In this paper we give a brief introduction to the Onsager pancake model for the fluid dynamics of a gas centrifuge and a history of the pancake computer code. We then present a comprehensive study of separative performance using the hypothetical gas centrifuge parameters of the so-called “Iguacu machine.” In particular, we present: (a) the optimal parameters (feed, wall pressure, temperature profile, scoop drive, position of product and waste orifices) for the maximum separative power for UF6; (b) study of the sensitivity for this machine of all these parameters around the optimum; (c) modeling of the scoop chamber (waste scoop) by a countercurrent flow through the chamber rather than sources/sinks; (d) general laws describing optimal parameters, such the wall pressure or position of product orifice, as functions of the speed, radius, and length; and (e) a general law for the optimum separative work. These results provide a benchmark for comparing the predictions of the pancake model with the predictions ...

Optimization Studies for Gas Centrifuges

The generalizations of the Onsager model for the radial boundary layer and the Carrier–Maslen model for the end-cap axial boundary layer in a high-speed rotating cylinder are formulated for studying the secondary gas flow due to wall heating and due to insertion of mass, momentum and energy into the cylinder. The generalizations have wider applicability than the original Onsager and Carrier–Maslen models, because they are not restricted to the limit , though they are restricted to the limit and a high-aspect-ratio cylinder whose length/diameter ratio is large. Here, the stratification parameter . This parameter is the ratio of the peripheral speed, , to the most probable molecular speed, , the Reynolds number , where is the molecular mass, and are the rotational speed and radius of the cylinder, is the Boltzmann constant, is the gas temperature, is the gas density at wall, and is the gas viscosity. In the case of wall forcing, 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. 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-order and second-order in the radial and axial directions for the Onsager equation, and fourth-order and second-order in the axial and radial directions for the Carrier–Maslen equation) are determined. 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 ‘diffuse reflection’ boundary conditions at 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 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 10 %, 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. The predictions of the generalized models are also significantly better than those of the original Onsager and Carrier–Maslen models, which are restricted to thin boundary layers in the limit of high stratification parameter.

The generalized Onsager model for the secondary flow in a high-speed rotating cylinder

A method is developed to compute the optimal multi-isotope separation in a gas centrifuge The method relies on three models: Onsager's pancake equation, diffusion equations written for each isotope, and an optimization routine Onsager's equation, well studied in the past for UF6, is adapted to multi-isotope gas mixtures, focusing on the three drives generating the countercurrent flow in practical centrifuges: feed drive, scoop drive, and linear wall thermal drive Diffusion equations are written for each isotope in the initial form of partial differential equations (PDE) and reduced to ordinary differential equations (ODE) by the radial averaging method These ODEs, linked with the solution of Onsager's equation through two parameters (flow profile efficiency and scaled countercurrent flow), are solved by an iteration method The optimization routine is based on a choice of a strategy and a method, using the solutions of the two proceeding models, to determine the optimal countercurrent driving

Multi-Isotope Separation in a Gas Centrifuge Using Onsager's Pancake Model

A finite element method for the onsager pancake equation

The Onsager model for the secondary flow field in a high-speed rotating cylinder is extended to incorporate the difference in mass of the two species in a binary gas mixture. The base flow is an isothermal solid-body rotation in which there is a balance between the radial pressure gradient and the centrifugal force density for each species. Explicit expressions for the radial variation of the pressure, mass/mole fractions, and from these the radial variation of the viscosity, thermal conductivity and diffusion coefficient, are derived, and these are used in the computation of the secondary flow. For the secondary flow, the mass, momentum and energy equations in axisymmetric coordinates are expanded in an asymptotic series in a parameter epsilon = (Delta m/m(av)), where Delta m is the difference in the molecular masses of the two species, and the average molecular mass m(av) is defined as m(av) = (rho(w1)m(1) + rho(w2)m(2))/rho(w), where rho(w1) and rho(w2) are the mass densities of the two species at the wall, and rho(w) = rho(w1) + rho(w2). The equation for the master potential and the boundary conditions are derived correct to O(epsilon(2)). The leading-order equation for the master potential contains a self-adjoint sixth-order operator in the radial direction, which is different from the generalized Onsager model (Pradhan & Kumaran, J. Fluid Mech., vol. 686, 2011, pp. 109-159), since the species mass difference is included in the computation of the density, viscosity and thermal conductivity in the base state. This is solved, subject to boundary conditions, to obtain the leading approximation for the secondary flow, followed by a solution of the diffusion equation for the leading correction to the species mole fractions. The O(epsilon) and O(epsilon(2)) equations contain inhomogeneous terms that depend on the lower-order solutions, and these are solved in a hierarchical manner to obtain the O(epsilon) and O(epsilon(2)) corrections to the master potential. A similar hierarchical procedure is used for the Carrier-Maslen model for the end-cap secondary flow. The results of the Onsager hierarchy, up to O(epsilon(2)), are compared with the results of direct simulation Monte Carlo simulations for a binary hard-sphere gas mixture for secondary flow due to a wall temperature gradient, inflow/outflow of gas along the axis, as well as mass and momentum sources in the flow. There is excellent agreement between the solutions for the secondary flow correct to O(epsilon(2)) and the simulations, to within 15 %, even at a Reynolds number as low as 100, and length/diameter ratio as low as 2, for a low stratification parameter A of 0.707, and when the secondary flow velocity is as high as 0.2 times the maximum base flow velocity, and the ratio 2 Delta m/(m(1) + m(2)) is as high as 0.5. Here, the Reynolds number Re = rho(w)Omega R-2/mu, the stratification parameter A = root m Omega R-2(2)/(2k(B)T), R and Omega are the cylinder radius and angular velocity, m is the molecular mass, rho(w) is the wall density, mu is the viscosity and T is the temperature. The leading-order solutions do capture the qualitative trends, but are not in quantitative agreement.

The generalized Onsager model for a binary gas mixture

An analysis is presented that describes a model of the flow field of a rotating compressible fluid in a cylinder with internal sources or sinks of mass, momentum or energy, A solution of the mathematical model is obtained using an expansion in eigenfunctions of the corresponding homogeneous equation. The internal sources or sinks produce countercurrent flows analogous to flows generated by boundary conditions in the classical analysis of the problem. The application of this model to the flow driven by a feed stream or a scoop is discussed. Some sample calculations are presented that illustrate the countercurrent flow produced by sources of mass, the three components of momentum, energy and a mass source/sink combination. Calculations simulating feed introduction and a tails-removal scoop have been performed and the fluid-dynamics solutions have been used to calculate the optimum separative performance of the example centrifuge.

G. Sanders

Papers

Rotating compressible flows with internal sources and sinks