Analysis of axial flow field & concentration field in a long counter-current gas centrifuge based on Berman-Olander theory and comparison with Karl-Cohen theory.

In this study the Berman-Olander analytical model governing the axial flow field in a high speed long counter-current gas centrifuge (large length to diameter ratio) have been studied due to imposed temperature gradient between the stationary central scoop post and the side-wall of the rapidly rotating outer cylinder to generate a convective secondary axial flow (known as axial counter-current flow) for stratification parameter A^2 =[ (M ω^2 r_2^2 )/(2 R_g T ) ] in the range 10 to 30, the radius ratio σ = (r_0 /r_2) in the range 0.08 to 0.15, the geometrical factor ν = [ σ^2 / (1 - σ^2) ] in the range 0.0075 to 0.024, and the parameter β = [ e^( A^ ν ) / { (A^2/ε) - (A^2 ν) – 1} ] in the range -0.099 to -0.036 [1 - 13]. Here, M is the molecular weight of the process gas, ω and r_2 are the angular velocity and inner radius of the rapidly rotating outer cylinder, R_g is the universal gas constant, T is the gas temperature, r_0 is the outer radius of the stationary central scoop post, and the parameter ε = [ (M g* )/(R_g T G ). The term ( ρg* ) arising in the governing equation represents the adjustable driving force for producing the axial counter-current flow, where ρ is the gas density, and the term G = (1/p) (∂p/∂z), represents the contribution to the axial momentum equation due to axial pressure gradient. The axial velocity profiles and axial mass flux profiles have been studied quantitatively considering two major simplified assumptions: the stratification parameter A^2 > 10, and stratification parameter A^2 > 20, which permit the evaluation of the integrals arising in the diffusion equation of separation theory, to estimate the axial variation of concentration of the desired isotope in a long counter-current gas centrifuge ((Pradhan & Kumaran, J. Fluid Mech., vol. 686, 2011, pp. 109-159); (Kumaran & Pradhan, J. Fluid Mech., vol. 753, 2014, pp. 307-359)). The net enrichment (N_P/N_W)net at total reflux condition [ product flow rate (P) = waste flow rate (W) = 0 ] as well as the net enrichment (N_P/N_W)net at half unit-CUT condition [ product flow rate (P) = waste flow rate (W) = ½ Feed flow rate (F) ] are determined. These solutions are compared with Karl-Cohen theory based on radial averaging technique. The comparison reveals that the Berman-Olander theory predicts lower net enrichment (N_P/N_W)net compared to Karl-Cohen theory and difference between them increases with the increase of peripheral speeds (Vθ = ω r_2) of the rapidly rotating outer cylinder. The K-integrals which appear in the net enrichment (N_P/N_W)net equation are evaluated knowing the form of the flow profiles (shape & magnitude) and results are analyzed for better physical understanding of the underlying phenomena. Here, N_P and N_W are the concentration of the product and waste stream respectively. An important finding is that the optimum enrichment (N_P /N_W)opt at half unit-CUT condition is strongly affected by the product to internal axial counter-current circulation ratio (P/L)", and, for a given product flow rate (P), the optimum enrichment (N_P /N_W)opt can be achieved by fine-tuning the desired internal axial counter-current circulation rate (L) through activation of mechanical, wall thermal, end-cap thermal, and feed drive.