# Interdisciplinary Applied Mathematics

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1.5272

2.5

0.5

regimes. The linear increase of the flowrate with Kn and complete description of rarefied    duct    flows    with    the    introduction    of    the    correction    factor

C(AR) are observed. The slope of the nondimensionalized mass flowrate increases gradually with Kn. This is attributed to the gradual change in the rarefaction coefficient as presented in Figure 4.28.

For the free-molecular scaling of the data we nondimensionalized the flowrate with

•    h?w AP

which gives the correct order of magnitude for the flowrate. The exact value of the free-molecular flowrate in rectangular ducts is given by (Thompson and Owens, 1975)

MFM(h, w) = rMMFM,

where

Г

(/i2+w2)3/2 h3 + w3 3    +    3    ‘

Here, h and w denote the height and the width of the rectangular duct. For the aspect ratios (AR) of 1, 2, and 4 the above relation results in 0.8387, 1.1525, and 1.5008 times the free-molecular mass flowrate MFM, respectively.

Nondimensionalizing the model with the free-molecular mass flowrate (Mfm), we obtain

M _ C(AR) . Мрм 6Кп

FIGURE 4.30. Free-molecular scaling of linearized Boltzmann solutions of (Sone and Hasegawa, 1987) for duct flows of various aspect ratio. Comparisons with the proposed model are also presented by lines corresponding to different aspect ratios.

_ 6Kn

1 H—= .

1 — 6KnJ

In Figure 4.30 we present the variation of the nondimensionalized flowrate as a function of Kn. The duct flow data are due to (Sone and Hasegawa, 1987), and the two-dimensional channel data (shown by AR = ж) are due to Sone (for Kn < 0.17) and Cercignani (Kn > 0.17) (Fukui and Kaneko, 1990). Comparisons are made against the linearized Boltzmann solutions. For duct flows, good agreement of the model with the numerical data in the entire flow regime is obtained. The model is also able to capture Knudsen’s minimum accurately. The parameters used in the model are given in Table 4.3. Note that ao is determined from the asymptotic constant limit of flowrate (4.36) as Kn ^ ж.

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