (such as aqueous solution and silicate melts), glass, and crystalline phases with isometric symmetry (such as spinel and garnet). Anisotropic diffusion medium includes crystalline phases with lower than isometric symmetry. That is, most minerals are diffu-sionally anisotropic. An isotropic medium in terms of diffusion may not be an isotropic medium in terms of other properties. For example, cubic crystals are not isotropic in terms of elastic properties. The diffusion equations that have been presented so far (Equations 3-7 to 3-10) are all for isotropic diffusion medium. 

For one-dimensional diffusion, if diffusion starts in the interior and has not reached either of the two ends yet, the diffusion medium is called an infinite medium. An infinite diffusion medium does not mean that we consider the whole universe as the diffusion medium. One example is the diffusion couple of only a few millimeters long (discussed later). In an infinite medium, there is no boundary, but one often specifies the values of C x= and C f=oo as constraints that must be satisfied by the solution. These constraints mean that the concentration must be finite as x approaches or +oo, and the concentrations at +oo or —CO must be the same as the respective initial concentrations. These obvious conditions often help in simplifying the solutions. 

If diffusion starts from one end (surface) and has not reached the other end yet in one-dimensional diffusion, the diffusion medium is called a semi-infinite medium (also called half-space). There is, hence, only one boundary, which is often defined to be at x = 0. This boundary condition usually takes the form of CU=o = g(t), (dC/dx) x=o=g f), or (dC/dx) x=o + aC x=o=g(t), where u is a constant. Similar to the case of infinite diffusion medium, one often also writes the condition C x=x, as a constraint. 

In experimental studies of diffusion, the diffusion-couple technique is often used. A diffusion couple consists of two halves of material each is initially uniform, but the two have different compositions. They are joined together and heated up. Diffusive flux across the interface tries to homogenize the couple. If the duration is not long, the concentrations at both ends would still be the same as the initial concentrations. Under such conditions, the diffusion medium may be treated as infinite and the diffusion problem can be solved using Boltzmann transformation. If the diffusion duration is long (this will be quantified later), the concentrations at the ends would be affected, and the diffusion medium must be treated as finite. Diffusion in such a finite medium cannot be solved by the Boltzmann method, but can be solved using methods such as separation of variables (Section 3.2.7) if the conditions at the two boundaries are known. Below, the concentrations at the two ends are assumed to be unaffected by diffusion. 

Note that neither initial nor boundary conditions have been applied yet. The above equation is the general solution for infinite and semi-infinite diffusion medium obtained from Boltzmann transformation. The parameters a and b can be determined by initial and boundary conditions as long as initial and boundary conditions are consistent with the assumption that C depends only on q (or ). Readers who are not familiar with the error function and related functions are encouraged to study Appendix 2 to gain a basic understanding. 

The diffusion distance concept is best defined for infinite and semi-infinite media diffusion problems. In these cases, C depends on x 4Dt), so if at time ti the concentration is Ci at Xi, then at time tz = 4fi the concentration is Ci at X2 = 2xi (because = Xi/(4Dfi) = 2 = 2/(4T>f2) ). This fact is often referred as the square root of time dependence. That is, the distance of penetration of a diffusing species is proportional to the square root of time. In other words, the concentration profile propagates into the diffusion medium according to square root of time. It can also be shown that the amount of diffusing substance entering the medium per unit area increases with square root of time. The square root dependence is often expressed as... 

Because the diffusion distance is proportional to the square root of time, instead of the first power of time, diffusion rate is a less well-defined concept. The rate of the diffusion front moving into the diffusion medium would be dx/dt, which is not a constant, but is proportional to Hence, there is no fixed... 

When the medium is finite, there will be two boundaries in the case of onedimensional diffusion. This finite one-dimensional diffusion medium will also be referred as plate sheet (bounded by two parallel planes) or slab. The standard method of solving for such a diffusion problem is to separate variables x and t when the boundary conditions are zero. This method is called separation of variables. As will be clear later, the method is applicable only when the boundary conditions are zero. 

More solutions in finite diffusion medium may be found in Appendix 3. 

If D is constant, an experimental diffusion profile can be fit to the analytical solution (such as an error function) to obtain D. If it depends on concentration and the functional dependence is known. Equation 3-9 can be solved numerically, and the numerical solution may be fit to obtain D (e.g., Zhang et al., 1991a Zhang and Behrens, 2000). However, if D depends on concentration but the functional dependence is not known a priori, other methods do not work, and Boltzmann transformation provides a powerful way (and the only way) to obtain D at every concentration along the diffusion profile if the diffusion medium is infinite or semi-infinite. Starting from Equation 3-58a, integrate the above from Po to 00, leading to... 

To use Equation 3-58d or 3-58e, it is necessary to know the interface position X = 0 (i.e., p = 0) because the value of the integration depends on the exact position of X = 0. For a semi-infinite diffusion medium with fixed interface, this is easy (x = 0 is the surface). However, for a diffusion couple, the location of the... 

Figure 3-14 Schematics of dividing the diffusion medium into N equally spaced divisions. Starting from the initial condition (concentration at every nodes at f = 0), C of the interior node at the next time step (f = At) can be calculated using the explicit method, whereas C at the two ends can be obtained from the boundary condition.

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