Dusty Gas Model DGM

Comparison between Plant (l) s actual output and simulated results with the dusty gas model (DGM) and the simplified models (A) and (B)... 

The dusty gas model (DGM) developed by Mason and collaborators also accounts for the viscous mechanisms in real porous systems [75], Within the framework of this model, permeability for the viscous flow is given by the following expression [75,76]... 

Studies with many types of porous media have shown that for the transport of a pure gas the Knudsen diffusion and viscous flow are additive (Present and DeBethune [52] and references therein). When more than one type of molecules is present at intermediate pressures there will also be momentum transfer from the light (fast) molecules to the heavy (slow) ones, which gives rise to non-selective mass transport. For the description of these combined mechanisms, sophisticated models have to be used for a proper description of mass transport, such as the model presented by Present and DeBethune or the Dusty Gas Model (DGM) [53], In the DGM the membrane is visualised as a collection of huge dust particles, held motionless in space. 

The constitutive equations of transport in porous media comprise both physical properties of components and pairs of components and simplifying assumptions about the geometrical characteristics of the porous medium. Two advanced effective-scale (i.e., space-averaged) models are commonly applied for description of combined bulk diffusion, Knudsen diffusion and permeation transport of multicomponent gas mixtures—Mean Transport-Pore Model (MTPM)—and Dusty Gas Model (DGM) cf. Mason and Malinauskas (1983), Schneider and Gelbin (1984), and Krishna and Wesseling (1997). The molar flux intensity of the z th component A) is the sum of the diffusion Nc- and permeation N contributions,... 

The dusty gas model (DGM) [21] is used most frequently to describe multi component transport in between the two limiting cases of Knudsen and molecular diffusion. This theory treats the porous media as one component in the gas mixture, consisting of giant molecules held fixed in space. The most important aspect of the theory is the statement that gas transport through porous media (or tubes) can be divided into three independent modes or mechanisms ... 

Several experimental techniques have been developed for the investigation of the mass transport in porous catalysts. Most of them have been employed to determine the effective diffusivities in binary gas mixtures and at isothermal conditions. In some investigations, the experimental data are treated with the more refined dusty gas model (DGM) and its modifications. The diffusion cell and gas chromatographic methods are the most widely used when investigating mass transport in porous catalysts and for the measurement of the effective diffusivities. These methods, with examples of their application in simple situations, are briefly outlined in the following discussion. A review on the methods for experimental evaluation of the effective diffusivity by Haynes [1] and a comprehensive description of the diffusion cell method by Park and Do [2] contain many useful details and additional information. 

Chapter 3 described a new model for transport through porous media, developed recently by Kerkhof [5] and called the binary friction model (BFM). It is of interest to see how this model can be applied to the description of available experiments and to compare the results with those of the dusty gas model (DGM). Kerkhof [5] took the experimental data of Evans et al. [6,7] for the permeation of He and Ar through a low-permeability porous graphite septum. The experimental set-up, similar to the Wicke-Kallenbach diffusion cell, is sketched in Figure 9.7. Of interest are the steady... 

At present two models are available for description of pore-transport of multicomponent gas mixtures the Mean Transport-Pore Model (MTPM)[4,5] and the Dusty Gas Model (DGM)[6,7]. Both models permit combination of multicomponent transport steps with other rate processes, which proceed simultaneously (catalytic reaction, gas-solid reaction, adsorption, etc). These models are based on the modified Maxwell-Stefan constitutive equation for multicomponent diffusion in pores. One of the experimentally performed transport processes, which can be used for evaluation of transport parameters, is diffusion of simple gases through porous particles packed in a chromatographic column. 

Adopting the dusty gas model(DGM) for the description of gas phase mass transfer and a Generalized Stefon-Maxwell(GSM) theory to quantify surface diffusion, a combined transport model has been applied. The tubular geometry membrane mass balance is given in equation (1). 

The mass transfer in the boundary layers can be described by a mass transfer coefficient. In the membrane phase, the diffusion of water vapor can be described by either of the four mechanisms, namely molecular/Knudsen diffusion model, or Poiseuille flow, or by the dusty gas model (DGM). Heat transfer coefficients are used to describe the heat transfer in the boundary layer on either side of the membrane. In the membrane heat transfer occurs through the vapor and by conduction. These aspects have been explained in detail in the following sections. 

Multicomponent diffusion in pores is described by the dusty-gas model (DGM) [38,44,46 8]. This model combines molecular diffusion, Knudsen diffusion, viscous flux, and surface diffusion. The DGM is suitable for any model of porous structure. It was developed by Mason et al. [42] and is based on the Maxwell-Stefan approach for dilute gases, itself an approximation of Boltzmann s equation. The diffusion model obtained is called the generalized Maxwell-Stefan model (GMS). Thermal diffusion, pressmn diffusion, and forced diffusion are all easily included in the GMS model. This model is based on the principle that in order to cause relative motion between individual species in a mixture, a driving force has to be exerted on each of the individual species. The driving force exerted on any particular species i is balanced by the friction this species experiences with all other species present in the mixture. Each of these friction contributions is considered to be proportional to the corresponding differences in the diffusion velocities. 

The Dusty Gas Model (DGM) is one of the most suitable models to describe transport through membranes [11]. It is derived for porous materials from the generalised Maxwell-Stefan equations for mass transport in multi-component mixtures [1,2,47]. The advantage of this model is that convective motion, momentum transfer as well as drag effects are directly incorporated in the equations (see also Section 9.2.4.2 and Fig. 9.12). Although this model is fundamentally more correct than a description in terms of the classical Pick model, DGM/Maxwell-Stefan models )deld implicit transport equations which are more difficult to solve and in many cases the explicit Pick t)q>e models give an adequate approximation. For binary mixtures the DGM model can be solved explicitly and the Fickian type of equations are obtained. Surface diffusion is... 

M.2 The Dusty Gas Model (DGM) (Mason and Malinauskas, 1983) is commonly used for describing diffusion in porous media. In this model the medium is modeled as giant molecules (dust) held motionless in space (u ugt = 0). Derive the DGM equations using the treatment given in Chapter 2 by taking the medium as the ( + l)th species in the mixture. Compare the results of your derivations with that in Mason and Malinauskas (1983). You may also refer to Wesselingh and Krishna (1990) for more information. 

For porous membranes the mass transport mechanisms that prevail depend mainly on the membrane s mean pore size [1.1, 1.3], and the size and type of the diffusing molecules. For mesoporous and macroporous membranes molecular and Knudsen diffusion, and convective flow are the prevailing means of transport [1.15, 1.16]. The description of transport in such membranes has either utilized a Fickian description of diffusion [1.16] or more elaborate Dusty Gas Model (DGM) approaches [1.17]. For microporous membranes the interaction between the diffusing molecules and the membrane pore surface is of great importance to determine the transport characteristics. The description of transport through such membranes has either utilized the Stefan-Maxwell formulation [1.18, 1.19, 1.20] or more involved molecular dynamics simulation techniques [1.21]. 

In the original BFM, as well as in the DFM and dusty gas model (DGM) discussed in the next section, the structure of the porous media is considered independent of the transport equations. The transport equations are first written with the pore-averaged fluxes N, and are cast per unit of pore surface area. The flux is then corrected to a flux per unit of membrane cross sectional area by multiplying the flux by a correction factor that includes the porosity A, and tortuosity factor [48, 49, 50] t... 

The dusty fluid model (DFM) shares some similarities with the BFM. It was derived based on the dusty gas model (DGM), which describes gas flow... 

In this context, the Dusty Gas Model (DGM) is conventionally used to describe gaseous molar fluxes through porous manbranes the most general form (again neglecting surface diffusion) is expressed as (Kast and Hohenthanner 2000)... 

The coupling of the ID SFR equations with the chemical processes in and on the catalytic plate is straightforward. AH models discussed in Section 2.3 can be coupled via the species mass fluxes at the boundary between fluid phase and catalytic plate (Karadeniz, 2014 Karadeniz et al., 2013). Even more sophisticated models for the description of mass transport and chemical reactions in porous media such as the dusty-gas model (DGM) and also energy balances can be implemented into the numerical simulation (Karadeniz, 2014). 

The permeance of porous stmctures is often expressed as / = a + (3p to represent a combination of Knudsen and viscous flow behavior. While such an expression may have little physical significance, it often describes experimental permeance data very well and can be used to analyze membrane defects and the dominant flow mechanism. The dusty gas model (DGM) (Mason, 1983) provides a fairly accurate and comprehensive treatment for the transport of mixtures in meso- and macroporous structures for aU flow regimes and mechanisms. Solutions of DGM equations, however, require finite-element methods (Benes et al., 1999). 

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