Knudsen regime, molecular diffusion

Effective diffusivity in Knudsen regime Effective diffusivity in molecular regime Knudsen diffusion coefficient Diffusion coefficient for forced flow Effective diffusivity based on concentration expressed as Y Dispersion coefficient in longitudinal direction based on concentration expressed as Y Radial dispersion coefficient based on concentration expressed as Y Tube diameter Particle diameter... 

Diffusion of A within the porous pellet takes place. If the pores arc very large this may be the normal type of molecular diffusion, but if the pore radius is smaller than the mean free path, a molecule will hit the pore wall more often than it hits its fellows, and this is the Knudsen regime of diffusion. Both types of diffusion can be described by Fick s law in which the flux is proportional to the concentration gradient, and if the diffusion coefficient is not in some sense large there may be large variations in the concentration of A within the pellet. Let r denote position within the catalyst particle then the concentration of A within the particle is a(r), a function of that position, and obeys the partial differential equation for diffusion with a(r) = as when r is a position on the exterior surface of the particle. Clearly, this is a complicated matter and we shall seek ways of simplifying it in Sec. 

It is evident from the above discussion that there must be a wide range of conditions under which both Knudsen and molecular diffusion are significant. Indeed, in a given adsorbent it is quite possible for molecular diffusion to be dominant in the larger pores while Knudsen flow is dominant in the smaller pores. Because of the dependence of mean free path on pressure, for any given adsorbent and adsorbate there will be a transition from molecular diffusion at high pressure to Knudsen flow at low pressures. In the intermediate regime both wall collisions and intermolecular collisions contribute to the diffusional... 

In equations (5) and (6), DM and DK are the molecular and Knudsen diffusivities, respectively, and e and x are the void fraction and the tortuosity of the porous solid, respectively. For pore dimensions significantly larger than the mean free path of the diffusant in the gas phase, the diffusivity is governed by molecular diffusion, but when the pore diameter becomes smaller than the mean free path, diffusion is properly described by Knudsen diffusion. When the pore diameter approaches that of the diffusing species, around 10, one enters the configurational regime. 

Viscous (Poiseuille) flow and molecular diffusion are non-selective. Nevertheless they play an important role in the macroporous substrate(s) supporting the separation layer and can seriously affect the total flow resistance of the membrane system. Mesoporous separation layers or supports are frequently in the transient-regime between Knudsen diffusion (flow) and molecular diffusion, with large effects on the separation factor (selectivity). 

The net diffusivity of component A within the pores of a catalytic pellet is obtained by adding mass transfer resistances for Knudsen diffusion and ordinary molecular diffusion, where convection reduces the resistance due to ordinary molecular diffusion but Knudsen flow occurs over length scales that are much too small for convective mass transfer to be important. This addition of resistances is constructed to simulate resistances in series, not parallel. Consider the trajectory of a gas molecule that collides with the walls of a channel or other gas molecules. In the pore-size regime where Knudsen and ordinary molecular diffusion are equally probable, these collisions occur sequentially, which suggests that gas molecules encounter each of these resistances in series. Hence, for binary mixtures. 

The three-halves power of dimensionless temperature in the expression for eA( ) is based on the temperature dependence of gas-phase ordinary molecular diffusion coefficients when the catalytic pores are larger than 1 p.m. In this pore-size regime, Knudsen diffusional resistance is negligible. The temperature dependence of the collision integral for ordinary molecular diffusion, illustrated in Bird et al. (2002, pp. 526, 866), has not been included in ea) ). The thermal energy balance given by equation (27-28), which includes conduction and interdiffu-sional fluxes, is written in dimensionless form with the aid of one additional parameter,... 

The influence of this mechanism can be enough significant in the Knudsen regime [8], when the pressure potential is very low and the flux through the pore is due to the molecular diffusion. In this... 

However, the specific features of the TAP system are demonstrated in pulse-response experiments under vacuum conditions in the Knudsen-diffusion regime, in which the diffusion process is well defined. This means that—unlike the molecular diffusion coefficient—the Knudsen diffusion coefficient does not depend on the composition of the gas mixture. 

To give an idea of the orders of magnitude at 300 K and 1 bar, A of air is 0.07 ptm. Thus the capillary must be smaller to get into the Knudsen regime. Alternatively, we may use a low pressure, for example, for a capillary of 0.1 mm, the pressure must be lower than 70 Pa. The molecules now collide much more frequently with the capillary walls than with other diffusing molecules. The (Knudsen) diffusion coefRcient for each species is proportional to the inverse square root of its molecular weight, and the flux through the capillary is ... 

Experimental figures very similar to that of Fig. 3.5.1.2.A-ld for the packed column technique are obtained. At a total pressure of 2.41 bar the effective diffusivity was calculated to amount to 3.665 x 10 m / m s. The diffusion covered both the Knudsen and molecular regime. By performing experiments at various total pressures, ranging from 1.51 to 3.01 bar, it became possible to distinguish between tk and The values found were 3.19 and 2.52, respectively. 

Diffusion in the continuum regime (A/rp)< 1 can be described by the usual molecular diffusion coefficient. However, the treatment of multi-component mixtures is very tedious except when they can be approximately considered as binary. Diffusion in the transition regime (A/r 1) has been described by effective diffusivities which are functions of the molecular and the Knudsen diffusivity [34]. 

Another very important feature of transient experiments in the TAP-2 reactor is that they can be performed in different diffusion regimes, which are determined by the reactor geometry and the number of molecules pulsed. When the pulse size is below 10 molecules, gas transport in the micro-reactor occurs via Knudsen diffusion. This means that any collisions between gas-phase molecules are strongly minimized. Therefore, pure heterogeneously-catalysed reactions can be investigated. Transient experiments with higher pulse sizes (molecular diffusion) provide important information about the contribution of gas-phase processes to the overall reaction studied. 

Helium permeance experiments on M2 sample at 273 K and 308 K indicate that the increase in permeability, in cm s" units (Table 2), is not proportional to the square root of the temperatinre, as defined by molecular flow (Knudsen regime). The higher value of the perm bility ratio, against the temperature square root ratio, suggests that helium flux caimot be described by the Knudsen approximation, but by the (xnresponding of activated diffusion. In this case, the presence of ultra micropores and the constrictions in the pores, hinder the molecular motion. As the temperature rises, the kinetic raiergy of the molecules increases and they can overcome the energy barrier of the diffusion. The phenomenon of activated diffusion is very common on micropore system and it can be expressed by an Arrhenius relation ... 

The value of Da,eff generally is not very sensitive to temperature. If diffusion is primarily in the molecular or Knudsen regime, Ddiff is only about 5-20 kJ/mol. On the other hand, the activation energies for chemical reactions are much higher, of the order of50-300 kJ/mol, If... 

For gases, if the pores are large and diffusion is completely in the molecular regime, then Da,p = T>a and >A,m/ A,eff = 10. However, if the pores are small and diffusion is in the transition or Knudsen regime, then DA,m/ A,eff can be substantially greater than 10. Values of 100 and even larger are possible. 

The fact that when the pores are small enough (such that the mean free path of the molecules becomes comparable to the dimension of the pore) the laws of molecular diffusion will no longer apply, but Knudsen diffusion will become important. In a physical sense, this means that in the Knudsen regime collisions between the gas molecules and the wall will become more frequent than collisions between gas molecules. 

For a binary gas mixture it is frequently necessary to measure two or even three diffusivities. For example, if diffusion in the porous medium during the reaction is Knudsen type then it will be necessary to measure the Knudsen diffusivity of each component. If the diffusion is in the transition regime then the investigator will need to measure the effective molecular diffusivity within the solid, in addition to the Knudsen diffusivities. Only in the purely molecular diffusion regime will one diffusivity suffice. 

The different situations that can be encountered analysing the permeation of molecules through a porous membrane are reported below. When the pore diameter of a porous solid is in the macropore range, collisions between the molecules will occur much more frequently than collisions with the wall. In this case molecular diffusion is the dominant mechanism. As the size of the pores decreases (mesoporous solid), the number of collisions with the wall increases and can become more frequent and important than the molecule-molecule colUsions. At this point, Knudsen diffusion takes over. When the pore diameter becomes comparable to the size of the molecules (microporous solid), the molecules continuously collide with the walls. When this happens, diffusion behaves as an activated process and the term configurational regime is used to describe it. 

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