Interdisciplinary Applied Mathematics

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a = стоtan 1 ^criKn^j ,    (4.34)


where a0 is determined to result in the desired free-molecular flowrate using (4.30) and ai = 4.0, в = 0.4. This model is shown in Figure 4.25 with lines.

FIGURE 4.25.    Variation    of    a    as    a    function    of    Knudsen    number.    Analytical    fit


to this variation is obtained by a = ao^ tan-1 (an Kn^3), (ai and f3 are free parameters).


Note that the values for a. and в are the same for both the experimental and numerical results presented in the figure, and that these are the only two undetermined parameters of the model.


In Figure    4.26    we    present    the    flowrate    variation    as a    function    of    Kn.


The data are obtained by the solution of linearized Boltzmann equations by (Loyalka and Hamoodi, 1990) for a very long pipe, so that h ^ X ^ L is maintained for all values of Kn. Knudsen’s two-parameter model is also presented. The experimental data presented in the figure were obtained by Tison for helium flow in finite-length pipes (L/a = 200). There are differences between the experimental and numerical data. For example, the experimental data have not reached the corresponding free-molecular flowrate limit. At Kn = 200, which is the highest Kn value in the experiments, L ж X, and therefore end effects are important and the expected mass flowrate is less than the theoretical free-molecular flowrate (Polard and Present, 1948). Since the analytical and experimental data show some differences, in the case of the experiments we found the value of a0 = 1.19 by using the experimental data at Kn = 200. Also, for the linearized Boltzmann solution we obtained a0 = 1.358 using equation (4.30). The model’s predictions for linearized Boltzmann solution and experimental data are also presented in the figure. The model describes the variation of the data very accurately, and it is successful in predicting the Knudsen’s minimum.

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