Knudsen flow

Wall Geometries. Rougher-than-rough waU geometries can reduce transmission probabUities in Knudsen flow by as much as 25% compared to the so-caUed rough-waU cosine reflection (34,35). For this and other reasons, conductance calculations that claim accuracy beyond a few percent may not be realistic. 

A Barrier Efficiency Eactor. In practice, diffusion plant barriers do not behave ideally that is, a portion of the flow through the barrier is bulk or Poiseuihe flow which is of a nonseparative nature. In addition, at finite pressure the Knudsen flow (25) is not separative to the ideal extent, that is, (M /Afg) . Instead, the degree of separation associated with the Knudsen flow is less separative by an amount that depends on the pressure of operation. To a first approximation, the barrier efficiency is equal to the Knudsen flow multiphed by a pressure-dependent term associated with its degree of separation, divided by the total flow. 

Pores Even porous membranes can give very high selectivity. Molecular sieve membranes exist that give excellent separation factors for gases. Their commercial scale preparation is a formidable obstacle. At the other extreme, UF,3 separations use Knudsen flow barriers, with aveiy low separation factor. Microfiltration (MF) and iiltrafiltra-tion (UF) membranes are clearly porous, their pores ranging in size from 3 nm to 3 [Lm. Nanofiltration (NF) meiTibranes have smaller pores. 

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. 

Process Description Gas-separation membranes separate gases from other gases. Some gas filters, which remove hquids or sohds from gases, are microfiltration membranes. Gas membranes generally work because individual gases differ in their solubility and diffusivity through nonporous polymers. A few membranes operate by sieving, Knudsen flow, or chemical complexation. 

Selective gas permeation has been known for generations, and the early use of p adium silver-alloy membranes achieved sporadic industrial use. Gas separation on a massive scale was used to separate from using porous (Knudsen flow) membranes. An upgrade of the membranes at Oak Ridge cost 1.5 billion. Polymeric membranes became economically viable about 1980, introducing the modern era of gas-separation membranes. Hg recoveiy was the first major apphcation, followed quickly by acid gas separation (CO9/CH4) and the production of No from air. 

The two BCs of the TAP reactor model (1) the reactor inlet BC of the idealization of the pulse input to tiie delta function and (2) the assumption of an infinitely large pumping speed at the reactor outlet BC, are discussed. Gleaves et al. [1] first gave a TAP reactor model for extracting rate parameters, which was extended by Zou et al. [6] and Constales et al. [7]. The reactor equation used here is an equivalent form fi om Wang et al. [8] that is written to be also applicable to reactors with a variable cross-sectional area and diffusivity. The reactor model is based on Knudsen flow in a tube, and the reactor equation is the diffusion equation ... 

Thus, in an isothermal system, the mass flow rate depends on the difference in pressures of the gas across the orifice and does not depend upon the thickness of the plate. One may define an area-normalized resistance, R, for mass transfer through the orifice using a generalization of Ohm s law, i.e., Resistance = force/ flux. For Knudsen flow, the force is the pressure difference (analogous to voltage difference in Ohm s law) and the flux is the mass flow per unit area of the hole (analogous to the electrical current density in Ohm s law). Thus, we have... 

As the pressure is lowered, slip occurs, and the flow mechanism is referred to as transition flow. At pressures so low that collisions between gas molecules are rare compared to the collisions between the gas and the tube wall, the flow is said to be Knudsen flow or free molecular flow. Free molecular flow prevails when Lla > 1. For air at 25°C, this condition means that we have free molecular flow when aPm on < 5. We now consider an intuitive derivation of the result for Fc in the free molecular flow region. 

Molecular diffusion and/or Knudsen flow of reactants from the exterior surface of the catalyst particle into the interior pore structure. 

The symbols refer to a single component. Since molecular collisions are rare events in Knudsen flow, flow and diffusion are synonymous and each component of a mixture behaves as if it alone were present. Numerical values of the Knudsen diffusivity for molecules of ordinary weight at ordinary temperatures range from 0.01 cm2/sec for pores with a radius of 10 A up to about 10 cm2/sec for pores with 10,000 A radii. 

Compare Knudsen flow through two parallel pores of different diameters with that through two equal pores of the same total cross section. 

Knudsen flow rate with a specified pressure gradient is... 

Compare Knudsen flows through a truncated cone of radii d0 and d1 with that through a uniform pore of the same length and volume and pressure difference. 

Knudsen flow varies as r3 and the cross section as r2. Take a total of 200 pores. On the plot, divide the area under the curve into 20 equal parts... 

The main emphasis in this chapter is on the use of membranes for separations in liquid systems. As discussed by Koros and Chern(30) and Kesting and Fritzsche(31), gas mixtures may also be separated by membranes and both porous and non-porous membranes may be used. In the former case, Knudsen flow can result in separation, though the effect is relatively small. Much better separation is achieved with non-porous polymer membranes where the transport mechanism is based on sorption and diffusion. As for reverse osmosis and pervaporation, the transport equations for gas permeation through dense polymer membranes are based on Fick s Law, material transport being a function of the partial pressure difference across the membrane. 

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. 

Type of gas flow Viscous flow Knudsen flow Molecular flow Molecular flow... 

Procedure For a given length (I) and internal diameter (d), the conductance C, which is independent of pressure, must be determined in the molecular flow region. To find the conductance C in the laminar flow or Knudsen flow region with a given mean pressure of p in the tube, the conductance value previously calculated for Cm has to be multiplied by the correction factor a determined in the nomogram C = C a. 

Compare Knudsen flow through these pores with that through an assembly of uniform pores with r = 10 and the same total cross section. 

In the region of Knudsen flow the effective diffusivity DeK for the porous solid may be computed in a similar way to the effective diffusivity in the region of molecular flow, i.e. Dk is simply multiplied by the geometric factor. 

Many heterogeneous reactions give rise to an increase or decrease in the total number of moles present in the porous solid due to the reaction stoichiometry. In such cases there will be a pressure difference between the interior and exterior of the particle and forced flow occurs. When the mean free path of the reacting molecules is large compared with the pore diameter, forced flow is indistinguishable from Knudsen flow and is not affected by pressure differentials. When, however, the mean free path is small compared with the pore diameter and a pressure difference exists across the pore, forced flow (Poiseuille flow see Volume 1, Chapter 3) resulting from this pressure difference will be superimposed on molecular flow. The diffusion coefficient Dp for forced flow depends on the square of the pore radius and on the total pressure difference AP ... 

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