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Question: 2b9 lowdensity phenomena in compressible tube flow23 fig 2b9 as...

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2B.9) Low-density phenomena in compressible tube flow23 (Fig. 2B.9). As the pressure is de- creased in the system studied in Example 2.3-2, deviations from Eqs. 2.3-28 and 2.3-29 arise. The gas behaves as if it slips at the tube wall. It is conventional2 to replace the customary no- slip boundary condition that vz = 0 at the tube wall by dv dr Oz (2B.9-1) in which ζ is the slip coefficient. Repeat the derivation in Example 23-2 using Eq. 2B.9-1 as the boundary condition. Also make use of the experimental fact that the slip coefficient varies in- versely with the pressure o/p, in which Lo is a constant. Show that the mass rate of flow is (2B.9-2) Rpavg in which Pavgー1(Po + ph) and Pavg is the average density calculated at pavg When the pressure is decreased further, a flow regime is reached in which the mean free path of the gas molecules is large with respect to the tube radius (Knudsen flow). In that regime GTR3) (2B.9-3) in which m is the molecular mass and K is the Boltzmann constant. In the derivation of this re- sult it is assumed that all collisions of the molecules with the solid surfaces are diffuse and not specular. The results in Eqs. 2.3-29, 2B.9-2, and 2B.9-3 are summarized in Fig. 2B.9. Free molecule flow or Knudsen flow t0 Po PL Slip flow Poiseuille flow -Fig. 2B.9 A comparison of the flow regimes Pavg in gas flow through a tube.

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