Fluxes modeling

The second type of approach to flux modeling, the so-called "dusty gas model," is developed in Chapter 3. In view of its completely different physical basis it is remarkable that its predictions are in complete agreement with those of the capillary model. 

Proposed flux models for porous media invariably contain adjustable parameters whose values must be determined from suitably designed flow or diffusion measurements, and further measurements may be made to test the relative success of different models. This may involve extensive programs of experimentation, and the planning and interpretation of such work forms the topic of Chapter 10, However, there is in addition a relatively small number of experiments of historic importance which establish certain general features of flow and diffusion in porous media. These provide criteria which must be satisfied by any proposed flux model and are therefore of central importance in Che subject. They may be grouped into three classes. 

One flux model for a porous medium—the dusty gas model- has already been described in Chapter 3. Although it is perhaps the most important and generally useful model currently available, it has certain shortcomings, and other models have been devised in attempts to rectify these. However, before describing these, we will review certain general principles to which all reasonable flux models must conform. 

As a result of the discussion in Chapters 1 to 6, we are now in a position to formulate certain conditions which must be satisfied by any acceptable model for the gaseous phase fluxes in a porous medium. These are very useful, as It turns out that they are sufficiently restrictive to determine completely the formulation of certain problems, without the need to appeal to any particular flux model. All the following conditions refer to isothermal systems. 

The simplest and most commonly used flux model is provided by the dusty gas equations (3.17)-(3.19), All the conditions (i)-(iv) above are satisfied by these equations, and the three parameters K, , and intro-... 

Since the void fraction distribution is independently measurable, the only remaining adjustable parameters are the A, so when surface diffusion is negligible equations (8.23) provide a completely predictive flux model. Unfortunately the assumption that (a) is independent of a is unlikely to be realistic, since the proportion of dead end pores will usually increase rapidly with decreasing pore radius. 

Chapter 10. EXPERIMENTAL CHARACTERIZATION AND TESTING OF FLUX MODELS... 

The purpose of all flux models is to express the fluxes in a porous medium in terms of gradients in pressure, composition and temperature. Isothermal flux models are therefore all of the general form... 

Experimentation in relation to flux models is therefore of two kinds. ... 

In general, tests have tended to concentrate attention on the ability of a flux model to interpolate through the intermediate pressure range between Knudsen diffusion control and bulk diffusion control. What is also important, but seldom known at present, is whether a model predicts a composition dependence consistent with experiment for the matrix elements in equation (10.2). In multicomponent mixtures an enormous amount of experimental work would be needed to investigate this thoroughly, but it should be possible to supplement a systematic investigation of a flux model applied to binary systems with some limited experiments on particular multicomponent mixtures, as in the work of Hesse and Koder, and Remick and Geankoplia. Interpretation of such tests would be simplest and most direct if they were to be carried out with only small differences in composition between the two sides of the porous medium. Diffusion would then occur in a system of essentially uniform composition, so that flux measurements would provide values for the matrix elements in (10.2) at well-defined compositions. 

In the investigations described above, the measurements have been interpreted in terms of the parameters of a well-defined flux model. However, it is appropriate at this point to illustrate the dangers of an imprecise... 

A further excellent demonstration of the need to interpret experimental results in terms of a specific flux model is provided by the dynamic test... 

Having discussed at some length the formulation and testing of flux models for porous media, we will now review v at Is, perhaps, their most Important application - the formulation of material balances In porous catalyst pellets. 

Though the solution procedure sounds straightforward, if tedious, practice difficulty is encountered immediately because of the implicit nature of the available flux models. As we saw in Chapter 5 even the si lest of these, the dusty gas model, has solutions which are too cumbersc to be written down for more than three components, while the ternary sol tion itself is already very complicated. It is only for binary mixtures therefore, that the explicit formulation and solution of equations (11. Is practicable. In systems with more than two components, we rely on... 

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... 

Hugo s approach can be extended without difficulty to apply throughout the whole range of pore sizes, but to accomplish this a specific and complete flux model must be used. To be definite we will assume that the dusty gas model is adequate, but the same reasoning could be applied to certain other models if necessary. The relevant flux relations are now equations (5.4). Applied to the radial flux components In one of our three simple geometries they take the form... 

T-Jhile the stoichiometric relations have rendered the above problem tractable by permitting an explicit solution of the dusty gas model flux relations, it should be pointed out that they do not lead to equally radical simplifications with all flux models. In the case of the Feng and Stewart models [49- for example, Che total flux of species r is formed by in-... 

The nearest thing to a complete justification of equations (11.3)i for pellets of arbitrary shape, is an argument given by W. E. Stewart [74], which does not depend on any particular choice of flux relations. In Chapter 10 it was pointed out that all isothermal flux models must have the general form... 

A proper resolution of Che status of Che stoichiometric relations in the theory of steady states of catalyst pellets would be very desirable. Stewart s argument and the other fragmentary results presently available suggest they may always be satisfied for a single reaction when the boundary conditions correspond Co a uniform environment with no mass transfer resistance at the surface, regardless of the number of substances in Che mixture, the shape of the pellet, or the particular flux model used. However, this is no more than informed and perhaps wishful speculation. 

The fluxing model was iniiially proposed by librhstctn and DeCrescciue, and kjes lopibd %. CBwbd s d ajiii cidhi -b ctoff... 

Zhao and Bi (2001b) concluded that the drift-flux model with zero drift velocity and Co = 1.2 - 0.2 Pg/Pl agrees with the measured gas velocities for the three tested miniature channels. 

Ishii (1977) One-dimensional drift-flux model and constitutive equations for relative motion between phases in various two-phase regimes. AML Report ANL-77-47 Ide H, Matsumura H, Tanaka Y, Fukano T (1997) Flow patterns and frictional pressure drop in gas-liquid two-phase flow in vertical capUlary channels with rectangular cross section, Trans JSME Ser B 63 452-160... 

Without using the formalism of the DIFF, both Ambrose and Norr (1993) and Tieszen and Fagre (1993) have pointed out from their experimental data that the carbonate 5 has a constant spacing from the average diet. Since the conversion of protein, or non-protein, to bioapatite-carbonate is metabolically more complicated than, for example, carbohydrate to protein, a simple physiological explanation may not be obvious. However, the interpretation of the DIFF does becomes clear in the light of a very simple flux model in Section 3. 

The area required for processing A = Qo — QVJ, where Qo Q is the perrneate volumetric flow, can be estimated by using the approximation / = 0.33/initiai + 0.67/finai (Cheryan, Ultrafiltration Handbook, Technomic, Lancaster, Pa., 1986) and a suitable flux model. An appropriate model relating flux to crossflow, concentration, and pressure is then applied. Pressure profiles along the retentate channel are empirically correlated with flow for spacer-filled channels to obtain A = APfQ/AT. 

System sizing involves integration of Eq. (20-69) using a flux model to give Eq. (20-71), where Vp is the permeate volume and J is the average flux. Note the direct tradeoff between area and process time. Table 20-19 shows the concentration and diafiltration steps separated and processing time. 

By incorporating a flux model [Eq. (20-73)], solute concentration [Eq. (20-77)], and polarization model [Eq. (20-60)] into the area cost... 

Equation (20-80) requires a mass transfer coefficient k to calculate Cu, and a relation between protein concentration and osmotic pressure. Pure water flux obtained from a plot of flux versus pressure is used to calculate membrane resistance (t ically small). The LMH/psi slope is referred to as the NWP (normal water permeability). The membrane plus fouling resistances are determined after removing the reversible polarization layer through a buffer flush. To illustrate the components of the osmotic flux model. Fig. 20-63 shows flux versus TMP curves corresponding to just the membrane in buffer (Rfouimg = 0, = 0),... 

The book is organized into eight chapters. Chapter 1 describes the physicochemical needs of pharmaceutical research and development. Chapter 2 defines the flux model, based on Fick s laws of diffusion, in terms of solubility, permeability, and charge state (pH), and lays the foundation for the rest of the book. Chapter 3 covers the topic of ionization constants—how to measure pKa values accurately and quickly, and which methods to use. Bjerrum analysis is revealed as the secret weapon behind the most effective approaches. Chapter 4 discusses experimental... 

The Chexal-Lellouche model (a flow regime-independent drift flux model)... 

The Ohkawa-Lakey model (a drift flux model with empirically derived coefficients) The Dix model (a drift flux model devised for analyzing boiling water reactors (BWRs) at operating conditions)... 

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