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Similar expressions can be obtained for the depleting branch:

Let assume that it is necessary to produce weapon-grade uranium (XP = 90% 235 U) from natural uranium (XF = 0, 71% 235 U) by the GDtechnology (ε′ = ε′′ = 0, 0043) with 235U content in the waste XW = 0, 2% 235 U . Then, the numbers of the enriching stages and the depleting stages are equal NP ≈ 1660 and NW ≈ 290, respectively. If reactor-grade uranium (XP = 4% 235 U) must be produced from natural uranium with the same 235U content in the waste (0,2%), then the number of the enriching stages decreases to NP ≈ 410 at the same number of the depleting stages ( NW ≈ 290 ).

1.2.4. Uranium enrichment in gas centrifuges

If a cylindrical vessel (centrifuge) containing a binary mixture of light and heavy gases rotates with angular velocity w, then the centrifugal force acts on the elementary volume of the gaseous mixture:

F1,2 (r) = g1,2 ×w2 × r;

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where γ1,2 − densities of the gaseous components; r – distance f rom cen-

ter of the vessel.

The pressures on the gaseous components can be determined with application of the following differential equation:

By using the Mendeleev-Clapeyron equation:

(M1, M2 – molecular masses of the gaseous components), den sities of the gaseous components can be determined:

Then, differential equation (5) can be re-written

where V – linear velocity.

Evidently, content of the light and heavy components in the gaseous mixture are proportional to the spatial pressure distribution:

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These formulas demonstrate that content of the heavy component (depleted uranium) is larger in peripheral region of the centrifuge, and, vice versa, content of the light component (enriched uranium) is larger in central region of the centrifuge. In this case, the single-stage enrichment factor can be determined from the following expressions:

a= X235 (0) / X238 (0) = exp (-MLIGHT × V2 (r)2 × R × T) =

(r)X235 (r) / X238 (r) exp (-MHEAVY × V2 (r)2 × R × T)

= exp (DM × V2 (r)2 × R × T);

e¢(r) = a(r) -1 » DM × V2 (r) ). 2 × R × T

As is seen, the single-stage enrichment gain ε′ of the GC-technology depends only on absolute, not relative like in the GD-technology, difference of molecular masses of the light and heavy gas components. Also, the single-stage enrichment gain is proportional to the squared linear velocity of the centrifuge rotation. The centrifuges of contemporary designs can rotate with linear velocities up to 500-700 m/s, i.e. near to the velocity of a bullet outgoing from the rifle tube. According to many numerical evaluations, the GC-technology can provide the following velocity-dependent values of the single-stage enrichment gain at the outer centrifuge radius r0:

e′(r0 ) = 0, 068 at V = 330 m / s; e′(r0 ) = 0, 098 at V = 400 m / s; e′(r0 ) = 0,152 at V = 500 m / s;

e′(r0 ) = 0,300 at V = 700 m / s.

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Nozzle

Fig. 1.4. Layout of the separation-nozzle technology

The hydrogen or helium auxiliary gas (gas-carriers) increases the flow velocity and, hence, it increases the centrifugal forces defining efficiency of the SN-process.

The SN-technology is profitably distinguished from the GCtechnology by the absence of the rotating details but it requires a very fine mechanical assemblage because of very little sizes of the splitting slits (decimal fractions of one millimeter). The single-stage enrichment gain in the SN-technology can reach ε′ ≈ 0, 025 at the specific energy consumption about 3000 kWh/SWU.

1.2.6. Laser technologies of isotope separation

The laser technologies of uranium enrichment rely on the slightly different excitation energies of electronic shells that surround 235U and 238U nuclei. Three extra neutrons in 238U nucleus caused the slight shift in the electron excitation energy scheme as compared with 235U nucleus. This energy shift can be used to excite selectively uranium atoms or uranium-containing molecules by the monochromatic laser light properly tuned to the required wavelength. The excited state of electronic shell can selectively enhance some physical or chemical processes with uranium-containing materials and, thus, promote isotope separation.

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The following conditions should be satisfied for successful implementation of the laser-induced isotope separation:

1.The energy spectrum of the excited electronic levels must contain a line belonging to one isotope only, and this line must be sufficiently far from other spectral lines of the desirable isotope and from all spectral lines of other isotopes.

2.Physical or chemical processes must be found which are able to separate the excited and non-excited uranium-containing components.

3.Laser-induced impact on the isotopic composition to be separated must be a main excitation mechanism, not inter-atomic or intermolecular collisions.

4.High-efficiency lasers must be developed and finely tuned to the appropriate wavelength.

Presently, the following two laser isotope separation technologies are under intense development and demonstration, namely atomic vapor laser isotope separation (AVLIS) and molecular laser isotope separation (MLIS).

The AVLIS-technology has been developed at the Lawrence Livermore National Laboratory (USA). The AVLIS technology includes the following stages:

1. Vacuum evaporation of uranium atoms at very high temperature ( 23000C). Beam of accelerated electrons knocks uranium atoms out of

uranium-rhenium alloy.

2. Irradiation by xenon laser (λ 3780 Å, ultraviolet range). are selectively excited.

3.Irradiation by krypton laser (λ 3500 Å, ultraviolet range). The excited 235U atoms are selectively ionized.

4.Collection of 235U ions on an electrically charged plate.

The MLIS-technology has been developed at the Los Alamos National Laboratory (USA). The MVLIS technology includes the following stages:

1. Expansion of gaseous uranium hexafluoride – hydr ogen composition through a hypersonic nozzle. As a result, uranium hexafluoride

cools down to about 30 K but it does not condense.

2. Irradiation by infrared laser (λ 1,6·105 Å). Molecules of are selectively excited.

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3. Irradiation by ultraviolet laser (l~3,08·104 Å). The excited molecules of 235UF6 are selectively dissociated with the formation of uranium pentafluoride 235UF5 and free fluorine:

2 × 235 UF6 ® 2 × 235 UF5 + F2 .

Uranium pentafluoride 235UF5 precipitates from the gas flow as a fine powder (so called, “laser snow”) that can be e asily collected.

The single-stage enrichment factors are very high for both laser technologies of uranium enrichment. They cover the range from 3 to 15, according to different experimental studies. Such high-efficiency technologies make it possible to use even the waste materials from GDand GC-processes containing about 0,2% 235U for production of reactorgrade uranium (about 3% 235U) by a single enrichment stage.

1.2.7. Chemical methods of isotope separation

The chemical methods of isotope separation are based on the preferential stability of certain isotopes in various immiscible chemical compounds. The isotope exchange reactions can occur, if two different chemical compounds of one multi-isotope chemical element enter into a contact. The isotope exchange reactions lead to the concentration of isotopes in those compounds where they can be more stable.

The following conditions must be satisfied for feasibility of the chemical isotope separation technologies:

1.The contacting compounds must be chemically stable together.

2.The contacting compounds must be separated by a relatively simple means (for example, organic and inorganic substances).

3.It is desirable for the chemical element to be of different valences in two contacting compounds.

The following examples of the chemical isotope separation are presented below:

1.Boron enrichment with isotope 10B:

BF3 + BF3O(CH3)2 ® 11BF3 + 10BF3O(CH3)2 ,

i.e. isotope 10B passes into the organic compound.

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Orbital radius R and angular frequency of spiral rotation can be derived from the following relationships:

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