Representations and Calculations

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. 

Other procedures and calculation techniques have been developed for both stagewise and differential permeation, such as those presented S-T Hwang and K. Kammermeyer, but they are not pursued here, inasmuch as the analogy is to be made specific to vapor-liquid mass transfer unit operations. In this way, the conventions and techniques already developed for mass transfer operations can be more readily utilized. Also note that the symbols and terminology used for membrane permeation have evolved through the years and vary from one author to another. 

The preceding assumes that perfect mixing takes place that is, there are no changes in flow rate and composition across or perpendicular to the face of the membrane surfaces. Furthermore, the flow rates and compositions at the cell are those of the streams leaving the cell. 

Of special consideration is the investigation of the cell as a continuum, first with point withdrawal of the permeate, then in both concurrent and countercurrent flow for the permeate and the reject phases. For this treatment, differential permeation is the mode of attack. The differential 

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For anisometric particles (e.g., needle- or platelet-shaped particles) mathematical statistics may likewise be applied [1.10]. The two-dimensional logarithmic normal distribution of the length L and breadth B of the particles also allows the representation and calculation of the characteristic parameters and mean values. The eccentricity of the calculated standard deviation ellipse (Fig. 2) is a measure of the correlation between the length and breadth of the particle. By using more than two... 

A difficulty with whatever the juxtaposition or arrangement is the mathematical means for representation and calculation. We will therefore be predominantly concerned with the necessary derivations and their simplifications. Of prime importance is the separation that can be achieved. Also of interest is the necessary sizing of the membrane area. 

It may be added, however, that the representations and calculations above pertain to non-equilibrium behavior for the membrane permeation of the components of gaseous systems. The same sort of notation may be adapted to liquid systems. 

Figure 9.23. A. N NMR spectra of polyborazilene precursor for the production of BN. Upper solid state MAS spectrum, middle solid-state CP MAS spectrum, lower liquid-state spectrum in tetrahydrofuran. B. Schematic representations and calculated N chemical shifts of the various environments in hexagonal BN. C. Observed and simulated N MAS NMR spectrum of polyborazilene showing the fitted components with assignments according to the environments of Figure 9.23B. The fitted peaks from the BHN2 sites are shown by full lines, those from BN3 sites by broken lines. From Gervais et al. (2001), by permission of the American Chemical Society.

Problems of uncertainty and inaccuracy can be addressed by using statistical and stochastic methods that have been described before [29,30]. The fuzzy logic approach provides a mathematical framework for representation and calculation of inaccurate data in AI methods [31,32]. Fuzzy logic is a superset of conventional (Boolean) logic that has been extended to handle the values between exactly true and exactly false. 

Membrane separations are an accepted means for separating noncondensable gases that is, gases that ordinarily condense only under low-temperature or cryogenic conditions. The technology might be in wider use if (1) better and more selective membrane materials were available and (2) the necessary mathematical representations and calculations were better spelled-out for the separations attainable. In fact, the one sometimes depends on the other. Of particular interest are ways in which separations could be enhanced using relatively nonselective membranes. 

Minimum reflux, by definition, is that condition whereby a zone of an infinite number of stages exists immediately on each side of the feed location. In the McCabe-Thiele method of graphical representation and calculation, both operating lines intersect with the locus of the feed partitioning, all at a point on the fC-value plot (or equilibrium curve). It is therefore impossible to calculate away from this point of intersection, in either direction. 

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