Knudsen mechanism

Micropore Diffusion. In very small pores in which the pore diameter is not much greater than the molecular diameter the diffusing molecule never escapes from the force field of the pore wall. Under these conditions steric effects and the effects of nonuniformity in the potential field become dominant and the Knudsen mechanism no longer appHes. Diffusion occurs by an activated process involving jumps from site to site, just as in surface diffusion, and the diffusivity becomes strongly dependent on both temperature and concentration. 

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. 

The pore diflusivity (Dp), which is assumed to follow Knudsen mechanism, takes Eq. 2 for non-overlapping cylindrical pore of radius r. 

Knudsen flow is characterized by the mean free path (A) of the molecules, which is larger than the pore size, and hence collisions between the molecules and the pore walls are more frequent than intermolecular collisions. A lower limit for the significance of the Knudsen mechanism has usually been set at dp> 20 A [28]. The classical Knudsen equation for diffusion of gas is... 

This example illustrates the following point. The variation of D with depends on the importance of bulk diffusion. At the extreme where the Knudsen mechanism controls, the composition has no effect on D. When bulk diffusion is significant, the effect is a function of a. For equimolal counterdiffusion, a = 0 and yJ has no influence on D. In our example, where a = 0.741, and at 10 atm pressure, D increased only from 0.044 to 0.050 cm /sec as y increased from 0.5 to 0.8. 

Diffusion rates for the H2-N2 system were measured by Rao and Smith for a cylindrical Vycor (porous-glass) pellet 0.25 in. long and 0.56 in. in diameter, at 25°C and 1 atm pressure. A constant-pressure apparatus such as that shown in Fig. 11-1 was used. The Vycor had a mean pore radius of 45 A, so that diffusion was by the Knudsen mechanism. The diffusion rates were small with respect to the flow rates of the pure gases on either side of the pellet. The average diffusion rate of hydrogen for a number of runs was 0.44 cm min (25°C, 1 atm). The porosity of the Vycor was 0.304. 

The geometric parameters t and e for the Knudsen mechanism may, in general, differ from those for the bulk mechanism due to the nature of the pore radius distribution. Again, however, it is doubtful in practice whether this amount of detail is necessary. 

Knudsen mechanism occurs when the average pore diameter is similar to the average free path of fluid moleeules. In this ease, the collisions of the molecules with the porous wall are very frequent and the flux of the component permeating through the membrane is ealeulated by means of the following equation [12] ... 

The permeation follows the Knudsen diffusion and viscous flow in the mesoporous and macroporous defects, eventually present, respectively. The Knudsen mechanism is present when the pore size is smaller than the mean free path of the diffusing molecules and the collisions among molecules of gaseous species are less frequent than their collisions with the pore wall. Equation 17.1 describes the permeance when the Knudsen diffusion takes place ... 

Thus, the time required for the system to attain steady state with the Knudsen mechanism is in the order of 30 seconds. It is emphasised at this point that this is the time required for the pure diffusion mechanism to reach steady state. In the presence of adsorption along the capillary, the time required will be longer because the adsorption process retards the penetration of concentration front through the capillary, that is more mass is supplied for the adsorption onto the capillary wall and hence more time is needed for the attainment of steady state. We will discuss this when we deal with diffusion and adsorption in Chapter 9. 

The capillary assumed so far is cylindrical in shape and its size is uniform along the tube. If the capillary size is not uniform, but either converging or diverging, the Knudsen diffusivity and flux given in eqs. (7.4-10) and (7.4-11) are still valid and they hold for every point along the capillary, provided that the diffusion process is still dominated by Knudsen mechanism. Evaluation of the Knudsen flux for such a capillary is dealt with in the next section. 

Note that the radius r in the above mass balance equation is a function of z as given in eq. (7.4-20a). Before solving this mass balance equation, we need to specify the flux expression and appropriate boundary conditions. We assume the flow is due to the Knudsen mechanism (that is the mean free path is longer than any radius along the diverging capillary), and hence the flux equation is given by (Table 7.4-3) ... 

We have discussed so far the Knudsen diffusivity and Knudsen flux for capillary as well as for porous medium. We have said that the flow of one species by the Knudsen mechanism is independent of that of the other species. The question now is in a constant total pressure system, is there a relationship that relates fluxes of all the species when the partial pressures are constrained by the constant total pressure condition. Let us now address this issue. 

Consider a system having n species and each species diffuses according to the Knudsen mechanism. The flux equation in a capillary for the component j is ... 

For this example, we calculate the Knudsen flux using eq. (7.4-13). The Knudsen diffusivity is = 0.837 cmVsec and the Knudsen flux is 4.5 X 10 mole/cmVsec. We see that the viscous flux in this case is very comparable to the Knudsen flux, and they must be accounted for in the calculation of the total flux. The reason for this significant contribution of the viscous flow is that the pressures used in this example are very high. For low pressure systems, especially those operated under sub-ambient pressure, the Knudsen mechanism is always dominating. 

The driving force to induce the flux by the Knudsen mechanism is ... 

We note from the two terms appearing in the bracket in the RHS of eq.(8.8-42) that the viscous mechanism is more important than the Knudsen mechanism at high pressures. 

The diffusion process in the macropore and mesopore follows the combination of the molecular and Knudsen mechanisms while the diffusion process inside the zeolite crystal follows an intracrystalline diffusion mechanism, which we have discussed in Section 10.2. The length scale of diffusion in the macropore is the dimension of the particle, while the length scale of diffusion in the micropore is the dimension of the zeolite crystal thus, although the magnitude of the intracrystalline diffusivity (in the order of 10 to 10 cmVsec) is very small compared to the diffusivity in the macropore the time scales of diffusion of these two pore systems could be comparable. 

We will first illustrate the time lag method with a simple case of non-adsorbing gas and conditions are chosen such that the transport mechanism is due to the Knudsen mechanism. Diffusion of oxygen, nitrogen, argon, krypton, methane and ethane through inert analcite spherical crystals (Barrer, 1953) at low pressure is an example of non-adsorbing gas with Knudsen flow. Conditions of the experiments are chosen such that the diffusion into the crystals does not occur and flow is restricted to the Knudsen mechanism around the individual crystallites in the bed. The time lag method can be used to complement with the steady state method by Kozeny (1927), Carman (1948) and Adzumi (1937). 

Since the pressures of the two ends of the medium are finite, viscous (Darcy) flow might be operating in addition to the Knudsen flow. To restrict the flow to only Knudsen diffusion, we must maintain the conditions of the experiment such that the Knudsen mechanism is dominating. This is possible when the pressure is... 

The temperature dependence of viscosity of many gases at moderate pressures is stronger than T , while the Knudsen diffusivity is proportional to T . Thus the parameter ttp decreases with an increase in temperature, suggesting that the Knudsen mechanism is gaining its dominance at high temperature. 

The mass balance equation written in the form of equation (12.5-4) is identical in form to eq. (12.2-3) for the case of nonadsorbing gas operating under the Knudsen mechanism. This means that the complete analysis of Section 12.2 or the Frisch s method of Section 12.3 is applicable to this case. The time lags for the receiving and supply reservoirs when the porous medium is initially free of any molecules are ... 

The DCMD flux will increase with an increase in the membrane pore size and porosity and with a decrease in the membrane thickness and pore tortuosity. In other words, to obtain a high DCMD permeability, the surface layer that governs the man-brane transport must be as thin as possible and its surface porosity as well as its pore size must be as large as possible. However, it must be mentioned here that there exists a critical pore size equal to the mean free path of the water vapor molecules for the given experimental DCMD conditions. In the DCMD process, air is always trapped within the membrane pores with pressure values close to the atmospheric pressure. Therefore, if the pore size is comparable to the mean free path of the water vapor molecules, the molecules of the water vapor collide with one another and diffuse among the air molecules. In this case, the vapor transport takes place via the combined Knudsen/molecular diffusion flow. On the other hand, if the pore size is smaller than the mean free path of the water vapor molecules, the molecule-pore wall collisions become dominant and the Knudsen type of flow will be responsible for the mass transport in DCMD. It should be noted that for the given experimental conditions, the calculated DCMD flux based on the Knudsen mechanism is higher than that based on the combined Knudsen/molecular diffusion mechanism. 

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