Interdisciplinary Applied Mathematics

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Next, we compare the flowrate model corrected as before by the rarefaction coefficient Cr (Kn) in a similar form obtained for channel flows with


Cr (Kn) = 1 + a Kn .


The volumetric flowrate for a pipe is


•    na4 dP    .


Q = ~8+“ n)


and the corresponding mass flowrate is


na4Po AP



Ad =



16p,oRTo L



+



8(a + b) П — 1



1 +



4 Kn



1- b Kn



[(П + 1) + 2(4 + a) Kno



Kno loge



П — 6Knc 16Kn„



(4.28)



(4.29)


Since b = — 1 is already determined from the linearized Boltzmann solution, the only parameter to be determined in the model is a. However, a should vary from zero in the slip flow regime to a constant asymptotic value (ao) in the free-molecular flow regime. It is possible to obtain the constant asymptotic value of a (as Kn ^ ж) by using the theoretical mass flowrate in the free-molecular flow regime given by equation (4.16) and the asymptotic value for the mass flowrate obtained by (4.30) while Kn ^ ж, as


64


“Kn-oo = «о = ^37r(1_ 4) J ■    (43°)


We can also compare the results with the formula derived by (Knudsen, 1909), which we normalize here with the corresponding free-molecular flow limit of equation (4.16):


(4.31)


MKn _ Зтг (1 +2.507(1/Kn) MFM ~~ 64Kn + V1 + 3.095(1/Kn)


where Kn is computed at the average pressure P = (P + PQ)/2. The constants 2.507 and 3.095 are taken from (Loeb, 1961), where details of derivation of Knudsen’s formula are presented. The same formula has also been used in (Tison, 1993), and (Loyalka and Hamoodi, 1990).

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