Combined diffusion-viscous flow

In the absence of any concentration polarization, and Cfi are equal to Cg and respectively. The extent of concentration polarization and its effects on the solvent flux and solute transport for porous membranes and macrosolutes/proteins can be quite severe (see Section 6.3.3). This model is often termed the combined diffusion-viscous flow model (Merten, 1966), and it can be used in ultrafiltration (see Sections 6.3.3.2 and 7.2.1.3). The relations between this and other models, such as the finely porous model, are considered in Soltanieh and Gill (1981). 

From the solute flux expression (3.4.91c) in the combined diffusion-viscous flow model, one can develop an expression for S,n,e or Rn e- Consider relation (3.4.93b) ... 

The transport of a sub-critical Lennard-Jones fluid in a cylindrical mesopore is investigated here, using a combination of equilibrium and non-equilibrium as well as dual control volume grand canonical molecular dynamics methods. It is shown that all three techniques yield the same value of the transport coefficient for diffusely reflecting pore walls, even in the presence of viscous transport. It is also demonstrated that the classical Knudsen mechanism is not manifested, and that a combination of viscous flow and momentum exchange at the pore wall governs the transport over a wide range of densities. 

Finely Porous Model. In this model, solute and solvent permeate the membrane via pores which connect the high pressure and low pressure faces of the membrane. The finely porous model, which combines a viscous flow model eind a friction model (7, ), has been developed in detail and applied to RO data by Jonsson (9-12). The most recent work of Jonsson (12) treated several organic solutes including phenol and octanol, both of which exhibit solute preferential sorption. In his paper, Jonsson compared several models including that developed by Spiegler eind Kedem (13) (which is essentially an irreversible thermodynamics treatment), the finely porous model, the solution-diffusion Imperfection model (14), and a model developed by Pusch (15). Jonsson illustrated that the finely porous model is similar in form to the Spiegler-Kedem relationship. Both models fit the data equally well, although not with total accuracy. The Pusch model has a similar form and proves to be less accurate, while the solution-diffusion imperfection model is even less accurate. 

The three modes of transport, Knudsen, viscous, and continuum diffusion were described by Graham in 1863. A combination of viscous flow and Knudsen flow leads to a phenomenon called viscous slip. This was observed experimentally by Knudt and Warburg (1875). Another combination between viscous flow and... 

We have presented above the three basic equations (8.8-1 to 8.8-3) for the case where bulk diffusion-Knudsen diffusion-viscous flows are simultaneously operating. What we will do in this section is to combine them to obtain a form which is useful for analysis and subsequent computation as we shall show in Chapters 9 and 10. 

In the intermediate range between viscous flow and Knudsen flow, that is, 0.05 pore wall. As a result, the velocity of gas molecules at the wall surface is not zero. This mechanism - combining both viscous flow and Knudsen diffusion - is thus called slip flow. The slip effect is negligibly small when r>> X but becomes significant when r is close to X. A correction has to be applied to the viscous flow with a wall velocity to describe the permeation flux as... 

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. 

This model assumes the diffusive flows combine by the additivity of momentum transfer, whereas the diffusive and viscous flows combine by the additivity of the fluxes. To the knowledge of the authors there has never been given a sound argument for the latter assumption. It has been shown that the assumption may result in errors for certain situations [22]. Nonetheless, the model is widely used with reasonably satisfactory results for most situations. Temperature gradients (thermal diffusion) and external forces (forced diffusion) are also considered in the general version of the model. The incorporation of surface diffusion into a model of transport in a porous medium is quite straightforward, since the surface diffusion fluxes can be added to the diffusion fluxes in the gaseous phase. 

As concerns the modeling of surface diffusion, Ulhom and co-workers [51] worked out a quantitative expression for the calculation of the surface-flow contribution to permeation fluxes, but its general applicability is questionable. In a more recent study, Jaguste and Bhatia [52] describe the combination of surface flow and viscous flow in a 7-AI2O3 porous media. 

Equations 3.S.b-4 to 6 can also be combined to give a single equation containing only the total flux resulting from both diffusive and viscous flow mechanisms ... 

The rest of the book is dedicated to adsorption kinetics. We start with the detailed description of diffusion and adsorption in porous solids, and this is done in Chapter 7. Various simple devices used to measure diffusivity are presented, and the various modes of transport of molecules in porous media are described. The simplest transport is the Knudsen flow, where the transport is dictated by the collision between molecules and surfaces of the pore wall. Other transports are viscous flow, continuum diffusion and surface diffusion. The combination of these transports is possible for a given system, and this chapter will address this in some detail. 

It is worthwhile at this point to remind the reader that the above conclusion for Knudsen diffusion is valid as long as the pressure is low or the capillary size is very small. When the capillary size is larger or the pressure is higher, the viscous flow will become important and the flow will be resulted due to the combination of the Knudsen and viscous flow mechanisms. This will be discussed in Sections 7.5 and 7.6. 

We have presented the various constitutive flux equations for the general case of combined bulk-Knudsen and viscous flow in the last section. Now let us illustrate its application to the simple case of transient flow of a diffusing mixture in a capillary exposed to an infinite environment. 

We have shown the essential features of the time lag in Section 12.2 using the simple Knudsen diffusion as an example, and a direct method of obtaining the time lag in Section 12.3. The diffusion coefficient dealt with in the Frisch s method in Section 12.3 is concentration dependent. In this section we will deal with a case where the transport through the porous medium is a combination of the Knudsen diffusion and the viscous flow mechanism. We shall see below that this case will result in an apparent diffusion coefficient which is concentration dependent, and hence it is susceptible to the Frisch s analysis as outlined in the Section 12.3. This means that the results of equations (12.3-21) are directly applicable to this case. 

The viscous flow mechanism is important when the pressure of the system is reasonably high. When this is the case, the constitutive flux equation describes a combined transport of Knudsen diffusion and viscous flow as ... 

A quantitative answer to above questions may be given through the theoretical modeling of non-isobaric, non-isothermal single component gas phase adsorption. External heat and mass transfer, intrapai ticle mass transport through Knudsen diffusion, Fickian diffusion, sorbed phase diffusion and viscous flow as well as intraparticle heat conduction are accounted for. Fig. 1 presents the underlying assumption on the combination of the different mass transport mechanisms in the pore system. It is shown elsewhere that the assumption of instantaneous... 

Viscoelasticity illustrates materials that exhibit both viscous and elastic characteristics. Viscous materials tike honey resist shear flow and strain linearly with time when a stress is applied. Elastic materials strain instantaneously when stretched and just as quickly return to their original state once the stress is removed. Viscoelastic materials have elements of both of these properties and, as such, exhibit time-dependent strain. Viscoelasticity is the result of the diffusion of atoms or molecules inside an amorphous material. Rubber is highly elastic, but yet a viscous material. This property can be defined by the term viscoelasticity. Viscoelasticity is a combination of two separate mechanisms occurring at the same time in mbber. A spring represents the elastic portion, and a dashpot represents the viscous component (Figure 28.7). 

In Fig.la a lower slope ((j)=2.25) and higher Intercept (g=1.32) Is obtained for the low pressure region as compared to the slope (((>=2.60) and Intercept (g=1.09) for high pressure region. Since from Eq(13), a value of g close to 1 implies adherence to the solution-diffusion model. It appears that at low pressures where g = 1.32>1.0, a combined model of viscous and diffusive flow Is operative. This correlates with previous SEM studies In our laboratory (unpublished), where mlcro-pln holes were postulated to exist In the skin. The presence of such m-LcAO-p-Ln hoZ 6 In the surface can be used to explain the high g-value. Above 10 atm, the DDS-990 membrane Is compressed or compacted and the mlcro-pln holes filled. Thus g = l.O l.O implies adherence to the solution-diffusion model. 

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