Title : Separation analysis in a high-speed rotating cylinder for a binary gas mixture.

Authors : Dr. Sahadev Pradhan and Prof. Viswanathan Kumaran

Affiliation : Department of Chemical Engineering, Indian Institute of Science, Bangalore- 560 012, India.

Abstract: The solutions of the species balance equations linked with the generalized Onsager model for the secondary gas flow in a high-speed rotating cylinder are compared with the direct simulation Monte Carlo (DSMC) simulations for a binary gas mixture. The concentration fields are obtained for the secondary gas flows generated by three different types of driving mechanism. These are: (a) wall thermal forcing, (b) inflow/outflow of gas along the axis, and (c) momentum source/sink inside the flow domain, for the stratification parameter *(A*) in the range (0.707- 3.535), and Reynolds number *(Re)* in the range (10^{2} − 10^{6} ) with aspect ratio (length / diameter) = 2, 4, 8. Two different types of cases have been considered, (a) no mass difference *(ε _{a} = (2 Δm/(m_{1} + m_{2} )) = 0*), and (b) with mass difference (ε

_{a}= 0.2 and 0.5) while calculating the secondary flow field in the analytical model. Here, the stratification prameter

*A =*√((m

*)/(2k*

_{av}Ω^{2}R^{2}_{B}

*T )),*and the Reynolds number

*Re = (ρ*, where

_{w}ΩR^{2})/μ*m*is the average molecular mass, Ω and

_{av}*R*are the angular velocity and radius of the cylinder,

*ρ*is the wall density,

_{w}*μ*is the gas viscosity and

*T*is the gas temperature.

The comparison between the numerical and analytical solution reveals that the boundary conditions in the numerical simulations and analytical model have to be matched with care. The commonly used ‘diffuse reflection’ boundary conditions at the solid walls in DSMC simulations result in a non-zero slip velocity as well as a 'temperature slip' (gas temperature at the wall is different from wall temperature). These have to be incorporated in the analytical model 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 the analytical model and numerical simulations, to within 15-20%, 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 with *ε _{a} = (2 Δm/(m_{1} + m_{2} )) = 0.5.*

The separation characteristics have been studied for two different cases, (a) with combination of wall thermal forcing, and inflow/outflow of gas along the axis, but no momentum source inside the flow domain, and (b) with momentum source inside the flow domain, for stratification parameter *(A)* in the range (0.707- 3.535), Reynolds number *(Re)* in the range (10^{2} − 10^{6} ), with aspect ratio (*Z/D* = 2, 4, 8) for the separation of a binary gas mixture with *ε _{a} = (2 Δm/(m_{1} + m_{2} )) = 0.2 and 0.5. *The overall separation factor (α) is found to be maximum at the total reflux conditions (F

^{∗}= 0), and decreases monotonically as the external feed rate (F

^{∗}) increases from 0 to 0.2.

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