As will be shown in Sec. 5.5, the separative capacity of a countercurrent gas centrifuge is proportional to its length L and increases rapidly as the peripheral speed va increases. Hence it is advantageous to run at the highest practical speed and to use centrifuges of the greatest practical length. An absolute limit to the speed is reached when tangential stresses caused by centrifugal forces equal the tensile strength of the rotor material. Limitations on practical values of the length are set by the need to avoid combinations of length, radius, and speed at which
the rotor experiences resonant vibrations. These two factors limiting centrifuge mechanical performance, which have the greatest effect on separation performance, will be discussed in this section. Many other relevant mechanical topics, such as design of bearings, motor drives, and damping mechanisms, are beyond the scope of this text.

Maximum peripheral speed. Consider a cylindrical shell of radius r and thickness dr, made of material of density p and rotating at angular velocity « rad/s. Figure 14.16 represents a volume element of the shell of height dz subtending an angle dd. The mass of the element

dm = prdrdzdd (14.145)

experiences a centrifugal force

ш1 rdm = pw1 r* drdzdd (14.146)

in the outward r direction. This must be balanced by the components in the opposite direction, Off sin(d6/2), of the tangential stresses oe acting on the two surfaces drdz offset by angle dd.

Figure 14.15 The Zippe centrifuge. (Adapted from Shacter et al. [S3.])

where umax is the maximum tangential speed, at which the tangential stress reaches the tensile strength a of the material.

Table 14.11 gives the density, tensile strength, and modulus of elasticity E of six possible high-tensile materials for centrifuge rotors. These properties are given in metric units and SI units.

The maximum tangential speed ranges from 400 m/s for aluminum alloy to 720 m/s for a carbon fiber-resin composite.

Conditions for resonant vibrations. Certain angular velocities со,- cause a thin, hollow cylinder to go into resonant longitudinal vibrations. If a centrifuge rotor is driven for any length of time at or near one of these angular velocities, rotational energy is used to increase the amplitude of longitudinal vibrations until the rotor or its bearings may be wrecked. Consequently, it is important to avoid tangential speeds at which a rotor of given length and radius will be in resonance. Texts on mechanical vibrations such as [D4] show that the longitudinal vibration frequency of a thin hollow cylinder of radius a, modulus of elasticity E, and length L, unrestrained at the ends, in the ith mode is

where the eigenvalues X,- are

і 1 2 3 4 5

X, 22.0 61.7 121.0 200.0 298.2 і is the number of loops into which the profile of the cylinder is displaced. Because

Vі СО/Г

the length-to-radius ratio at which rotors of each of the materials run at maximum tangential speed umax would be in resonant vibration is

The last part of Table 14.11 gives values of L/a for the first five resonances in cylinders of the five materials operated at the maximum speed, at which tangential stress equals the tensile strength of the material. At lower speeds u, the critical L/a ratio is obtained by multiplying the values of Table 14.11 by /vmaxlv.

Rotors that are shorter than the first critical length are said to be subcritical. Such rotors do not need special means to avoid resonant speeds. Rotors that are longer than the first critical length are called supercritical. They must be operated at speeds away from resonance

and must be provided with drives of sufficient power to accelerate them quickly through resonant speeds and brakes of sufficient power dissipation to decelerate them quickly. All of Groth’s rotors listed in Table 14.10 have length-to-diameter ratios below the first critical at the listed peripheral speeds. However, if ZG7 had been made of titanium and operated at its maximum peripheral speed of 442 m/s, it would have run between the first and second resonance.

Power consumption. Because the separation performed by the gas centrifuge is a thermodynamically reversible process, the minimum energy input necessary to separate an isotopic mixture is merely the small difference in free energy between the separation products and the feed. The actual energy input is thousands of times greater because it is dominated by the work necessary to overcome mechanical friction in bearings, aerodynamic drag, and pressure drops in gas circulation. These energy inputs are specific to details of centrifuge and plant design and cannot be estimated from principles of the separation process, as was possible for gaseous diffusion. The U. S. DOE stated [U3] that the power consumption of a centrifuge plant per unit separative capacity would be around 4 percent of the power consumption of a gaseous diffusion plant.

The comparatively low power consumption of a gas centrifuge plant is its greatest advantage over competing processes. The relatively low separative capacity of a single centrifuge is its greatest disadvantage.

Means for estimating the separative capacity of a centrifuge will be developed in Sec. 5.5.