The generalized Onsager model and DSMC simulations of high-speed rotating flow with swirling feed

摘要

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 and found excellent agreement (with a difference of less than 15%) between the predictions of the analytical model and the DSMC simulations, provided the boundary conditions in the analytical model are accurately specified. 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 (3-21)%, reduces the axial spreading of the feed gas by (4-24)%, minimizes the formation of small secondary vortices near the feed zone, and increases the overall axial mass flux by (16 - 27)%, and thereby enhance the efficiency of the feed drive for the centrifugal gas separation process.