Dusty fluid model

Higler AP, Krishna R, Taylor R. Nonequilibrium modeling of reactive distillation a dusty fluid model for heterogeneously catalyzed processes. Ind Eng Chem Res 2000 39 1596-1607. 

Maxwell-Stefan (dusty gas) approach by taking the membrane to be the additional component in the mixture. When the model is extended to account for thermodynamic nonidealities (what may be considered to be a dusty fluid model) almost all membrane separation processes can be modeled systematically. Put another way, the Maxwell-Stefan approach is the most promising candidate for developing a generalized theory of separation processes (Lee et al., 1977 Krishna, 1987). 

The dusty fluid model as developed by Krishna and Wesselingh [8] is a modification of the dusty gas model so as to be able to model liquid phase diffusion in porous media. For a non-ideal mbcture we have... 

A schematic diagram of the unit cell for a vapor-Uquid-porous catalyst system is shown in Fig. 9.9. Each cell is modeled essentially using the NEQ model for heterogeneous systems described above. The bulk fluid phases are assumed to be completely mixed. Mass-transfer resistances are located in films near the vapor-liquid and liquid-solid interfaces, and the Maxwell-Stefan equations are used for calculation of the mass-transfer rates through each film. Thermodynamic equilibrium is assumed only at the vapor-liquid interface. Mass transfer inside the porous catalyst may be described with the dusty fluid model described above. 

The model of TMT is one of the few models solely targeted at predieting conductivity behaviour of a membrane, and in contrast to the model of SZG, is based on physical rather than purely empirical considerations [22]. It is in this vein that they invoke the dusty fluid model (DFM) to model transport in the membrane. Before considering the model of TMT we examine the baek-ground of the DFM and the binary friction model (BFM). 

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... 

A. Higler, R. Krishna, R. Taylor, Nonequilibrium Modeling of Reactive Distillation A Dusty Fluid Model for Heterogeneously Catalyzed Processes, Ind. Eng. Chem. Res., 2000, 39, 1596-1607. 

Grew, K.N. and Chiu, W.K.S. (2010) A dusty fluid model for predicting hydroxyl anion conductivity in alkaline anion exchange membranes. Journal of the Electrochemical Society, 157, B327-B337. 

All these different mechanisms of mass transport through a porous medium can be studied experimentally and theoretically through classical models (Darcy s law, Knudsen diffusion, molecular dynamics, Stefan-Maxwell equations, dusty-gas model etc.) which can be coupled or not with the interactions or even reactions between the solid structure and the fluid elements. Another method for the analysis of the species motion inside a porous structure can be based on the observation that the motion occurs as a result of two or more elementary evolutions that are randomly connected. This is the stochastic way for the analysis of species motion inside a porous body. Some examples that will be analysed here by the stochastic method are the result of the particularisations of the cases presented with the development of stochastic models in Sections 4.4 and 4.5. 

For laminar flow, the characteristic time of the fluid phase Tf can be deflned as the ratio between a characteristic velocity Uf and a characteristic dimension L. For example, in the case of channel flows confined within two parallel plates, L can be taken equal to the distance between the plates, whereas Uf can be the friction velocity. Another common choice is to base this calculation on the viscous scale, by dividing the kinematic viscosity of the fluid phase by the friction velocity squared. For turbulent flow, Tf is usually assumed to be the Kolmogorov time scale in the fluid phase. The dusty-gas model can be applied only when the particle relaxation time tends to zero (i.e. Stp 1). Under these conditions, Eq. (5.105) yields fluid flow. This typically happens when particles are very small and/or the continuous phase is highly viscous and/or the disperse-to-primary-phase density ratio is very small. The dusty-gas model assumes that there is only one particle velocity field, which is identical to that of the fluid. With this approach, preferential accumulation and segregation effects are clearly not predicted since particles are transported as scalars in the continuous phase. If the system is very dilute (one-way coupling), the properties of the continuous phase (i.e. density and viscosity) are assumed to be equal to those of the fluid. If the solid-particle concentration starts to have an influence on the fluid phase (two-way coupling), a modified density and viscosity for the continuous phase are generally introduced in Eq. (4.92). 

The dusty gas model is often used as the basis for the calculation of a catalyst effectiveness factor in chemical reactor analysis. The extension to non-ideal fluids noted above and its application in RD modeling has not been used as often, partly because of the somewhat greater uncertainty in the parameters that appear in the equations [1, 11]. 

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). 

Solving this flow model for the velocity the pressure is calculated from the ideal gas law. The temperature therein is obtained from the heat balance and the mixture density is estimated from the sum of the species densities. It is noted that the viscous velocity is normally computed from the pressure gradient by use of a phenomenologically derived constitutive correlation, known as Darcy s law, which is based on laminar shear flow theory [139]. Laminar shear flow theory assumes no slip condition at the solid wall, inducing viscous shear in the fluid. Knudsen diffusion and slip flow at the solid matrix separate the gas flow behavior from Darcy-type flow. Whenever the mean free path of the gas molecules approaches the dimensions of pore diameter, the individual gas molecules are in motion at the interface and contribute an additional flux. This phenomena is called slip flow. In slip flow, the layer of gas next to the surface is in motion with respect to the solid surface. Strictly, the Darcy s law is valid only when the flow regime is laminar and dominated by viscous forces. The theoretical foundation of the dusty gas model considers that the model is applied to a transition regime between Knudsen and continuum bulk diffusion. To estimate the combined flux, the model is based on the assumption that the combined flux can be expressed as a linear sum of the Knudsen flux and the convective flux due to laminar flow. 

If conditions (1) and (2) are met, but the interstices are much smaller than the mean free path of the molecules, then the dusty gas model of Mason and coworkers [23] can be used to calculate a mean particle size from permeametry measurements. There is a major experimental difficulty involved in using permeametry at very small particle sizes, however. The fluid may channel through fissures in the agglomerate, which, although large compared with the interstices, may well pass unnoticed to the eye. 

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