Permeability constant, equations

An equation (also referred to as the constant field equation, the Goldman-Hodgkin-Katz equation, and the GHK equation) which relates the membrane potential (Ai/r) to the individual permeabilities of the ions (and their concentrations) on both sides of the membrane. Thus,... 

Permeability is an essential property of the materials that constitute the membrane and is independent of membrane thickness [23], Additionally, permeability can be described as the product of the diffusion coefficient and the solubility constant (see Equation 10.3) and is temperature dependent. Permeability can then be represented by the following Arrhenius-type expression (see Section 5.6.1)... 

During the constant rate period shown in Figure 14.1, either the boundary layer mass or heat transport is rate controlling. The flow of liquid to the surface of the green body to keep it wet is governed by the permeability equation for the flow of liquid relative to the ceramic particles [8,9], written as Fick s second law, dC/dt = V(DfVC), for diffusion considering (1 — )[= e = volume fiaction of liquid] to be the... 

Pautz and Crocker (20) adapted Barkman and Davidson s internal cake permeability equation at a constant injection flow rate and derived... 

The overall ability of a core to carry flux also depends on its size and shape, and its cross-sectional area. This is described by a quantity called permeance. The basic relationship of permeance to permeability in a core is defined in Eq. (10.2), where P is the permeance, p is the permeability of the material, A is the cross-sectional area of the core, and Z is the mean length of the flux path in the core. This equation assumes uniform flux distribution in the core and constant permeability inside the core. It does not take into account the variations in the length of the flux path from the inside of the core to the outside. The reciprocal of permeance is reluctance... 

TABLE II. Values of the permeability ratios a and 3 estimated from the constant field equation fitted to the intracellular potassium data. 

To make progress, we consider constant permeabilities (a log r function is not available for heterogeneous reservoirs). This leads to the simpler equation... 

Modeling formation heterogeneities. Rock heterogeneities such as internal filter cake, or damaged zones, are easily modeled by allowing kr to vary with X. If so, the differential equation d2p(x)/dx2 = 0 no longer applies, as it is derived for constant permeabilities only. Instead, we must consider... 

In our studies, critical Deborah numbers are rather constant (around 1) for different relaxation times at constant permeability, whereas for constant fluid properties and different permeabilities the critical Deborah numbers vary between 1 and 2. This suggests that the longest relaxation time obtained from dynamic oscillation measurements for practical solutions may successfully be used in calculations indicating the onset of an excess pressure increase, but that the approximations of the stretching rate, as given in equation 2, is somewhat dubious. 

Incompressible Single Phase Flow In an incompressible flow without sources, and with constant permeability the pressure is a solution of Laplace s equation. 

The definition given in Equation D.9 also includes the value of/cM in terms of the permeability of free space, Mo- The unit for charge in the SI unit system becomes I [As] and is defined as the coulomb [Gj. Using Equation D.6, the proportionality constant in Equation D.4 becomes ... 

Hite s treatment is based on equations (5.18) and (5.19) which describe the dusty gas model at the limit of bulk diffusion control and high permeability. Since temperature Is assumed constant, partial pressures are proportional to concentrations, and it is convenient to replace p by cRT, when the flux equations become... 

Although microporous membranes are a topic of research interest, all current commercial gas separations are based on the fourth type of mechanism shown in Figure 36, namely diffusion through dense polymer films. Gas transport through dense polymer membranes is governed by equation 8 where is the flux of component /,andare the partial pressure of the component i on either side of the membrane, /is the membrane thickness, and is a constant called the membrane permeability, which is a measure of the membrane s ability to permeate gas. The ability of a membrane to separate two gases, i and is the ratio of their permeabilities,a, called the membrane selectivity (eq. 9). 

The permeability varies with temperature according to equation 12 where is a constant, E is the activation energy for permeation, E. is the gas constant, and Tis the absolute temperature. 

The temperature dependence of the permeability arises from the temperature dependencies of the diffusion coefficient and the solubility coefficient. Equations 13 and 14 express these dependencies where and are constants, is the activation energy for diffusion, and is the heat of solution... 

However, before proceeding with the description of simulation data, we would like to comment the theoretical background. Similarly to the previous example, in order to obtain the pair correlation function of matrix spheres we solve the common Ornstein-Zernike equation complemented by the PY closure. Next, we would like to consider the adsorption of a hard sphere fluid in a microporous environment provided by a disordered matrix of permeable species. The fluid to be adsorbed is considered at density pj = pj-Of. The equilibrium between an adsorbed fluid and its bulk counterpart (i.e., in the absence of the matrix) occurs at constant chemical potential. However, in the theoretical procedure we need to choose the value for the fluid density first, and calculate the chemical potential afterwards. The ROZ equations, (22) and (23), are applied to decribe the fluid-matrix and fluid-fluid correlations. These correlations are considered by using the PY closure, such that the ROZ equations take the Madden-Glandt form as in the previous example. The structural properties in terms of the pair correlation functions (the fluid-matrix function is of special interest for models with permeabihty) cannot represent the only issue to investigate. Moreover, to perform comparisons of the structure under different conditions we need to calculate the adsorption isotherms pf jSpf). The chemical potential of a... 

For highly permeable molecules it is useful to consider the flux ionization constant, pfCa - , which refers to the pH value where the resistance to transport across a permeation barrier is 50% due to the ABL and 50% due to the membrane [21]. The approximate hyperboUc log-log equation (which is accurate when Pq is at least 10 times greater than Pabl)... 

The gas permeability constants (P) are generally expressed by the amount of the gas at standard temperature and pressure normalized for the thickness, membrane area, time and differential pressure of gas as in the following equation ... 

The airflow equations presented above are based on the assumption that the soil is a spatially homogeneous porous medium with constant intrinsic permeability. However, in most sites, the vadose zone is heterogeneous. For this reason, design calculations are rarely based on previous hydraulic conductivity measurements. One of the objectives of preliminary field testing is to collect data for the reliable estimation of permeability in the contaminated zone. The field tests include measurements of air flow rates at the extraction well, which are combined with the vacuum monitoring data at several distances to obtain a more accurate estimation of air permeability at the particular site. 

Since, in general, lower permeability media exhibit higher capillary-pressure suction, we argue that it is more difficult to stabilize foam when the permeability is low. Indeed the concept of a critical capillary pressure for foam longevity can be translated into a critical permeability through use of the universal Leverett capillary-pressure J-function (.13) and, by way of example, the constant-charge model in Equation 2 for II ... 

---