Interdisciplinary Applied Mathematics

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Due to    the    lack of    detailed    experimental    data,    we    do    not    have    exact


values for    the    pressure    ratio    П,    and    thus    we    cannot    use    equation    (4.30)


directly. Instead, we approximately integrate the volumetric flowrate equation (4.28) by multiplying it by the average density in order to obtain the mass flowrate. The flow conditions are evaluated at an average state. For example, the average pressure is defined as P = (P; + PQ)/2 so that (dP/dx) = АР/L, and Kn is evaluated at P. The corresponding mass flowrate becomes


Ml



7та4Р ДР 8/a0RT L



(1 + a Kn)



4Kn 16Kn



(4.32)


Nondimensionalizing this relation with the theoretical free-molecular flow limit (4.16), we obtain the following relation


Ml


Mfm



37Г


64Kn



(1 + aKn)



_ 4Kn


1 H—=


1 —6Kn



(4.33)


Comparison of equations (4.31) and (4.33) shows that both models predict the same limit in the free-molecular flow regime (Kn ^ж) if the value of ao is chosen according to equation (4.30) (ao = 1.358 for pipe flows, i.e. a < A < L).


Knudsen’s formula is often used to describe the flow for the entire flow regime, including the slip flow regime. Considering that


Mc 37Г —


—:- = —КП,


Mfm    64


Knudsen’s formula can be written for the slip flow regime (Kn < 0.1) as


Мкп


ж



1 +



64Kn


37Г



2.507


3.095



+ O(Kn2),


where the subscript “C” stands for continuum predictions. This relation shows that Knudsen’s formula is not accurate for the slip flow regime, since the first-order variation of flowrate from the corresponding continuum limit should be

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