Interdisciplinary Applied Mathematics

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   Temperatures at 300 K and 400 K, and


Knudsen number at Kn = 0.365, 0.122, 0.052.


The pressure in both tanks is initially atmospheric. Thermal creep effects cause pumping of the fluid toward the hot tank, increasing the pressure in the hot tank and lowering the pressure in the colder one. This pressure


TABLE 5.1. Pressure differences due to the thermal creep effects obtained by numerical simulation and from the analytical formula.


Gas


Kn


APanaly (Pa)


APnumer i^Pft)


Air


0.052


342.0


336. 0


Air


0.122


1482.0


1409.0


Air


0.365


9151.0


8832.0

difference eventually starts flow in the middle of the channel in the direction from hot to cold (high pressure to low pressure), resulting in zero average mass flowrate    in the channel    as    the    steady    state    is reached.    For    zero    net


mass flowrate, equation (5.2) can be written as


dP =    £


dx h2P ) 1 + 6[^(Kn — Kn2)]’


It is    possible    to    integrate    this    equation    approximately    using    average    val


ues of viscosity (Д), pressure (P = PlP2), and Knudsen number (Kn), resulting in


Pi — P‘2



9t2R    -T2)


h2p j 1 + 6 (Kn — Kn2



(5.3)


where viscosity and Knudsen number are evaluated at average temperature (T = TlT2) and average pressure (P). Equation (5.3) shows that the pressure drop between two tanks can be increased by either decreasing the channel thickness (h) or the average pressure (P). In other words:


• Thermal creep effects can be significant in rarefied flows where the pressure is low or in microflows in atmospheric pressures where the typical dimensions are on the order of a micron or lower.

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