Diffusion permeability equations

Figure 10. Calculated and measured metal permeabilities for different microorganisms. Permeabilities have been calculated using data given in each of the original references (P. tricornutum, Zn, [4] T. pseudonana, Mn, [4] C. kesslerii, Zn, [90] C. kesslerii, Pb, [160] C. pyrenoidosa, Cd, [205] C. reinhardtii, Ag, Mn, [91] T. oceanica, T. pseudonana, Zn, [311] T. oceanica, E. huxleyi, Cd, [214] T. pseudonana, Mn, [92]). The grey circles correspond with observed permeabilities, calculated by dividing the observed internalisation flux by the concentration of free ion (P = /int/[M]). The range of limiting diffusive permeabilities, / ,)] "njrl and / ,)] "n.lx, as calculated by equations (55) and (57), are given by the black rectangle...

On the Energy of Activation. The temperature dependence of many processes, such as diffusion, permeability, and partition coefficients, has been represented often in terms of the Arrhenius plots (52). It would appear that ki in Equation 14 also could be treated in this manner by defining... 

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... 

Transport Properties. Important transmembrane transport parameters of the fibers are Lp, the hydraulic conductivity Pm, the diffusive permeability for a given solute o, the solute reflection coefficient and R, the solute rejection. These coefficients appear in the following equations, which are assumed to be valid at the steady state at each position Z along the fiber wall ... 

Myoglobin, cytochrome-C, inulin, and vitamin B-12 were the solutes studied in saline, calf serum, and BSA systems at 37 C and pH 7.4. Observed solute rejections were corrected to intrinsic values by using uniform-wall-flux boundary layer theory for the developing and fully-developed asymptotic regions. The Splegler-Kedem equation ( ) for rejection versus volume flow was used to calculate reflection coefficients and diffusive permeabilities for each solute. There was no significant difference between rejection parameters measured in saline and protein solutions. 

The presence of an unstirred layer which may adhere to a given cell membrane can be treated operationally as a barrier with its own permeability property in series with the actual membrane. Its importance in membrane transport processes depends essentially on the permeability of the membrane itself relative to that of the unstirred layer to the particular molecule being transported. Consequently, only molecules which permeate membranes at high rates are affected, since diffusion in the unstirred layer is quite rapid. Water transfer across human red cell membrane and those of most other cells and tissues studied falls within this category. Dainty [22] has given the following equation by which the apparent diffusion permeability coefficient may be corrected for the effect of an unstirred layer of thickness, 8 ... 

Permeability equations for diffusion in solids. In many cases the experimental data for diffusion of gases in solids are not given as diffusivities and solubilities but. as permeabilities, Pm, in m of solute gas A at STP (0°C and 1 atm press) diffusing per second per m cross-sectional area through a solid 1 m thick under a pressure difference of 1 atm pressure. This can be related to Pick s equation (6.5-2) as follows. 

If the condition. Equation 3.47 is not fulfilled, that is, diffusion permeability of phase 1 is high enough and its nuclei are competitive, the concentration preparation time becomes equal to the incubation time. Equation 3.46. A rigorous theory of nucleation in a concentration gradient and its influence on the incubation period and phase competition was developed by Gusak, Desre, and Hodaj in the series of papers [7, 33-40]. 

Separation of metal ions using SLMs has evoked considerable interest. Danesi studied the transport of americium (29) using CMPO as the extractant in a flat sheet SLM in a co-transport system and verified his permeability equations (Equations 12 and 13). When the americium concentration was low (10 M), the transport followed Equation 12 and when it was high (10 M), Equation 13 described the transport behavior. Their work (55) with two different thicknesses of PP membranes indicated that for a thin membrane (25 fim flat sheet), the permeation was aqueous diffusion controlled while with a thicker membrane (430 [im flat sheet), both aqueous and membrane diffusion were controlling factors. 

When an amperometric electrode is used as the transducer of a biosensor, there is a consumption of reaction products this is the major difference from a potentiometric electrode. The diffusion-reaction equations ((3) and (4)) still apply, assuming that the product concentration at the transducer-active layer interface is zero ([P] = 0). This hypothesis corresponds to the maximal sensitivity of the biosensor. The flow of product towards the transducer can be limited by mass transfer, either in the semi-permeable membrane, which separates the enzyme from the sample medium [139], or in the active enzymatic layer [140]. When the semi-permeable membrane is the limiting factor of a diffusion process, all of the product formed is... 

Let us now turn attention to situations in which the flux equations can be replaced by simpler limiting forms. Consider first the limiting case of dilute solutions where one species, present in considerable excess, is regarded as a solvent and the remaining species as solutes. This is the simplest Limiting case, since it does not involve any examination of the relative behavior of the permeability and the bulk and Knudsen diffusion coefficients. 

The disposable parameters in these equations are the permeability 0, the surface diffusion factor y, the tortuosity function . (a) (which also... 

At the opposite limit of bulk diffusion control and high permeability, all flux models are required to he consistent with the Stefan-Maxwell relations (8.3). Since only (n-1) of these are independent, they are insufficient to determine all the flux vectors, and they permit the problem to be formulated in closed form only when they can be supplemented by the stoichiometric relations (11.3). At this limit, therefore, attention must be restricted from the beginning to those simple pellet shapes for ich equations (11.3) have been justified. Furthermore, since the permeability tends to infininty, pressure gradients within the pellet tend to zero and... 

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... 

As a particular case of this result, it follows that the stoichiometric relations are always satisfied in a binary mixture at the limit of bulk diffusion control and Infinite permeability (at least to the extent that the dusty gas equations are valid), since then all the binary pair bulk diffusion coefficients are necessarily equal, as there is only one of them. This special case was discussed by Hite and Jackson [77], and the reasoning set out here is a straightforward generalization of their treatment. 

When bulk diffusion controls and the d Arcy permeability is large, corresponding to pores of large diameter, the flux relations for a binary mixture reduce to a limiting form given by equation (3.29) and its companion obtained by interchanging the suffixes 1 and 2, namely... 

In section 11.4 Che steady state material balance equations were cast in dimensionless form, therary itancifying a set of independent dimensionless groups which determine ice steady state behavior of the pellet. The same procedure can be applied to the dynamical equations and we will illustrate it by considering the case t f the reaction A - nB at the limit of bulk diffusion control and high permeability, as described by equations (12.29)-(12.31). 

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). 

Permeability P, can be expressed as the product of two terms. One, the diffusion coefficient, reflects the mobility of the individual molecules in the membrane material the other, the Henry s law sorption coefficient, reflects the number of molecules dissolved in the membrane material. Thus equation 9 can also be written as equation 10. 

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... 

When paint films are immersed in water or solutions of electrolytes they acquire a charge. The existence of this charge is based on the following evidence. In a junction between two solutions of potassium chloride, 0 -1 N and 0 01 N, there will be no diffusion potential, because the transport numbers of both the and the Cl" ions are almost 0-5. If the solutions are separated by a membrane equally permeable to both ions, there will still be no diffusion potential, but if the membrane is more permeable to one ion than to the other a diffusion potential will arise it can be calculated from the Nernst equation that when the membrane is permeable to only one ion, the potential will have the value of 56 mV. 

Furthermore, for calculating the effective coefficient of quasi-diffusion in a composite (D) with the corresponding limitation of the entire process of heterogeneous mass-exchange, equations reported in Section 5.1 may be used. The high kinetic permeability of cellosorbents for large organic ions are listed in Table 16. 

Theoretical aspects of mediation and electrocatalysis by polymer-coated electrodes have most recently been reviewed by Lyons.12 In order for electrochemistry of the solution species (substrate) to occur, it must either diffuse through the polymer film to the underlying electrode, or there must be some mechanism for electron transport across the film (Fig. 20). Depending on the relative rates of these processes, the mediated reaction can occur at the polymer/electrode interface (a), at the poly-mer/solution interface (b), or in a zone within the polymer film (c). The equations governing the reaction depend on its location,12 which is therefore an important issue. Studies of mediation also provide information on the rate and mechanism of electron transport in the film, and on its permeability. 

When a two- or higher-phase system is used with two or more phases permeable to the solute of interest and when interactions between the phases is possible, it would be necessary to apply the principle of local mass equilibrium [427] in order to derive a single effective diffusion coefficient that will be used in a one-equation model for the transport. Extensive justification of the principle of local thermdl equilibrium has been presented by Whitaker [425,432]. If the transport is in series rather than in parallel, assuming local equilibrium with equilibrium partition coefficients equal to unity, the effective diffusion coefficient is... 

For heterogeneous media composed of solvent and fibers, it was proposed to treat the fiber array as an effective medium, where the hydrodynamic drag is characterized by only one parameter, i.e., Darcy s permeability. This hydrodynamic parameter can be experimentally determined or estimated based upon the structural details of the network [297]. Using Brinkman s equation [49] to compute the drag on a sphere, and combining it with Einstein s equation relating the diffusion and friction coefficients, the following expression was obtained ... 

A semi-permeable membrane, which is unequally permeable to different components and thus may show a potential difference across the membrane. In case (1), a diffusion potential occurs only if there is a difference in mobility between cation and anion. In case (2), we have to deal with the biologically important Donnan equilibrium e.g., a cell membrane may be permeable to small inorganic ions but impermeable to ions derived from high-molecular-weight proteins, so that across the membrane an osmotic pressure occurs in addition to a Donnan potential. The values concerned can be approximately calculated from the equations derived by Donnan35. In case (3), an intermediate situation, there is a combined effect of diffusion and the Donnan potential, so that its calculation becomes uncertain. 

Palm et al. [578] derived a two-way flux equation which is equivalent to Eq. (7.13), and applied it to the permeability assessment of alfentanil and cimeti-dine, two drugs that may be transported by passive diffusion, in part, as charged species. We will discuss this apparent violation of the pH partition hypothesis (Section 7.7.7.1). 

The method preferred in our laboratory for determining the UWL permeability is based on the pH dependence of effective permeabilities of ionizable molecules [Eq. (7.52)]. Nonionizable molecules cannot be directly analyzed this way. However, an approximate method may be devised, based on the assumption that the UWL depends on the aqueous diffusivity of the molecule, and furthermore, that the diffusivity depends on the molecular weight of the molecule. The thickness of the unstirred water layer can be determined from ionizable molecules, and applied to nonionizable substances, using the (symmetric) relationship Pu = Daq/ 2/iaq. Fortunately, empirical methods for estimating values of Daq exist. From the Stokes-Einstein equation, applied to spherical molecules, diffusivity is expected to depend on the inverse square root of the molecular weight. A plot of log Daq versus log MW should be linear, with a slope of —0.5. Figure 7.37 shows such a log-log plot for 55 molecules, with measured diffusivities taken from several... 

Permeability of an FML is evaluated using the Water Vapor Transmission test.28 A sample of the membrane is placed on top of a small aluminum cup containing a small amount of water. The cup is then placed in a controlled humidity and temperature chamber. The humidity in the chamber is typically 20% relative humidity, while the humidity in the cup is 100%. Thus, a concentration gradient is set up across the membrane. Moisture diffuses through the membrane, and with time the liquid level in the cup is reduced. The rate at which moisture is moving through the membrane is measured. From that rate, the permeability of the membrane is calculated with the simple diffusion equation (Fick s first law). It is important to remember that even if a liner is installed correctly with no holes, penetrations, punctures, or defects, liquid will still diffuse through the membrane. 

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