The generalized Onsager model for the radial boundary layer and of the generalized Carrier-Maslen model for the axial boundary layer at the end-caps in a high-speed rotating cylinder ((S. Pradhan & V. Kumaran, J. Fluid Mech., 2011, vol. 686, pp. 109-159); (V. Kumaran & S. Pradhan, J. Fluid Mech., 2014, vol. 753, pp. 307-359)), are extended to incorporate the angular momentum of the feed gas for a swirling feed for single component gas and binary gas mixture. For a single component gas, the analytical solutions are obtained for the sixth-order generalized Onsager equations for the master potential, and for the fourth-order generalized Carrier-Maslen equation for the velocity potential. In both cases, the equations are linearized in the perturbation to the base flow, which is a solid-body rotation. The equations are restricted to the limit of high Reynolds number and (length/radius) ratio, but there is no limitation on the stratification parameter. The linear operators in the generalized Onsager and generalized Carrier-Maslen equations with swirling feed are still self-adjoint, and so the eigenfunctions form a complete orthogonal basis set. However, the differential operator depends explicitly on the stratification parameter, and so it is necessary to evaluate the eigenvalues and eigenfunctions numerically. For the case of mass/momentum/energy insertion into the flow, the separation of variables procedure is used, and the appropriate homogeneous boundary conditions are specified so that the linear operators in the axial and radial directions are self-adjoint. The discrete eigenvalues and eigenfunctions of the linear operators (sixth and second order in the radial and axial directions for the generalized Onsager equation, and fourth and second order in the axial and radial directions for the generalized Carrier-Maslen equation) are determined. The solutions for the secondary flows with swirling feed are determined in terms of these eigenvalues and eigenfunctions. These solutions are compared with direct simulation Monte Carlo (DSMC) simulations. The comparison reveals that the boundary conditions in the simulations and analysis have to be matched with care. The commonly used â??diffuse reflectionâ?? boundary conditions at solid walls in DSMC simulations result in a non-zero slip velocity as well as a â??temperature slipâ?? (gas temperature at the wall is different from wall temperature). These have to be incorporated in the analysis in order to make quantitative predictions. In the case of mass/momentum/energy sources within the flow, it is necessary to ensure that the homogeneous boundary conditions are accurately satisfied in the simulations. When these precautions are taken, there is excellent agreement between analysis and simulations, to within 15%. Next, we study the generalized Onsager and generalized Carrier-Maslen model for a binary gas mixture (V. Kumaran & S. Pradhan, J. Fluid Mech., 2014, vol. 753, pp. 307-359)) with swirling feed, incorporating the angular momentum of the feed gas in a high speed rotating cylinder. In the base state, the flow is considered to be a solid body rotation with constant temperature, but the variation in the mole fraction is taken into account. For the secondary flow, the mass, momentum and energy equations in axisymmetric coordinates are expanded in an asymptotic series in a parameter ÎµÂ = (Î?m/mav)
, where Î?m is the difference in the molecular masses of the two species, and the average molecular mass mav
is defined as mav = ((Ïw1m1 + Ïw2m2)/Ïw)
, where Ïw1
are the mass densities of the two species at the wall, and Ïw = Ïw1Â + Ïw2
. The equation for the master potential and the boundary conditions are derived correct to O(Îµ2)
. The results of the Onsager hierarchy, up to O(Îµ2)
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 O(Îµ2)Â
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 (2Î?m/(m1 +m2))
is as high as 0.5, and the scaled angular momentum of the feed gas FÎ¸* = FÎ¸/(Î¸miÂ Î©)
is as high as 0.2. Here, the Reynolds number Re = (ÏwÂ Î©R2)/Î¼
, the stratification parameter A =â??(mÎ©2 R2/2kBT),Â
are the rotational speed and radius of the cylinder, m
is the molecular mass, Ïw
is the wall density, Î¼
Â is the gas viscosity, T
is the gas temperature, kB
is the Boltzmann constant, and Î¸miÂ
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.
Key words:High speed rotating flow, Swirling feed, DSMC Simulations, Rarefied gasÂ flow.
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.