Rate of momentum transfer

When Che diameter of the Cube is small compared with molecular mean free path lengths in che gas mixture at Che pressure and temperature of interest, molecule-wall collisions are much more frequent Chan molecule-molecule collisions, and the partial pressure gradient of each species is entirely determined by momentum transfer to Che wall by mechanism (i). As shown by Knudsen [3] it is not difficult to estimate the rate of momentum transfer in this case, and hence deduce the flux relations. 

Equations (2.10), (2.18) and (2.24) provide the flux relations in situations where each of the three separate mechanisms of momentum transfer dominates. However, there remains the problem of finding the flux relations in "intermediate" situations where all three mechanisms may be of comparable importance. This has been discussed by Mason and Evans [7], who assumed first that the rates of momentum transfer due to mechanisms (i) and (ii) should be combined additively. If we write equation (2.10) in the form... 

The right hand side represents the rate of momentum transfer from species r by mechanism (i) and, combining this with the rate of transfer by mechanism (ii) as given by equation (2.IS), we obtain... 

The Prandtl number, Cu/k = (p/p)/(k/pC), compares the rate of momentum transfer through friction to the thermal diffusivity or the transport of heat by conduction. 

Consider the propagation of a one-dimensional normal shock wave in a gas medium heavily laden with particles. Select Cartesian coordinates attached to the shock front so that the shock front becomes stationary. The changes of velocities, temperatures, and pressures of gas and particle phases across the normal shock wave are schematically illustrated in Fig. 6.12, where the subscripts 1, 2, and oo represent the conditions in front of, immediately behind, and far away behind the shock wave front, respectively. As shown in Fig. 6.12, a nonequilibrium condition between particles and the gas exists immediately behind the shock front. Apparently, because of the finite rate of momentum transfer and heat transfer between the gas and the particles, a relaxation distance is required for the particles to gain a new equilibrium with the gas. 

As the rate of momentum transfer is equal to a force, momentum balances are equivalent to force balances. 

Equations 3.13 are known as the Stefan-Maxwell equations and are valid when the total pressure and temperature gradients as well as external forces can be neglected. They have the physical meaning that the rate of momentum transfer between two species is proportional to their concentrations and to the difference in their velocities. The molar average velocities of the species v, and v are defined in a such way that the molar fluxes of the various species are... 

In the less turbulent flow through the straight channel of a monolith, momentum transfer from the fluid to the wall is less effective and in the case of two-phase, countercurrent annular flow, momentum transfer between gas and liquid will also be less than in the interstitial channels of a packed bed. The lower rates of momentum transfer, which is the reason for the higher permeability of monoliths, should in principle improve the possibility for achieving countercurrent flow of gas and liquid at realistic velocities. 

The rate of momentum transfer = (rate of collisions along any axis) X (momentum transfer per collision)... 

For most cases of interest involving doughnut beams, the radial mode index p = 0 and the Laguerre polynomial = 1. If an area element dA is highly absorbing, the rate of momentum transfer to it will be given by the Poynting vector ... 

Similarly, the total rate of momentum transfer through plane AB... 

The reason why only n-1 equations in the Stefan-Maxwell equation are independent is not surprising since these equations describe only momentum exchange between pairs of species, and they lack the necessary boundary conditions defining the rate of momentum transfer to the capillary walls (Burghdardt, 1986). 

This also works for the case of elastic scattering of identical particles in identical quantum states K aa aa must be multiplied by a factor of 2 to get the rate of momentum transfer (k scatters to kg k) since two atoms scatter per collision event. Gao ° has also described the formal theory for collisions of cold atoms taking into account identical particle symmetry. 

Longer cilia should be able to clear mucus faster because they can generate a greater forward velocity. In smaller airways the cilia are generally shorter and fewer in number than in the large bronchi, and even though the cilia beat frequency may be comparable, the rate of momentum transfer to the mucus is proportionately less. 

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