# (416d) The Generalized Onsager Model and Dsmc Simulations of High-Speed Rotating Flow with Swirling Feed

#### AIChE Annual Meeting

#### 2016

#### 2016 AIChE Annual Meeting

#### Engineering Sciences and Fundamentals

#### Mathematical Modeling of Transport Processes

#### Tuesday, November 15, 2016 - 4:15pm to 4:35pm

*ÎµÂ = (Î?m/m*, where Î?m is the difference in the molecular masses of the two species, and the average molecular mass

_{av})*m*is defined as

_{av}*m*, where

_{av}= ((Ï_{w1}m_{1}+ Ï_{w2}m_{2})/Ï_{w})*Ï*and

_{w1}*Ï*are the mass densities of the two species at the wall, and

_{w2Â }*Ï*. The equation for the master potential and the boundary conditions are derived correct to

_{w}= Ï_{w1Â }+ Ï_{w2}*O(Îµ*. The results of the Onsager hierarchy, up to

^{2})*O(Îµ*Â are compared with the results of DSMC simulations for a binary hard-sphere gas mixture for secondary flow due to a inflow/outflow of gas along the axis, mass/momentum/energy sources in the flow, with applied linear wall temperature profile, incorporating the angular momentum of the feed gas for a swirling feed. There is excellent agreement between the solutions for the secondary flow correct toÂ

^{2})*O(Îµ*and the simulations, to within 15 %, even when the stratification parameter is as low as 0.707, the Reynolds number is as low as 100 and the aspect ratio length/diameter) of the cylinder is as low as 2, and the secondary flow velocity is as high as 0.2 times the maximum base flow velocity, and the ratio of the mass difference and the average mass

^{2})Â*(2Î?m/(m*is as high as 0.5, and the scaled angular momentum of the feed gas

_{1}+m_{2}))*F*is as high as 0.2. Here, the Reynolds number

_{Î¸}* = F_{Î¸}/(Î¸_{miÂ }**Î©**)*Re = (Ï*, the stratification parameter

_{wÂ }Î©R^{2})/Î¼*A =â??(mÎ©*whereÂ

^{2}R^{2}/2k_{B}T),Â*Î©*and

*R*are the rotational speed and radius of the cylinder,

*m*is the molecular mass, Ï

*is the wall density,*

_{w}*Î¼*Â is the gas viscosity,

*T*is the gas temperature,

*k*is the Boltzmann constant, and

_{B}*Î¸*is the moment of inertia of the rotating cylinder. The major advantages of the swirling feed associated with the angular momentum of the feed gas, is that it results in a reduction of the angular momentum loss of the rotating gas due to feed injection near the feed point by (3 â?? 21)%, reduces the axial spreading of the feed gas by (4 â?? 24)%, minimizes the formation of small secondary vortices near the feed zone, and increases the overall axial mass flux by (16 â?? 27)%, and thereby enhance the efficiency of the feed drive for the centrifugal gas separation process.

_{mi}Â**Key words:**High speed rotating flow, Swirling feed, DSMC Simulations, Rarefied gasÂ flow.

**References**Â :

1. S. Pradhan and V. Kumaran. The generalized Onsager model for the secondary flow in a high-speed rotating cylinder.Â *J. Fluid Mech*., 2011, 686, 109 - 159.

2. V. Kumaran and S. Pradhan. The generalized Onsager model for a binary gas mixture.Â *J. Fluid Mech*., 2014, 753, 307 - 359.

3. H. G. Wood and J. B. Morton. Onsagerâ??s pancake approximation for the fluid dynamics of a gas centrifuge.Â *J. Fluid Mech*., 1980, 101, 1 - 31.

4. H. G. Wood and G. Sanders. Rotating compressible flows with internal sources and sinks.Â *J. Fluid Mech*., 1983, 127, 299 - 311.

5. H. G. Wood and R. J. Babarsky. Analysis of a rapidly rotating gas in a pie-shaped cylinder.Â *J. Fluid Mech*., 1992, 239, 249 - 271.

6. H. G. Wood and J. A. Jordan and M. D. Gunzburger. The effect of curvature on the flow field in rapidly rotating gas centrifuges.Â *J. Fluid Mech*., 1984, 140, 373 - 395.

7. D. R. Olander. The theory of uranium enrichment by the gas centrifuge. *Prog. Nucl. Energy.*, 1981, 8, 1 - 33.

8. G. A. Bird. Molecular Gas Dynamics and the Direct Simulation of Gas Flows. *Clarendon Press*, 1994.

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