Steady state permeability

This section is concerned with measurement techniques of the diffusivity and solubility from which the permeability can easily be calculated. In the following analysis we restrict ourselves to the measurement of constant values of D. Concentration- and position-dependent diffusivities are analyzed in Crank and Park (1968) and Crank (1975). Generally, the techniques are for permeability, steady-state and time lag techniques and for diffusivity, sorption and desorption kinetics and concentration-distance curves. For self-diffusivity in polymer melts the techniques are (Tirrell, 1984) nuclear magnetic resonance, neutron scattering, radioactive tracer, and infrared spectroscopy. 

The same device was used in section 11.7 when discussing steady states at the limit of bulk diffusion control and high permeability. 

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). 

Permeable Catalysts. Vol. I, The Theory of the Steady State, Clarendon Press, Oxford, 1975. 

The anode and cathode chambers are separated by a cation-permeable fluoropolymer-based membrane (see Membrane technology). Platinum-electroplated high surface area electrodes sold under the trade name of TySAR (Olin) (85,86) were used as the anode the cathode was formed from a two-layer HasteUoy (Cabot Corp.) C-22-mesh stmcture having a fine outer 60-mesh stmcture supported on a coarse inner mesh layer welded to a backplate. The cell voltage was 3.3 V at 8 kA/m, resulting ia a 40% current efficiency. The steady-state perchloric acid concentration was about 21% by weight. 

Because mass flow bins have stable flow patterns that mimic the shape of the bin, permeabihty values can be used to calculate critical, steady-state discharge rates from mass flow hoppers. Permeabihty values can also be used to calculate the time required for fine powders to settle in bins and silos. In general, permeabihty is affected by particle size and shape, ie, permeabihty decreases as particle size decreases and the better the fit between individual particles, the lower the permeabihty moisture content, ie, as moisture content increases, many materials tend to agglomerate which increases permeabihty and temperature, ie, because the permeabihty factor, K, is inversely proportional to the viscosity of the air or gas in the void spaces, heating causes the gas to become more viscous, making the sohd less permeable. 

Under conditions of nonlimiting interfacial kinetics the normalized steady-state current is governed primarily by the value of K y, which is the relative permeability of the solute in phase 2 compared to phase 1, rather than the actual value of or y. In contrast, the current time characteristics are found to be highly dependent on the individual K. and y values. Figure 16 illustrates the chronoamperometric behavior for K = 10, log(L) = —0.8 and for a fixed value of Kf.y = 2. It can be seen clearly from this plot that whereas the current-time behavior is sensitive to the value of Kg and y, in all cases the curves tend to be the same steady-state current in the long-time limit. This difference between the steady-state and chronoamperometric characteristics could, in principle, be exploited in determining the concentration and diffusion coefficient of a solute in a phase that is not in direct contact with the UME probe. 

In the second category, SECMIT has been used to probe the relative permeability of oxygen between water and DCE or NB, with no supporting electrolyte present in any phase. Under the conditions employed, direct voltammetric measurements in the organic phase would be impractical due to the high solution resistivity (DCE or NB) or limitations of the solvent window available (NB). Figure 24 shows the steady-state current for the... 

Thus, the rate of change for the cumulative mass of diffusant passing through a membrane per unit area, or the flux of diffusant, j, may be evaluated from the steady-state portion of the permeation profile of a drug, as shown in Eq. (3). If the donor concentration and the steady-state flux of diffusant are known, the permeability coefficient may be determined. 

The steady state separation factor (a) for the two gases (C02 and CH4) is defined as the ratio of their individual permeabilities. 

This equation teaches us that the total stead-state flux (total rate of permeation across a membrane in the steady state of permeation), dM/dt, is proportional to the involved area (A) and the concentration differential expressed across the membrane, AC. In an experiment, flux is the experimentally measured parameter while A and AC are fixed in value when setting up an experiment. The value of the permeability coefficient, Ptotai, is what is calculated upon completion of an experiment using Eq. (8). The permeability coefficient, besides having the specific attributes ascribed to it, is... 

In PAMPA measurements each well is usually a one-point-in-time (single-timepoint) sample. By contrast, in the conventional multitimepoint Caco-2 assay, the acceptor solution is frequently replaced with fresh buffer solution so that the solution in contact with the membrane contains no more than a few percent of the total sample concentration at any time. This condition can be called a physically maintained sink. Under pseudo-steady state (when a practically linear solute concentration gradient is established in the membrane phase see Chapter 2), lipophilic molecules will distribute into the cell monolayer in accordance with the effective membrane-buffer partition coefficient, even when the acceptor solution contains nearly zero sample concentration (due to the physical sink). If the physical sink is maintained indefinitely, then eventually, all of the sample will be depleted from both the donor and membrane compartments, as the flux approaches zero (Chapter 2). In conventional Caco-2 data analysis, a very simple equation [Eq. (7.10) or (7.11)] is used to calculate the permeability coefficient. But when combinatorial (i.e., lipophilic) compounds are screened, this equation is often invalid, since a considerable portion of the molecules partitions into the membrane phase during the multitimepoint measurements. 

In this section we want to discuss unsteady diffusion across a permeable membrane. In other words, we are interested in how concentration and flux change before reaching the steady state discussed in Section IV.B. The membrane is initially free of solute. At time zero, the concentrations on both sides of the membrane are increased, to C and c2. Equilibrium between the solution and the membrane interface is assumed therefore, the corresponding concentrations on the membrane surfaces are Kc, and Kc2. Fick s second law is still applicable ... 

Given the low permeability of the antioxidant across MDCK cell monolayers and its large membrane partition coefficient, efflux kinetic studies using drug-loaded cell monolayers cultured on plastic dishes could yield useful information when coupled with the following biophysical model. The steady-state flux of drug from the cell monolayer is equal to the appearance rate in the receiver solution ... 

The significance of the permeability depends upon the particular system at hand. If the hydrogel forms a barrier between a drug reservoir and a solution, the flux across the hydrogel membrane at steady state is simply... 

Figure 17 Permeability of uranine ( ), dextran 4.4K (O), and dextran 150K (A) through cross-linked poly(/V-isopropyl acrylamide-co-butyl methacrylate, 95 5 mol%) membrane. Error bars represent standard deviation in the slope of the curve of the receiver concentration of solute as a function of time at steady state. (From Ref. 37.)...

High pressure equipment has been designed to measure foam mobilities in porous rocks. Simultaneous flow of dense C02 and surfactant solution was established in core samples. The experimental condition of dense CO2 was above critical pressure but below critical temperature. Steady-state CC -foam mobility measurements were carried out with three core samples. Rock Creek sandstone was initially used to measure CO2-foam mobility. Thereafter, extensive further studies have been made with Baker dolomite and Berea sandstone to study the effect of rock permeability. 

In a reactive transport model, the domain of interest is divided into nodal blocks, as shown in Figure 2.11. Fluid enters the domain across one boundary, reacts with the medium, and discharges at another boundary. In many cases, reaction occurs along fronts that migrate through the medium until they either traverse it or assume a steady-state position (Lichtner, 1988). As noted by Lichtner (1988), models of this nature predict that reactions occur in the same sequence in space and time as they do in simple reaction path models. The reactive transport models, however, predict how the positions of reaction fronts migrate through time, provided that reliable input is available about flow rates, the permeability and dispersivity of the medium, and reaction rate constants. 

Problem P7.06.02 reproduces one result from the literature. There it is apparent that in some ranges of the parameters, effectiveness can be much greater than unity, and also that at low values of Thiele modulus several steady states are possible. When both external and internal adiabatic diffusion occur, moreover, other studies find that half a dozen or more steady states can exist. Those kinds of findings involve much computer work. A book by Aris (Mathematical Theory of Diffusion and Reaction in Permeable Catalysts,... 

As far as can be ascertained, no performance standards exist for this product. In the absence of such standards, the existing standards for automotive vehicles were used as guidelines. By using the most stringent standard, the SHED test, a petrol permeation rate of approximately 3.3 g/m2 for 24 h at 40°C can be estimated. With a single-fluorination treatment a pipe already exceeds this standard with a steady state permeation rate of 1.7 g/m2 per 24 h at 50°C. Since it is a known fact that permeability increases drastically with a rise in temperature, a permeability ofless than 0.17 g/m2 per 24h at 30°C is expected for a single fluorination treatment. 

This expression has been interpreted as the total resistance , being the sum of the diffusion (ro/Asi) and adsorption + internalisation (l/kKn) resistances [11,32] or a combination of permeabilities [19]. If the couple adsorption + internalisation is much more effective than diffusion (kKh > T>m/to), then r// 0 and we recover the steady-state maximum uptake flux for a spherical... 

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