# Interdisciplinary Applied Mathematics

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FIGURE 4.28. Variation of a as a function of Kn for various aspect ratio ducts.

Knudsen regime. We consider flows in ducts with aspect ratio (AR = w/h = width/height) of 1, 2, and 4. The data are obtained by linearized Boltzmann solution in ducts with the corresponding aspect ratios (Sone and Hasegawa, 1987). Our previous analysis was valid for the two-dimensional channels, where we reported flowrate per channel width. For duct flows, three-dimensionality of    the flow    field    (due    to    the    side    walls    of    the    duct)

must be considered. In no-slip duct flows the flowrate formula developed for two-dimensional channel flows is corrected in order to include the blockage effects of the side walls. According to this, the volumetric flowrate in a duct with aspect ratio AR for no-slip flows is (see (White, 1991), p. 120)

Q = C (AR)

wh3

~l2/j

dP

dx J

where C(AR) is the correction factor given as

C(AR)

1

192(AR)

n5

E

i=1,3,5

tanh(*7r/2(AR))

%

(4.35)

With this correction, aspect ratios of 1, 2, and 4 ducts correspond to 42.17%, 68.60%, and 84.24% of the theoretical two-dimensional channel volumetric flowrate for no-slip flows, respectively. According to the new model, the volumetric flowrate for rarefied gas flows in ducts is

Q = C (AR)

wh3

12/xq

dP

dx

(1 + a Kn)

6Kn

^ 1 —b Kn

where the correction factor C(AR) is independent of Knudsen number. The variation of a as a function of Kn is calculated by using the correction factors (C(AR)), the linearized Boltzmann solutions in (Sone and Hasegawa, 1987), and our    model.    This    variation    is    given    in Figure    4.28.    The    rarefac

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