Models Based on Thin Layer Approximation

Pressure gradients in the x and y directions do not depend on 2, hence the integration of the components of the equation of motion, subject to conditions (5.42), yields 

Integration of llie velocity components, given by liquation (5.43), with re.spect to gap-wise direction (i.e. z) yields the volumetric Dow rates per unit width m the. V and V directions as 

Wc now obtain the integral of the continuity equation for incompressible fluids with respect to the local gap height hr this flow domain 

Svibstitiition from Equation (5.44) for the integrals in Equation (5.47) gives 

---

The last equation demonstrates that the starting point for the solution of the problem is the calculation of ci(double layer (this makes low-frequency dielectric dispersion [LFDD] measurements a most valuable electrokinetic technique). Probably, the first theoretical treatment is the one due to Schwarz [61], who considered only surface diffusion of counterions (it is the so-called surface diffusion model). In fact, the model is inconsistent with any explanation of dielectric dispersion based on double-layer polarization. The generalization of the theory of diffuse atmosphere polarization to the case of alternating external fields and its application to the explanation of LFDD were first achieved by Dukhin and Shilov [20]. A full numerical approach to the LFDD in suspensions is due to DeLacey and White [60], and comparison with this numerical model allowed to show that the thin double-layer approximations [20,62,63] worked reasonably well in a wider than expected range of values of both and ku [64]. Figure 3.12 is an example of the calculation of As. From this it will be clear that (i) at low frequencies As can be very high and (ii) the relaxation of the dielectric constant takes place in the few-kHz frequency range, in accordance with Equations (3.56) and (3.57). 

In general. Equation 1.50 is nonlinear, as Rqrr and j, which appear in the source terms (1.45) and (1.46), exponentially depend on temperature. However, the CCL is thin, and the temperature variation along x is usually very small. To a good approximation, the CCL temperature in the source terms can be taken to be constant. This leads to a simple general expression for the heat flux produced in the CCL, which will be derived in the section Heat Flux from the Catalyst Layer in Chapter 4. This expression can be used in the modeling of application-relevant cells and stacks, in which quite significant in-plane temperature gradients may arise based on a nonuniform distribution of reactants. 

To date, it has been documented that ILs can be adsorbed onto various electrode surfaces. For example, Nanjundiah et al. found that several ILs used as electrolytes can induce double-layer capacitance phenomena on the surface of an Hg electrode and obtained the respective capacitance values for various ILs. Hyk and Stojek have also studied the IL thin layer on electrode surfaces and suggested that counterions substantially influence the distribution of IL. Kornyshev further discussed IL formations on electrode surfaces, suggesting that IL studies should be based on modern statistical mechanics of dense Coulomb systems or density-functional theory rather than classical electrochemical theories that hinge on a dilute-solution approximation. There are three conventional models that describe the charge distribution of an ion near a charged surface the Helmholtz model, the Gouy-Chapman model, and the Stern model. In the case of ILs, it remains controversial which model can best explain and lit the experimental data. 

---