Viscous and Knudsen Flows

To distinguish between the viscous and Knudsen gas-phase flows (Section 5.8.2), the ratio between the mean free path, X, and the characteristic length of the flow geometry, L, is defined as 

A viscous flow takes place when the effects of viscosity become significant. This type of flow can be categorized as either a laminar flow or a turbulent flow [72], A laminar flow is one with no considerable mixing of neighboring fluid particles, apart from molecular motion. In a turbulent flow, the quantities which typify the flow exhibit a random variation with the time and space coordinates. The quantity used to predict the type of flow regime is the Reynolds number (Re), which is a dimensionless parameter defined as [72] 

L is the characteristic length, for example, pipe diameter, that is, the length of the flow field v is the flow velocity r is the dynamic viscosity 

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Relative contributions of viscous and Knudsen flow some data As will be discussed in Section 9.3, small contributions of viscous flow to the total flow in the transition region can have a considerable effect on the selectivity in separations. Therefore some typical data are given in Table 9.2. for N2 as a reference gas. Note that for light gases (H2) the contribution of the viscous flow differs considerably from that given in Table 9.2. 

Permeation data of macroporous supports and mesoporous layers for N2 at 20°C and an average pressure p = 1 bar in the tramsition region of viscous to Knudsen flow. The fraction of the viscous flow (b) to the total flow is given by Fr, the remainder is the Knudsen contribution (a)... 

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... 

Transition between the Viscous Flow and Knudsen Flow... 

In the last sections, you have learnt about the basic analysis of bulk flow, bulk flow and Knudsen flow using the Stefan-Maxwell approach. Very often when we deal with diffusion and adsorption system, the total pressure changes with time as well as with distance within a particle due to either the nonequimolar diffusion or loss of mass from the gas phase as a result of adsorption onto the surface of the particle. When such situations happen, there will be an additional mechanism for mass transfer the viscous flow. This section will deal with the general case where bulk diffusion, Knudsen diffusion and viscous flow occur simultaneously within a porous medium (Jackson, 1977). 

Transition between the viscous flow and Knudsen flow 380... 

Types of membranes. Early membranes were limited in their use because of low-selectivities in separating two gases and quite low permeation fluxes. This low-flux problem was due to the fact that the membranes had to be relatively thick (1 mil or 1/1000 of an inch or greater) in order to avoid tiny holes which reduced the separation by allowing viscous or Knudsen flow of the feed. Development of silicone polymers (1 mil thickness) increased the permeability by factors of 10 to 20 or so. 

In other words, the pore radius should be larger than 15(X) or even 10,(X)0 nm (1.5 or 10 /im) in order for viscous flow to be effective. These pores are extremely large and can be regarded as defective pinholes on the membranes that may arise when the membranes arc not carefully prepared. Therefore, the gas-phase flow mechanisms in the ordinary gas separation membranes are restricted to either the slip or Knudsen flow mechanisms. Looking into the flux equations (Equations 6.95 and 6.94) applicable to the slip and Knudsen flow mechanisms, both and are proportional to p2 P3 Therefore, if the latter flow mechanisms govern the total flux (0,). the permeability Aa deflned by Equation 6.114 should become independent of the operating pressure. On the other hand, according to the flux equation applicable to surface flow mecha-... 

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... 

FIGt 22-48 Transport mechanisms for separation membranes a) Viscous flow, used in UF and MF. No separation achieved in RO, NF, ED, GAS, or PY (h) Knudsen flow used in some gas membranes. Pore diameter < mean free path, (c) Ultramicroporoiis membrane—precise pore diameter used in gas separation, (d) Solution-diffusion used in gas, RO, PY Molecule dissolves in the membrane and diffuses through. Not shown Electro-dialysis membranes and metallic membranes for hydrogen. 

Knudsen s result for free molecular flow in a tube is given by Eq. (73). With/ = 1, Knudsen s result and the second term in Eq. (84) differ only by a numerical factor. In Knudsen s result, the numerical factor is 2/3 in Eq. (84), the corresponding factor is n/8. Thus, except for a modest difference in the numerical factor, the slip term in Eq. (84) is the Knudsen free molecular flow term, and transition flow in a tube appears as a mixture of free molecular flow and viscous flow. That is, the total flow behaves approximately as a sum of two parallel flow mechanisms. 

Three types of flow are mainly encountered in vacuum technology viscous or continuous flow, molecular flow and - at the transition between these two - the so-called Knudsen flow. 

The transitional range between viscous flow and molecular flow is known as Knudsen flow. It is prevalent in the medium vacuum range = d. 

Flow in the range between viscous and molecular types (sometimes termed Knudsen flow) is also mentioned but quantified only for tubes. It is, however, an additional check for vacuum technologists who tend only to look at the limiting cases to obtain upper and lower values for parameters. 

The numerical value of the conductance of a component in a vacuum system depends on the type of flow in the system. Gas flow in simple, model systems (e.g. tubes of constant circular cross-section, orifices, apertures) was considered for viscous flow (Examples 2.6-2.8) and molecular flow (Examples 2.9-2.11). The chapter concluded with two illustrations (Examples 2.13, 2.14) of Knudsen (intermediate) flow through a tube. 

There are three basic types of gas flow turbulent, viscous, and molecular. The type of flow passing through a given system is dependent on both the mean free path (MFP) of the molecule(s) and the diameter of the container (tube) through which they are flowing. A useful formula when talking about MFP is the Knudsen number (Kn), defined in Eq. (7.6). 

The difference between the DGM and BFM is in the description of molecule-wall collisions. In contrast to the DGM, the BFM does not separate viscous flow and Knudsen diffusion. According to the BFM they are the limiting forms of the same phenomena, described by the second term in the right-hand side of Equation 3.24. For large Knudsen numbers, that is for small pressures and/or small pore sizes,... 

Schofield showed in his model Equation 19.30 (or Equation 19.23) that Toc PAP for viscous flux and /oc AP for Knudsen flux, and obtained a correlation Jm = aP AP that approximates the Knudsen-Poiseuille transition of DGM. However, the Schofield model has the advantage that the exponent b indicates the extent of Knudsen diffusion and Poiseuifle flow contributes to the permeability, while such approximations are not possible from K(, and Bq used in the DGM. Schofield tested his model on membranes having different pore sizes ranging from 0.10 to 0.45 p,m and estimated the values of in a range of 0.1-0.6, which suggest that both the mechanisms (Knudsen and viscous) play an important role in MD flux. On the other hand, Schofield s model has two main disadvantages compared to the DGM model. One is, the components a and b are dependent upon the gas used. The other is, reference pressure P f is chosen in such a way that the dimensionless pressure becomes close to unity (Pr 1), and a is evaluated at P f, hence the parameters a and b also depend on the reference pressure chosen. 

Because of the intricate composition, various transport modes contribute to the supply of O2 to the reaction sites, involving viscous flow, Knudsen diffusion and ordinary molecular diffusion [103]. The relative importance of the distinct mechanisms depends decisively on the psd. Dominating transport though micropores favors Knudsen flow, whereas molecular diffusion with considerably larger diffusion constants will... 

One method which is known under the name of permeametry [131] or Poiseuille-Knudsen method [124] is based on the law of gas permeability in a porous media in the two flow regimes molecular flow (Knudsen) and laminar or viscous flow (Poiseuille). According to Darcy s law, the gas flux through a membrane with a thickness / can be written as / = KAP/l, where K is the permeability coefficient and AP (AP = Pi - P2) the pressure difference across the membrane. If the membrane pore diameter is comparable to the mean free path of the permeating gas, K can be expressed as a stun of a viscous and a non-vis-cous term... 

Transition flow occurs when viscous flow and Knudsen diffusion both play a role, that is in the region with Knudsen number values around unity. Estimates of the value of Kn can be made with the help of the gas kinetic expression for X ... 

In Eig. 5.13 profiles of axial and radial velocity are shown at three axial positions for the annulus, the membrane and the packed bed. The profiles correspond to an a-alumina membrane with a relatively low permeability (Bo = 9.5 x 10" m ) and a mean pore diameter of dp = 3pm (compare with Table 5.1). The Knudsen number is approximately 10 for this membrane at the given conditions, so that only viscous flow occurs. Because the annulus is empty and the flow within it is laminar, the resulting profile for the axial velocity component is parabolic. Compared to the annulus flow, the axial velocity component in the membrane is... 

The Reynolds number in microreaction systems usually ranges from 0.2 to 10. In contrast to the turbulent flow patterns that occur on the macroscale, viscous effects govern the behavior of fluids on the microscale and the flow is always laminar, resulting in a parabolic flow profile. In microfluidic reaction systems, where the characteristic length is usually greater than 10 pm, a continuum description can be used to predict the flow characteristics. This allows commercially written Navier-Stokes solvers such as FEMLAB and FLUENT to model liquid flows in microreaction channels. However, modeling gas flows may require one to take account of boundary sUp conditions (if 10 < Kn < 10 , where Kn is the Knudsen number) and compressibility (if the Mach number Ma is greater than 0.3). Microfluidic reaction systems can be modeled on the basis of the Navier-Stokes equation, in conjunction with convection-diffusion equations for heat and mass transfer, and reaction-kinetic equations. 

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. 

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