Asymptotic penetration

When compression ceased (i.e. during relaxation), the tissue receded in the hole until an asymptotic penetration was attained after around 100 seconds (Fig. 7). 

The penetration theory holds for the region where t is much less than L2jD, the film theory for the region where t is much greater than L2/D. This comparison is shown in Fig. 8, which clearly shows that the film and penetration theories are asymptotes of the film-penetration model. 

Extraction is a process in which the solvent penetrates the dried herb and dissolves the soluble components of the plant material. These then diffuse out into the free solvent surrounding the plant material particles. This process eventually reaches a state of equilibrium (Theoretically, complete equilibrium is only achieved at infinite time and the approach to completion is an asymptotic curve Figure 12.4.) In practice, about 90% of available solids come into solution within 24 horn s for a typical leaf herb at ambient temperature, although harder material, such as some dried woody roots and barks, will take rather longer to achieve a satisfactory degree of extraction. 

These compounds are able to diffuse through both the membrane and aqueous pores and/or tight junctions. The global absorption rate constant is described by the sum of two absorption rate values which represent two different pathways the penetration into the lipoidal membrane, and across the aqueous pores. B, I , a, and o are constants that depend on the experimental technique used to obtain the absorption rate constants, P is the lipophilicity parameter, and km and kp are the asymptotic values for the membrane and the paracellular way, respectively. For compounds with a molecular weight higher than 250 Da the contribution of the aqueous pathway is negligible, collapsing the equation to the one hyperbola model already described for colon. 

For real molecules A and B the charge clouds have exponential tails so that there is always some overlap and the expansion (16) is an asymptotic series . Still, for the long range the multipole approximation to AE can be quite accurate, if properly truncated (for instance, after the smallest term). For shorter distances, the penetration between the molecular charge clouds becomes significant, the screening of the nuclei by the electrons becomes incomplete even for neutral molecules, and the power law for AE 5 is modified by contributions which increase exponentially with decreasing R. These penetration contributions we define as ... 

At the low-frequency part of the spectrum with an increase of the probe length the field asymptotically tends to that in a uniform medium with the formation resistivity. Minimum of curves of apparent conductivity, Oa/o ii is related with the fact that quadrature component of the field becomes equal to zero. For relatively shallow penetration of borehole filtrate into the formation (02/01 2) for a > 16 the influence of the invasion zone on the signal value for a two-coil probe does not exceed 25%. 

---