Membrane calculations

Using these equations, Lowe and Walmsley [48] have calculated the dissociation constants for sugar binding at the extracellular surface of the membrane (K s = b a in Fig. 2) and at the cytoplasmic surface (K. = elf = bid) x [dgich]) from the estimated rate constants for carrier re-orientation and the measured Michaelis constants. The dissociation constant for binding at the extracellular surface of the membrane, calculated in this way, is approximately lOmM and is largely unaffec-... 

Strandberg E, Esteban-Martin S, Salgado J, Ulrich AS (2009) Orientation and dynamics of peptides in membranes calculated from 2H-NMR data. Biophys J 96 3223-3232... 

Below we present a well-known calculation of membrane potential based on the classical Teorell-Meyer-Sievers (TMS) membrane model [2], [3]. The essence of this model is in treating the ion-selective membrane as a homogeneous layer of electrolyte solution with constant fixed charge density and with local ionic equilibrium at the membrane/solution interfaces. In spite of the obvious idealization involved in the first assumption the TMS model often yields useful results and represents in fact the main tool for practical membrane calculations. We shall return to TMS once again in 4.4 when discussing the electric current effects upon membrane selectivity. In the case of our present interest, the simplest TMS model of membrane potential for a 1,2 valent electrolyte reads... 

Figure 5.5 Water permeability as a function of sodium chloride permeability for membranes made from cellulose acetate of various degrees of acetylation. The expected rejection coefficients for these membranes, calculated for dilute salt solutions using Equation (5.6),...

Figure 11.18 Flux through a facilitated transport membrane calculated using Equation (11.20) ([A] 0 and DRA[R](m)tot/f 1)...

PPO membranes, calculated from gas adsorption/desorption Isotherms. 

Cumulative pore volume and pore size distribution for PPO membranes, calculated from the thermograms. (The numbers 0.15 and 0.20 Indicate the casting thickness (nm) during preparation of the membranes). 

Predictability of Membrane Performance. New membranes were placed in the cells as before and an experiment was done with a reference solute (NaCl). With the use of the transport equations (eq. (2), (3), (6), and (7)) and the correlation of k with A, eq. (14), ( AM/X6)jjg(.2 was determined. The appropriate (AAff/RT) j s were used from Table VI to determine C aC] each membrane. Calculations of (PR) and f for several salts at various concentrations and pressures were made and compared to the experimental results with the new membranes and these are summarized in Figure 4 and Tables VIII, and IX. The satisfactory agreement between predicted and experimental results obtained indicates the practical utility of the correlations and parameters generated in this work. 

The function Vp/(1-Vp), where Vp is the volume fraction of polymer in a water swollen material, is plotted as the abscissa in Figure 2. The denominator of this term is therefore the volume fraction of water in the membrane, calculated from sorption results. This function was developed by Yasuda and co-workers to treat diffusion in various hydrophilic polymers (12). Their equation ... 

Figure 9.21 The effect of agent concentration on the uranium flux through a coupled transport membrane calculated from Equations 13 and 24. The measured flux data are shown for comparison.17 (Membrane Celgard 2400/Alamine 336 dissolved in Aromatic 150. Feed 0.2% Uranium, pH 1.0. Product pH 4.5).

Fig. 1 Electron density profile for a polymersome membrane calculated from SAXS tmd ceirtoon to schematize the conformation of the hydro- phobic and hydrophihc blocks. Seeding relations for the hydrophobic membrane, t, the hydrophilic corona thickness, d, and the area per molecule, a, as...

At T = 25°C, z = 1, we have qo = 282 A. If we put a = 2 A, = 3, we have y = 10 °. In this case the conductance of the BLM would be much below the values given above. Apparently we have failed to take into account a number of factors reducing the hydrophobic barrier height, viz., small membrane thickness, ion pair generation, and ion hydration. Consider these effects one by one. The finite membrane thickness can be accounted for if we allow for repulsive forces acting on the ion near the membrane/water interface. At the center of the membrane calculation shows the energy to go down by... 

In the geometric representation of the composition of the membrane, the volume fraction of each component in each layer is described quantitatively by corresponding geometric parameters. In the electrochemical part of the model, each layer is treated as a set of two resistors and all sets, whose number equals the number of layers, are arranged in series forming an equivalent electrical circuit. Summation of the resistances of the layers expressed with appropriate equations leads to the final formula on specific conductivity of the membrane. Calculations based on the model require measurements of the conductivity of the membrane in contact with electrolyte solutions of different concentration. 

Fig. 4. Composition of two-component PP-g-PAA membranes calculated according to electrochemical model. Volume fraction of PAA in dry membrane A - 26%, B - 36%, C - 48%...

Table 2, Composition of PP-g-PAA membranes calculated according to the three-layer model. 

Fig. 9 Water content profiles in the membrane, calculated in the hydraulic permeation model, at various fuel-cell current densities. A typical value of the parameter / that determines the onset of membrane dehydration near the anode was estimated as / 5-10Acm for Nafion 117 [11,16]...

Table 5.7. Average pore size and geometric standard deviation for various PES UF membranes calculated from solute separation data and from AFM images...

Table 3. Pore diameter of membrane calculated from measured separation factors of helium and each gas and the Hard Sphere Model (angstroms)...

The single-stage membrane unit becomes equivalent to a so-called flash vaporization. The flash vaporization calculation itself is straightforward, with the vapor and liquid phases assumed at equilibrium, and is presented in a number of references." " The limits correspond to the dew-point and bubble-point calculations for vapor-liquid equilibrium, which are special or limiting cases for the flash vaporization calculation. It is the object, therefore, to adapt the membrane calculation to the techniques for the flash vaporization calculation and thereby take advantage of the relative simplicity of the latter. 

And so on, all up and down the line, whereby a condition of essentially constant permeate rates may be maintained from stage to stage and, at the same time, essentially constant reject rates. This simplification is akin to constant molal overflow as assumed in distillation calculations and makes multistage membrane calculations manageable. 

The same is true in utilizing membrane calculations. As indicated by S. Y. Lee and B. S. Minhas, the observed permeabilities for the components of a mixture may be markedly less than that measured for the individual pure components. This indicates that there is a role for efficiency ratings, which may be stagewise or pointwise (as in the case of differential permeation) or an overall figure may be used. 

Consider, therefore, Figure 4.6, which denotes a graphical membrane calculation based on the McCabe-Thiele method for distillation. The ordinate, y here, denotes the composition of the permeate phase(s) V. The abscissa, x here, denotes the composition of the reject phase(s) L. Constant values of V and L are assumed throughout, equivalent to a condition of constant molal overflow. To represent the equations, a continuum is... 

A distinction is made, however, in that, in distillation, the temperature and phase equilibrium compositions vary. For membrane calculations, on the other hand, the relation between the reject and permeate... 

As a footnote, considerable simplification in the mathematical separation representations could result by assuming that the respective molar flow rates remain constant throughout the membrane unit. Such is the practice in distillation calculations, where there is mass transfer in both directions. The assumption is similarly made in absorber or stripper calculations, where only one key component is involved. This condition, called constant molal overflow in distillation and absorber and stripper derivations and calculations, may also be accommodated in the case of multistage or cascade membrane calculations, as derived and utilized... 

FIGURE 13. Temperature dependence of log E for MBM-14 membrane. Calculated curves 1 to 5 are compared with experimental data. 

F. Pastushenko, Yu. A. Chizmadzhev, and V. B. Arakelyan, Electric breakdown of bilayer lipid membranes. Calculation of the membrane lifetime in the steady-state diffusion approximation, Bioelectrochem. Bioenerget. , 6 53 (1979). 

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