Wall collision frequency

When bounding walls exist, the particles confined within them not only collide with each other, but also collide with the walls. With the decrease of wall spacing, the frequency of particle-particle collisions will decrease, while the particle-wall collision frequency will increase. This can be demonstrated by calculation of collisions of particles in two parallel plates with the DSMC method. In Fig. 5 the result of such a simulation is shown. In the simulation [18], 2,000 representative nitrogen gas molecules with 50 cells were employed. Other parameters used here were viscosity /r= 1.656 X 10 Pa-s, molecular mass m =4.65 X 10 kg, and the ambient temperature 7 ref=273 K. Instead of the hard-sphere (HS) model, the variable hard-sphere (VHS) model was adopted in the simulation, which gives a better prediction of the viscosity-temperature dependence than the HS model. For the VHS model, the mean free path becomes ... 

The first possibility is that the attractive potential associated with the solid surface leads to an increased gaseous molecular number density and molecular velocity. The resulting increase in both gas-gas and gas-wall collision frequencies increases the T1. The second possibility is that although the measurements were obtained at a temperature significantly above the critical temperature of the bulk CF4 gas, it is possible that gas molecules are adsorbed onto the surface of the silica. The surface relaxation is expected to be very slow compared with spin-rotation interactions in the gas phase. We can therefore account for the effect of adsorption by assuming that relaxation effectively stops while the gas molecules adhere to the wall, which will then act to increase the relaxation time by the fraction of molecules on the surface. Both models are in accord with a measurable increase in density above that of the bulk gas. 

It should be recognized that the rates of adsorption are also limited by the wall collision frequency, as given by... 

We are free to place our test plane A at any point along the x axis, and the flux through the plane Zyy will be the same. Thus by letting the test plane coincide with one of the walls, we have derived the desired result the gas-wall collision frequency is Zw of Eq. 10.61 or 10.62. 

The Knudsen number, Kn, is frequently used to characterize different transport regimes in terms of the ratio of X to the critical apparatus dimension, L (Kn = X/L) (Stechelmacher, 1986). The dimension L is typically chosen such that Kn represents the ratio of intermolecular to molecule-wall collision frequencies. For gas flow between the source and the substrate regions, L is the length of the transport chamber or the... 

The experimental data in Table l-II show that decreasing the volume by one-half doubles the pressure (within the uncertainty of the measurements). How does the particle model correlate with this observation We picture particles of oxygen bounding back and forth between the walls of the container. The pressure is determined by the push each collision gives to the wall and by the frequency of collisions. If the volume is halved without changing the number of particles, then there must be twice as many particles per liter. With twice as many particles per liter, the frequency of wall collisions will be doubled. Doubling the wall collisions will double the pressure. Hence, our model is consistent with observation Halving the volume doubles the pressure. 

For polyatomic gases in porous media, however, the relaxation rate commonly decreases as the pore size decreases [18-19]. Given that the relaxation mechanism is entirely different, this result is not surprising. If collision frequency determines the Ti, then in pores whose dimensions are in the order of the typical mean free path of a gas, the additional gas-wall collisions should drastically alter the T,. For typical laboratory conditions, an increase in pressure (or collision frequency) causes a proportional lengthening of T1 so the change in T, from additional wall collisions should be a good measure of pore size. 

Figure 3.5.2 shows the results obtained using M-5 and TS-500 samples with S/V values of 3.03 x 107 and 3.28 x 107 m 1, respectively, and porosities of 0.936 and 0.938, respectively. Note the significant deviation of the relaxation behavior from that ofbulk CF4 gas (dotted lines in Figure 3.5.2). The experimental data were first fitted to the model described above, assuming an increase in collision frequency due purely to the inclusion of gas-wall collisions, assuming normal bulk gas density. However, this model merely shifts the T) versus pressure curve to the left, whereas the data also have a steeper slope than bulk gas data. This pressure dependence can be empirically accounted for in the model via the inclusion of an additional fit parameter. Two possible physical mechanisms can explain the necessity of this parameter.

The one-dimensional velocity distribution function will be used in Section 10.1.2 to calculate the frequency of collisions between gas molecules and a container wall. This collision frequency is important, for example, in determining heterogeneous reaction rates, discussed in Chapter 11. It is derived via a change of variables, as above. Equating the translational energy expression 8.9 with the kinetic energy, we have... 

As the volume of a fixed sample of gas is decreased at a constant temperature, the pressure increases. Since the temperature is constant, the average velocity of the gas particles remains constant. Constrained to a smaller volume, the collision frequency of the molecules with the walls of the container increases. Therefore, the pressure increases. 

Diffusion in macropores occurs mainly by the combined effects of bulk molecular diffusion (as in the free fluid) and Knudsen flow, with generally smaller contributions from other mechanisms such as surface diffusion and Poiseuille flow. Knudsen flow, which has the characteristics of a diffusive process, occurs because molecules striking the pore wall are instantaneously adsorbed and re-emitted in a random direction. The relative importance of bulk and Knudsen diffusion depends on the relative frequency of molecule-molecule and molecule-wall collisions, which in turn depends on the ratio of the mean free path to pore diameter. Thus Knudsen flow becomes dominant in small pores at low pressures, while in larger pores and at higher pressures diffusion occurs mainly by the molecular mechanism. Since the mechanism of diffusion may well be different at different pressures, one must be cautious about extrapolating from experimental diffusivity data, obtained at low pressures, to the high pressures commonly employed in industrial processes. 

The collision frequency with a wall is then given by Z = P/(2nmkBT)v2 while the mean free path X is given by (4.20.12)... 

The resultant solutions are still quite complicated but explicit, and w e have ignored first-order radi( al termination processes which may take place on the walls. The factor of 2, above, derives from the symmetry number occurring in the collision frequencies. The approximation is e.xpected to be poor when relating reactions of different order or different M dependence. 

The collision frequency (collisions per unit of time) with the two walls that are perpendicular to the x axis is given by... 

Consider separate 1.0-L samples of He(g) and UF6collision frequency with the vessel walls ... 

The first attempt to resolve this problem was the radiation hypothesis of Perrin [1], in which the molecule is assumed to be energized by absorbing thermal radiation (in the infra-red) emitted by the walls of the vessel. However it was rapidly realized that the intensity of thermal radiation was quite insufficient to explain the observed rates of reaction [2], although interestingly, under conditions of extremely low collision frequency in interstellar space, Perrin s radiation mechanism is now believed to be significant [3]. 

Boyle s law relates pressure and volume of a gas, and Charles s law states the relationship between a gas s temperature and volume. What is the relationship between pressure and temperature Pressure is a result of collisions between gas particles and the walls of their container. An increase in temperature increases collision frequency and energy, so raising the temperature should also raise the pressure if the volume is not changed. 

A sample of 2.00 mol argon is confined at low pressure in a volume at a temperature of 50°C. Describe quantitatively the effects of each of the following changes on the pressure, the average energy per atom in the gas, the root-mean-square speed, the rate of collisions with a given area of wall, the frequency of Ar-Ar collisions, and the mean free path ... 

In a gas sample the molecules are very far apart and do not attract one another significantly. Each kind of gas molecule acts independently of the presence of the other kind. The molecules of each gas thus collide with the walls with a frequency and vigor that do not change even if other molecules are present (Figure 12-11). As a result, each gas exerts a partial pressure that is independent of the presence of the other gas, and the total pressure is due to the sum of all the molecule-wall collisions. 

Consider the flow of an aerosol through a 4-in. duct at a velocity of 50 ft/sec. Compare the coagulation rate by Brownian motion and laminar shear in the viscous sublayer, near the wall. Present your results by plotting the collision frequency function for particles with dp = 1 /im colliding with particle.s of other sizes. Assume a temperature of 20 C. Hint In the viscous. sublayer, the velocity distribution is given by the relation... 

Temperature gradient Collision frequency per unit volume of monodisperse particles Collision frequency per unit volume of particles with all test particles Dimensionless function of wall potential, Eq. (8.1.12b)... 

---