Charge saturation model

In particular, from the analysis in the very low frequency region, the limiting capacitance Q can be evaluated (via the limiting resistance R ). Practical cases have given values of Cj, of about 20mFcm (Panero et al, 1989), i.e. values of the same order as those obtained by cyclic voltammetry, thus confirming the validity of the charge saturation model. 

More sophisticated approaches to describe double layer interactions have been developed more recently. Using cell models, the full Poisson-Boltzmann equation can be solved for ordered stmctures. The approach by Alexander et al shows how the effective colloidal particle charge saturates when the bare particle charge is increased [4o]. Using integral equation methods, the behaviour of the primitive model has been studied, in which all the interactions between the colloidal macro-ions and the small ions are addressed (see, for instance, [44, 45]). 

Figure 5. The critical absolute permeability necessary to sustain the stability of a static foam as a function of liquid saturation. Calculations are for the constant-charge electrostatic model.

According to Eq. (13.52), saturation charge is directly proportional to the square of the particle diameter and the external electric field. Particle charging depends also on the composition of the particle, which is taken into account by the relative dielectric constant e,. It is worth noticing that the field charging model should not be applied for small particles (dp < 0.5 pm). 

The modification by method 2 is more acceptable. Although several types of modifications have been reported, Abraham and Liszi [15] proposed one of the simplest and well-known modifications. Figure 2(b) shows the proposed one-layer model. In this model, an ion of radius r and charge ze is surrounded by a local solvent layer of thickness b — r) and dielectric constant ej, immersed in the bulk solvent of dielectric constant ),. The thickness (b — r) of the solvent layer is taken as the solvent radius, and its dielectric constant ej is supposed to become considerably lower than that of the bulk solvent owing to dielectric saturation. The electrostatic term of the ion solvation energy is then given by... 

P vapor pressure (PEM modeling atm) p capillary pressure (CCL modeling atm) p8 total gas pressure (CCL modeling atm) p liquid pressure (CCL modeling atm) q coulombic charge (C) q" saturated vapour pressure (CCL atm) q vapor pressure (CCL modeling atm)... 

There are not many models that do transients, mainly because of the computational cost and complexity. The models that do have mainly been discussed above. In terms of modeling, the equations use the time derivatives in the conservation equations (eqs 23 and 68) and there is still no accumulation of current or charging of the double layer that is, eq 27 still holds. The mass balance for liquid water requires that the saturation enter into the time derivative because it is the change in the water loading per unit time. However, this treatment is not necessarily rigorous because a water capacitance term should also be included,although it can be neglected as a first approximation. 

---