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A gas centrifuge is used to separate gases of different mass. The centrifuge creates an artificial gravity, like earths gravity force, `mg`, due to very high speed rotation. It is well known that the centripetal force that needs to be exerted to a keep particle on a rotating disc is:
`bb "F"=-(m omega^2 r) bb hat "r" \ \ \ (1)`
where `omega` is the rate of rotation of the disc, `r` is the radial location of the particle, m is the mass of the particle and `bb hat "r"` is a unit vector along the radial direction. We can convert the force in equation 1 to a potential by integrating it by `dr`.`V(r)=(m omega^2 r^2)/2 \ \ \ (2)`
We now need to convert equation 2 so that the lowest potential energy is at the outer bound radius `r_B`.`V(r)=(m omega^2 (r_B-r)^2)/2 \ \ \ (3)`
Thus, instead of the potential energy expression `mgh` where g is gravity and h is height above the earth's surface, which we assign to particles in our atmosphere, we use a slight modification of the the expression in equation 2 since we expect the density to be highest at large r. A Maxwell-Boltzmann expression for the density distribution, `n(h)` of particles in earth's atmosphere is:`n(h)=n_0 exp(-(mgh)/E_(avg)) \ \ (4)`
where `n_0` is the density at earth's surface, and `E_(avg)` is the average thermal kinetic energy of the atmosphere at elevation `h`. Similarly we should expect the density of particles of mass `m` in the centrifuge to be:`n(r)=n(r_B) exp(-(m omega^2 (r_B-r)^2)/(2E_(avg))) \ \ (5)`
where `E_(avg)` is the average kinetic energy of particles of all masses.For this animation, we will use just 2 different masses, `m_L` and `m_S`, where `L` and `S` denote large and small.
In order to induce rotation of a gas in a cylinder in a physically valid way, it is necessary to have the inner surface of the cylinder rough. For this animation I have chosen to make the cylinder surface rough by using a sinusoidal corrugation. The corrugation has two parameters: amplitude and the number of periods on the inner boundary. The parameters are both adjustable by the use of sliders. The rate of rotation `omega` of both inner and outer corrugated cylinder is also adjustable.
The explanation of how the particles are reflected from the corrugations as well as many other details of this animation are explained in Laminar Flow of a Gas By the expression "laminar flow" we expect that the boundary particles' rotational speeds will be almost the same as its adjacent rough cylinder. As one might expect, the average particle energy as well as the average angular momentum increases as the corrugations rotate. Essentially the corrugations act like "fans" that continuously increase both the rotational energy of the particles and their random directional (thermal) energy.In order to keep the particles within the cylinder annulus, the animation stops when the average particle speed becomes greater than 1.0 pixel per second. Occasionally particles briefly escape the annulus. The number of escapees is also printed on Canvas 1.
Some important results are printed on Canvas 1. Most important is the Averagde Tangential Speed since this is what causes heavier particles to go toward the outside of the annulus. You should note that the Average Tangential Speed is never as large as the Outer Bound Tangential Particle Speed. This is due to the fact that the Outer Bound Particles are forced to rotate at the speed of the outer corrugations. while the internal particles are a gas and rotate at that speed Occasionally because of their "viscosity". The term viscosity is the shear force that exists due to radially varying speeds. Viscosity will increase as the particle radius increases since the time between collisions will decrease. Typical ratios of large mass particles numbers to small mass particle numbers in the outer bounds are 430 to 250. The ratio is limited by the initial random (thermal) motion as well as the random motion induced by the rotating corrugations. In any case the learner can see that a gas centrifuge is a viable means to separate gas atoms that have significantly different masses. The stepped (histogram) plots are done by counting the number of particles at defined radii and placing these numbers in bins. After the rotation has run, the learner can easily see, near the outer annulus bounds, that the red stepped curve values are much larger than the blue stepped curve values. This is what we expect taking into account centrifugal forces. Similarly, near the inner annulus bounds, the blue stepped curve values exceed the red stepped curve values as also expected. The smooth curves are plots of the expected Maxwell-Boltzmann distribution of equation 5 Vs radius `r`. The red smooth curve is always set to full scale at the outer radius while the blue curve maximum value is the ratio of the average outer bound small mass tangential speed to the outer bound large mass tangential speed. The smooth curves agree qualitatively with the histogram curves.
One of the major uses of gas centrifuges: Wikipedia Gas Centrifuge is separating uranium gases to purify the U-235 isotope where the abundance of U-235 is only about 0.72% of natural uranium. The other major isotope is U-238 which has only about a 1.3% mass difference. So this is a rather daunting task and requires many centrifuges working in tandem.