# Interdisciplinary Applied Mathematics

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w

1

—nv

4

2(h + w)dx.

The intermolecular collision frequency in the flow volume is

fg = —nhw dx,

A

where v is the mean thermal speed and n is the number density. Assuming that w ^ h, the ratio of intermolecular collisions to total collisions becomes

(4.21)

f9 1

fg + fw 1 + Kn

This analysis resulted in a correction to the continuum-based flowrate models, similar to the variable diffusion coefficient model presented earlier, with the only difference of ^ in front of the Kn term.

In general, the increased rarefaction effects in our flowrate model can be taken into account by introducing a correction expressed as rarefaction

coefficient Cr (Kn), which is a function of Knudsen number. The flowrate is then obtained as

Q

h3 dP 12p, dx h3 dP 12^q dx

6Kn

+1-6Kn 6Kn

+16Kn

Cr (Kn),

(4.22)

where Cr (Kn) is a general function of Knudsen number. A possible model for Cr is suggested by the aforementioned analysis (equations (4.20) and (4.21)) in the form

Cr (Kn) = 1 + a Kn,    (4.23)

where a is a parameter. If we assume that a is constant in the entire Knudsen regime, the flowrate in the slip flow regime will be erroneously enhanced, resulting in

= l + (6 + a)Kn+C>(Kn2),

MC

where MC corresponds to continuum mass flowrate. This model becomes inaccurate for a nonzero value of a in the slip flow regime. Moreover, in the free-molecular flow regime, for very long channels (L ^ X ^ h) there are no physical values for a, since the flowrate increases logarithmically with Kn.    For    finite-length    channels    the    flowrate    is    asymptotic    to a    constant

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