Mean free path plates

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 ... 

Three general flow regimes may be anticipated for the flow over a flat plate shown in Fig. 12 12. First, the continuum flow region is encountered when the mean free path A is very small in comparison with a characteristic body dimension. This is the convection heat-transfer situation analyzed in preceding chapters. At lo wer gas pressures, when A L, the flow seems to slip along the surface and u 4= 0 at y = 0. This situation is appropriately called slip flow. At still lower densities, all momentum and energy exchange is the result of... 

As a first example of low-density heat transfer let us consider the two parallel infinite plates shown in Fig. 12-14. The plates are maintained at different temperatures and separated by a gaseous medium. Let us neglect natural-convection effects. If the gas density is sufficiently high so that A — 0, a linear temperature profile through the gas will be experienced as shown for the case of A. As the gas density is lowered, the larger mean free paths require a greater distance from the heat-transfer surfaces in order for the gas to accommodate to the surface temperatures. The anticipated temperature profiles are shown in... 

Fig. 12-14 Effect of mean free path on conduction heat transfer between parallel plates (a) physical model (b) anticipated temperature profiles.

We have made the additional approximation of assuming that the number of collisions Z at any point is independent of d, the distance between plates. This is justifiable if the mean speed 5 F L/d, where F L/d is the difference in velocity between two layers of gas separated by a mean free path. Und( r such conditions the molecular density in each layer is constant and most collisions then take place between molecules that have essentially the same relative Maxwellian distribution. When this condition is not satisfied, there will be important density gradients and thermal gradients, so that the entire analysis does not apply. This condition is the equivalent of saying that the velocity of the moving plate is small compared to the velocity of sound. 

If we consider a system in which the gas density is so low that the mean free path is of the same order of magnitude or greater than the distance between the plates, the entire mechanism of transport changes from transport through the collision between gas molecules to direct transport by collisions of molecules with the plates, since the average molecule now makes no collisions in going from one plate to the other. We can now calculate the momentum transport directly from our formula for the number of molecules striking a unit surface per second (Sec. VII.6). This rate is [Eq. (VII.6.6)] ... 

When the gas density between two plates at different temperatures is such that the mean free path of the gas is much greater than the distance between the plates, the transport of thermal energy is directly by molecular impacts upon the plates. This process can be analyzed by following the procedure used for momentum transport at low densities (Sec. VIII.5). 

Copper and nickel were deposited from metal foil wrapped around a hot tungsten filament. Chromium was evaporated from a chrome plated tungsten wire. XPS measurements were made to determine the metal coverage as well as the electronic structure at the interface. The metal coverage was determined by substituting the experimentally measured areas under the XPS curves, core hole cross sections ( ), and electron mean free path in both the metal and the polymer and an instrument response... 

In conventional parallel plate channel the intermolecular collisions dominate, because the characteristic length 2 y 1 is much larger than the molecular mean free path. The velocity of the fluid at the surface is zero u[yl]=0. 

In micro parallel plate channel the interactions between the fluid and the wall become significant, because the molecular mean free path is comparable to 2 y 1. The gas slip along the wall with a finite velocity in the axial direction as described by Maxwell in 1890 [13]. The kinetic theory of gases gives the following boundary condition at the surface of the channel [28] ... 

In this derivation of the expression for we have assumed that the pressure is high enough so that X is much smaller than the distance separating the two plates. At very low pressures where X is much larger than the distance between the plates, the molecule bounces back and forth between the plates and only rarely collides with another gas molecule. In this case the mean free path does not enter the calculation, and the value of depends on the separation of the plates. At these low pressures the thermal conductivity is proportional to the pressure, since it must be proportional to A, and X does not appear in the formula to compensate for the pressure dependence of N. 

Debye length (3.1.10b), mean free path of a gas molecule (3.1.114), filter coefficient (7.2.187), parameter for a dialyzer (8.1.399), parameter for a distillation plate/stage (8.3.38), latent heat of vaporization/condensation molecular conformation coordinate (3.3.89c) electrode spacings (7.3.18) retention parameter for species i (7.3.213), ionic equivalent conductance of ion i (3.1.108r) value of Xi for a cation value of A, for an anion value of A, at infinite dilution (Table 3.A.8) defined by (5.4.100) equivalent conductance of a salt (an electrolyte) (3.1.108s)... 

Gas scattering Scattering of a high velocity atom by collision with gas molecules. See also Gas scatter plating Mean free path Thermalization. 

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