Uranium extraction by TBP may in some cases become poorer in the presence of other extractable components because of depletion of free TBP by components other than uranium. Such behavior is illustrated by the U02(N03)2-HN03-TBP system analyzed above, as shown by the data [Ml ] in Fig. 4.7 for the distribution coefficient of uranium as affected by nitric acid concentration. For acid concentration less than about 5 M, the uranium distribution coefficient is greater the higher the acid concentration, because of the salting effect of nitrate ion from the acid. At acid concentrations greater than about 5 M, increasing acid concentration inhibits uranium extraction, because enough nitric acid has been extracted so that less free TBP is available to form the extractable complex with uranium. Similar effects have been observed in the extraction of other elements with TBP [Ml].

In designing multistage extraction systems for extractive separations by TBP, or by other extractants that can change appreciably in noncomplexed concentration as a result of extraction, it is necessary to perform analyses similar to Eq. (4.15) through Eq. (4.24) for each of the extractable species present in other than trace quantities to determine the distribution coefficients for each of the species in each of the contacting stages [G6, H2, L3]. Such design procedures are illustrated in Sec. 6.6 for the separation of hafnium from zirconium by TBP extraction from a nitric acid solution.

Figure 4.7 Effect of nitric acid concentration on extraction of uranyl nitrate with TBP. (Data of McKay [Ml].)

Solutions of uranyl salts can be readily prepared by dissolving U03 in the appropriate acid. Uranyl nitrate, sulfate, acetate, fluoride, chloride, bromide, and iodide all are very soluble, the solutions having a characteristic yellow-green fluorescence. Uranyl nitrate can be made by dissolving uranium metal or any of the oxides in nitric acid. It crystallizes from solution as the well-formed yellow uranyl nitrate hexahydrate U02(N03)2’6H20, often called UNH.

Uranium in solution as uranyl nitrate can be purified by addition of hydrogen peroxide, which selectively precipitates pale yellow uranyl peroxide U02(02)*2H20. Addition of sodium hydroxide to uranyl nitrate solution precipitates sodium diuranate Na2U207. Addition of ammonium hydroxide precipitates ammonium diuranate:

2U02(N03)j + бЫН^ОН -*• (NH^UjO, + 4NH4N03 + 3H20

Uranyl ion forms complexes with many anions. Table 5.13 gives equilibrium constants for complex formation at 20°C and an ionic strength around 1, arranged in order of increasing complex stability. At high concentrations of fluoride, sulfate, or carbonate ion, the complex uranyl ions then present are much less extractable by organic solvents than uranyl nitrate. The very stable uranyl sulfate and uranyl carbonate complex anions are strongly held by anion-exchange resins and are often used to recover and purify uranium from solutions obtained in leaching uranium ores (Sec. 8 of this chapter). The complex uranyl carbonate anions are very soluble in aqueous solutions of sodium or ammonium carbonate. This property has been used to separate uranium from elements such as radium, iron, or lead, which form insoluble precipitates with carbonate ion.

Reference [II] summarizes the somewhat conflicting data on the system thorium-carbon. From maxima in the melting point curve, it is concluded that two compounds exist:

The dicarbide, either by itself, mixed with uranium dicarbide, or in solid solution with uranium dicarbide, is used as fuel material in some versions of high-temperature gas-cooled reactors. Like uranium carbides, the thorium carbides react rapidly with water or moist air and must be protected from moisture in storage and fuel fabrication.

ThC2 and (Th, U)C2 particles are made by granulating oxides and graphite flour in the proper proportions for the reaction

Th02 + 4C -»■ ThC2 + 2CO

reacting the granules at high temperature, and then melting to consolidate and spheroidize the particles.

History. In 1947, Fischer and co-workers [FI, F2] described a solvent extraction method for separating hafnium from zirconium in which an aqueous solution of sulfates containing ammonium thiocyanate was extracted with diethyl ether containing thiocyanic acid. Hafnium concentrates preferentially in the organic phase; in one reported experiment zirconium in the aqueous phase contained 035 percent hafnium, while the organic phase zirconium contained more than 5 percent. Six to eight batch laboratory separations concentrated hafnium from 0.5 percent in zirconium to 70 to 90 percent.

In a study of methods for separating hafnium from zirconium, the Oak Ridge National Laboratory [02] concluded that solvent extraction of the thiocyanate was the method best suited for commercial use. The aqueous phase recommended was an HC1 solution of the oxychlorides (Zr, Hf)OCl2, the solution obtained when the tetrachlorides are dissolved in water. The organic solvent recommended was methylisobutyl ketone (hexone) containing around 2.3 mol thiocyanic acid (HCNS) per liter. Hexone was preferred over diethyl ether because it is less volatile and less flammable. A hafnium-zirconium separation factor of about 5 is obtained in this system, with hafnium concentrating in the organic phase. A plant to separate 24 kg of natural zirconium per hour into hafnium and reactor-grade zirconium was built at the Y-12 plant of the Union Carbide Corporation at Oak Ridge in 1951 [R1 ]; its performance has been analyzed by Googjn [Gl], In 1952 operation was transferred to a similar plant built at the Albany, Oregon, station of the U. S. Bureau of Mines; its construction and performance have been described by McClain and Shelton [Ml ]. Another similar plant was used by the Carborundum Metals Corporation at Akron, New York. Both of these plants have since been shut down. They have been superseded by a larger plant of Teledyne Wah Chang Albany Corporation at Albany, Oregon, with a capacity of around 400 kg hafnium-free zirconium per hour. This hexone-thiocyanate separation process has also been used in France and England [Jl] and was studied by Fischer et al. [F3] in Germany.

U. S. Bureau of Mines plant. Figure 7.6 is a process flow sheet for the zirconium-hafnium separation portion of the U. S. Bureau of Mines zirconium plant at Albany, Oregon [Ml]. Commercial-grade zirconium tetrachloride containing about 2 w/o hafnium was dissolved in water together with ammonium thiocyanate (NH4CNS) and NH4OH, to make a feed solution containing around 120 g zirconium + hafnium per liter that was 1.0 to 1.1 M in HQ and 2.7 to

2.9 M in NH4CNS. The zirconium and hafnium were in the form of thiocyanate complexes that could be extracted from the aqueous solution by a solution of thiocyanic acid HCNS in hexone. This aqueous solution was fed at a rate of 189 liters/h to a solvent extraction system consisting of spray columns made of sections of Pyrex glass pipe 10.2 cm in diameter. At the feed point the feed joined the aqueous stream flowing at the rate of 76 liters/h from the scrubbing section В and entered the extracting section C, 62.8 m in total length, made up of four shorter columns. Here countercurrent extraction by a solution of HCNS in hexone reduced the hafnium content of zirconium to 0.004 w/o.

Hexone flowing from the extracting section to the scrubbing section В contained almost all the hafnium in the feed and about 30 percent of the zirconium. The scrubbing section consisted of three columns of 10.2 cm Pyrex pipe 45.4 m in total length. Countercurrent flow

of 76 liters/h of 3.6 to 3.9 M HC1 scrub solution transferred most of the zirconium from the hexone back to the aqueous phase and reduced the zirconium content of the extract to 2 w/o.

Hafnium was removed from the extract by stripping with 5 /V H2 S04 in the stripping column A, which was 15.2 m long.

HCNS was recovered from the aqueous product stream by extraction with pure hexone in the HCNS recovery column D. The pure hexone was prepared by reacting a hexone solution of HCNS from the scrubbed solvent tank E with 28% NH4OH in a cooled NH4CNS recovery reactor F. The 6 to 7 M NH4CNS thus recovered was recycled to feed.

A rough estimate of the number of theoretical plates in the scrubbing and extracting sections can be obtained from the material flow sheet for these sections, Fig. 7.7. The distribution coefficients for zirconium and hafnium assigned to the two sections were obtained from the following conditions:

1. The hafnium/zirconium separation factor for this system of 5 implies that Z)Hf/-Dzr = 5.

2. Tests on the columns of the Y-12 plant reported by Googin [Gl] showed that distribution coefficients were in the following ranges:

Scrubbing, Hf 0.5

Zr 0.012-0.10 Extracting, Hf 0.5-1.0

Zr 0.13-0.17

3. The height of an equivalent theoretical plate (HETP) in each section should be about the same.

With these assumptions the distribution coeffients and number of theoretical plates in the two sections are as given in Fig. 7.7 (Prob. 7.1). Assignment of constant distribution coefficients to each section is an oversimplification, but does permit semiquantitative representation of the more complex actual system.

Two significant points may be noted. (1) The concentrations of zirconium and hafnium in the feed are approximately equal to the corresponding concentrations in the aqueous stream entering the extracting section, a condition that minimizes loss of separation at this point. (2) The fraction of feed zirconium recycled through the scrubbing section is (14.73 g/literX530 liters)/(l 17.6 g/literX189 liters) = 0.35; McClain and Shelton state that about 30 percent was recycled.

Materials of construction. It is necessary to use materials of construction for this separation that resist corrosion at the high concentrations of HC1 used in the process. The Bureau of Mines plant [Ml] used glass columns, glass — or rubber-lined equipment, and rubber tubing connectors. The British plant [J1) used polythene mixer-settlers.

Zirconium purification and conversion to zirconium dioxide. The zirconium-product stream leaving the HCNS recovery column D of Fig. 7.6 contained most of the metal impurities in the ZrCl4 feed other than hafnium. Purified zirconium was obtained by precipitating Zr(OH)4 at a pH low enough to prevent precipitation of other metal hydroxides. The precipitation procedure used by the Bureau of Mines was as follows. Zirconium content of the raffinate was diluted to 19 g/liter. To every cubic meter of diluted raffinate were added 5.7 liters of concentrated 33 N sulfuric acid, followed by sufficient 28% ammonium hydroxide to bring the pH to 1.2 to 1.6 at a temperature controlled at 88°C. The basic zirconium sulfate precipitate was filtered off. The precipitate was twice reslurried with 28% ammonium hydroxide and filtered off to complete conversion to zirconium hydroxide. The hydroxide was dried in a rotary stainless steel drier at 350 to 400°C and converted to Zr02 in a rotary Thermalloy-40 retort at 700° C.

In summary, actinides in the oxidation stage M(III) form trivalent ions in aqueous solution with chemical properties similar to those of the trivalent rare earths, e. g., lanthanum. The M(IV) actinides form tetravalent ions with properties characteristic of Th4+. The M(V) actinides form M02+ ions, and the M(VI) form M0224 ions whose properties are characteristic of UVI022+. The chemical properties of uranium and thorium are discussed in Chaps. 5 and 6, respectively. The following sections summarize the properties of protactinium, neptunium, plutonium, americium, and curium, which are the bred actinides important in reprocessing thorium and uranium fuels.

ТаЫе 9.8 Standard oxidation-reduction potentials for oxidizing and reducing agents in add solutions’*’

Figure 9.1 Oxidation-reduction diagrams. Formal and standard potentials in volts. Calculated or uncertain couples are listed in parentheses.

_09I Cttf oil—

— 1229__

_ 10.1) „ „ (-0.1) „ „ (-0.4) „ „-2 (-0.6)

Ru — RU2O3 — RuQ2——————————— ■- RUO4 ————-

||____ 0O4___ I I_____

0.05

I I I II I

|t_oi275_N2Hj-oA4i-NH3OH+-LK — Nz -|77 NgOig)—‘^ — NCXgl’^^-H N02—^—————————————————————— N2Q,(g) N03 [L l]

-^H2N202-^-

Mn-LLS Mn

L

Figure 9.1 Oxidation-reduction diagrams. Formal and standard potentials in volts. Calculated or uncertain couples are listed in parentheses. (Continued)

MJPa. The isotope 232 Pa is a beta emitter with a half-life of 1.31 days. It is formed in irradiated thorium by (n, i) reactions in 231 Pa. Most of it undergoes beta decay into 72-year 232 U, which is an important radioactive contaminant in the 233 U recovered from irradiated thorium, as discussed in Chap. 8.

M3Pa. The isotope 233 Pa is an important intermediate nuclide in the formation of 233 U, in the chain originating by neutron capture in 232 Th. In this sense, 233 Pa is functionally analogous to

23,Np in the chain leading to 23,Pu by neutron capture in 238U. An important difference, however, is the relatively long 27.0-day half-life of 233Pa. As a result, 233Pa persists much longer in irradiated thorium fuel and may contribute significantly to actinide radioactivity during thorium fuel reprocessing.

Because of epithermal resonance absorption of neutrons in 233 Pa, its effective cross section in a thermal-neutron spectrum is much greater than the 2200 m/s cross section listed in Table 9.9. During thorium irradiation 233Pa may exist in sufficient concentration that its destruction by chain-branching neutron absorption can reduce the rate of formation of 233 U. For this reason, thorium-uranium breeder reactors tend to optimize at lower neutron fluxes, and at lower specific power, than do uranium-plutonium breeders.

233Pa can be recovered from irradiated thorium fuel as a separated actinide, with possible value because of its subsequent decay to form isotopically pure 233 U.

234Pa. The isotope 234 Pa is formed in irradiated thorium from (n, y) reactions in 233 Pa and, to a limited extent, from the decay of 234mPa, the daughter of 234Th.

234mPa. The isotope 234mPa is formed by the beta decay of 234Th, and by (n, 7) reactions in 233Pa. A small fraction (0.13 percent) of the decays of 234mPa are isomeric transitions to 234Pa, and the rest are beta transitions to 234 U.

2.3 Steps in Purex Process

The Purex process has become the process quite generally used for reprocessing slightly enriched uranium fuel from power reactors. For this reason, it will be described in more detail in this chapter than other fuel separation processes.

The principal steps in the Purex process as applied to fuel clad with stainless steel or zircaloy are shown schematically in Fig. 10.2. Each of these steps will be described in more detail later in Sec. 4.

In preparation for dissolution, step 1, cladding is opened to permit subsequent dissolution of the oxide fuel. For steel or zircaloy this is done by mechanical shearing or sawing. Off-gases from decladding contain up to 10 percent of the radiokrypton and xenon in the fuel and some of the I4C02,t tritium, and other volatile fission products. If voloxidation (Sec. 4.3) is used after decladding to remove tritium, more of the other volatile radionuclides will then be evolved also.

^14C is produced primarily by the (n, p) reaction on

Figure 10.2 Principal steps in Purex process.

Fuel and cladding next are charged to a dissolver where, in step 2, they are reacted with hot nitric acid. This dissolves oxides of uranium and most other elements in the fuel while leaving cladding essentially unreacted. Dissolver off-gases are primarily a mixture of steam, air, and oxides of nitrogen, NO*. Off-gases also contain all of the radiokrypton and xenon remaining in the fuel after voloxidation and any 14 C convertible to I4C02 or other volatile form. Practically all of the radioiodine can also be driven into the dissolver off-gases.

In NO* absorption, step 15, the off-gases are cooled and scrubbed with water to recover oxides of nitrogen for conversion to nitric acid and recycle to the dissolver and to other process steps.

Gases from NO* absorption are combined with off-gases from decladding and treated in step 16 for as complete removal of radioactivity as practical. Substantially complete retention of radioiodine and entrained liquids and solids is required; technology for retaining them is fully developed. Retention of radiokrypton has been mandated by the U. S. Environmental Protection Agency in reprocessing fuel irradiated after 1982. Separation and packaging of
radiokrypton from dissolver off-gases has been demonstrated on a pilot-plant scale and is practiced at the Idaho Chemical Processing Plant. Retention of tritium and 14 C may also be required in the future, but the requisite technology is not yet completely worked out.

Gadding hulls from the dissolver are washed with water, monitored to show that most of the fission products and fuel have been removed, packaged, and shipped to waste storage.

In feed preparation, step 3, the acidity of the dissolver solution is adjusted to the desired value around a pH of 2.5, and plutonium is brought into its most extractable valence of 4, usually by controlled addition of N204 or, formerly, sodium nitrite.

In primary decontamination, step 4, uranium and plutonium are separated from more than 99 percent of the fission products by solvent extraction with 30 v/o (volume percent) TBP in a paraffinic hydrocarbon diluent.

In partition, step 5, plutonium is separated from uranium by reducing plutonium to the organic-insoluble, trivalent state with a reductant strong enough to act on plutonium but not so strong as also to reduce uranium. Ferrous sulfamate was used in early plants; tetravalent uranium, hydroxylamine, or controlled cathodic reduction is now preferred. In some plants, reduction of plutonium and its return to the aqueous phase is carried out in a single step. In others, plutonium and uranium are first returned to the aqueous phase, then plutonium is reduced, and finally uranium is separated by reextraction into the organic phase.

Impure plutonium nitrate is purified, step 6, by one or more additional solvent extraction cycles, plus ion exchange in some plants.

Purified plutonium nitrate is converted to the preferred product form, Pu02, in step 7 either by evaporation to dryness and calcination or by precipitation as the oxalate or peroxide and calcination.

Impure uranyl nitrate from step 5 is purified in step 8 by one or more additional cycles of solvent extraction and, in some plants, by treatment with silica gel or hydroxamic acid.

Purified uranyl nitrate solution from step 8 is evaporated to dryness and calcined to U03 in step 9. Nitric acid vapors are condensed and recycled.

If the U03 product is to be reenriched, it is converted to UF6 in step 10, either by one of the processes described in Qiap. 5, Secs. 9.4 through 9.7, or by direct reaction with fluorine.

Low-level aqueous wastes from steps 6 and 8 are processed for further recovery of plutonium and uranium, then concentrated for recovery of water and nitric acid. High-level aqueous wastes from step 4 are concentrated by evaporation, with recovery of condensed nitric acid in step 11.

After a period of interim storage as liquid, step 12, to permit the rate of heat generation to decrease, high-level wastes are solidified in step 13. The duration of interim storage as liquid can be reduced if the spent fuel is stored as solid an equivalent time before reprocessing.

In a well-designed reprocessing plant, materials are recycled to the maximum extent practicable, to minimize the volume of effluents and reduce the cost of chemicals. Water and nitric acid evaporated from products and wastes are recycled. Nitrogen oxides are converted to nitric acid and reused. Solvent stripped of uranium and plutonium is cleaned of degradation products and contaminants in step 14 and reused.

The technology for the more important of these steps will be described in Secs. 4.3 through 4.13.

Figure 10.29 shows the principal steps in applying the Purex process to irradiated LMFBR fuel, step 7 of Fig. 10.28. The flow scheme and the compositions and locations of solvent, scrubbing, and stripping streams have been taken from the process flow sheet of a 1978 Oak Ridge report [Oil] describing a planned experimental reprocessing facility designed for 0.5 MT of uranium-plutonium fuel or 0.2 MT of uranium-plutonium-thorium fuel per day. As that report gave process flow rates only for the uranium-plutonium-thorium fuel, Fig. 10.29 does not give flow rates for the uranium-plutonium fuel of present interest. This flow sheet shows the codecontamination step, in which fission products are separated from uranium and plutonium; the partitioning step, which produces an aqueous stream of partially decontaminated

plutonium; and the uranium stripping step, which produces an aqueous stream of partially decontaminated uranium. The proposed facility has additional solvent extraction cycles, not shown, for completing decontamination.

Codecontamination. The codecontamination section consists of the HA extraction section equipped with short-contact-time centrifugal contactors and the HS scrubbing section equipped with pulse columns. In the HA section, uranium and plutonium in the aqueous feed and reflux from the HS section are extracted into the organic stream containing 30 v/o TBP. In the HS section any ruthenium extracted by TBP is scrubbed into the aqueous phase with З M HN03. Then any zirconium-niobium in the TBP is scrubbed with 0.3 M HN03. Scrubbing is at 50°C to enhance decontamination of ruthenium.

Stripping. This flow sheet uses hydroxylamine^ to reduce plutonium to inextractable Pu(III). Because the reduction of plutonium by hydroxylamine takes place almost entirely in the aqueous phase and requires many minutes for completion, before reduction it is necessary to return the uranium and plutonium in the organic phase leaving the HS contactor to the aqueous phase. This is done in two stages in the HC stripping column. In the bottom stage, plutonium is stripped with 0.3 M HN03. Lower acid concentration must be avoided because plutonium polymer would then form. In the top stage, solvent, now containing no plutonium, is stripped of uranium by 0.012 M HN03. Some hydroxylamine is added to this strippant to start reduction of plutonium in the HC unit.

Plutonium reduction. Reduction of plutonium to Pu(III) is completed by adding concentrated hydroxylamine (with hydrazine as holding reductant) to the aqueous raffinate leaving the HC column. The mixture must be held long enough, half an hour or more [B2], to complete the rather slow reduction to Pu(III). To hasten the reaction, the hydroxylamine concentration should be high and the nitric acid concentration as close to 0.3 M as possible without risking plutonium polymer formation.

Partitioning. Feed for partitioning is made З M in HN03 by addition of 13 M HN03 to reducing reactor effluent, to enhance extraction of uranium from inextractable Pu(III). In the plutonium partitioning pulse column 1A uranium is extracted from Pu(III) by 30 v/o TBP. In the uranium scrubbing section IB, any Pu(III) that may have been extracted with uranium and traces of extracted fission products are scrubbed with two aqueous streams, 3.1 M HN03 to remove ruthenium and 0.31 MHN03 to remove zirconium-niobium.

Uranium stripping. Uranium in solvent leaving the IB column is stripped into the aqueous phase by counterflowing 0.01 M HN03 in the 1C column. This is run at 50°C to reduce the uranium distribution coefficient.

Prevention of criticality. Because the plutonium content of feed to this LMFBR solvent extraction flow sheet is 10 times that of the Barnwell plant, Sec. 4.14, extra precautions must be taken to prevent criticality in the dissolver; the HA, HS, HC, and 1A contactors; and the plutonium reduction reactor. Addition of sufficient soluble poison to the feed will prevent criticality in the dissolver, feed adjustment tanks, and centrifugal HA contactors. The other sections of the plant processing plutonium must either have small enough dimensions to be

^Hydroxylamine is used for plutonium reduction instead of cathodic reduction as in the Barnwell flow sheet Fig. 10.11, because the plutonium/uranium ratio in this LMFBR fuel is 10 times that in LWR fuel and because electrolytic reduction has not been demonstrated for this high plutonium content.

subcritical (small-diameter columns or small-diameter or thin-slab tanks) or be provided with neutron-absorbing inserts such as boron steel plates or borosilicate glass Raschig rings. General procedures for guarding against criticality are discussed in Sec. 8.

Geologic containment. The salt domes in the northern part of Germany where the tentative site for a waste repository is located are one example of a geologic containment under considera­tion. They are more than a hundred million years old. It was only after the formation of these salt domes that America and Europe began to separate, forming the Atlantic Ocean, and that the Alps came into being. The salt domes withstood numerous geologic catastrophes without changing their shape or location. The area was three times covered by ocean water and dried again, vulcanism developed all over Germany, and later several glaciers moved across the salt. Nothing, however, happened to the salt formations, thereby proving that they were in perfect equilibrium with the geologic environment. As long as this equilibrium is not disturbed by human activities, it is extremely unlikely that the salt domes will undergo any changes. The utilization as waste repository will influence this equilibrium mechanically by mining the salt and thermally by charging it with heat sources. Geologists and mining engineers, however, have no doubt that this can be done without serious disturbance of the equilibrium. This gives a very high degree of confidence in the long-term integrity of the salt formation.

Miration of radionuclides. Even in the event of intrusion of groundwater into the waste repository, the low solubility of the waste, the slow motion of water at depth, the sorptive capacity of the soil, and the distance from the repository to water used by people provide additional protection against contamination of the environment.

Burkholder [B8] and others [FI] have developed models for analyzing the consequences of accidental intrusion of underground water into a geologic repository for HLW. In an example calculation [B8] it is assumed that the geologic medium has sorptive properties typical of U. S. western desert subsoil, that the waste material dissolves at the slow rate of from 0.03 to 0.003 percent per year, that the solution percolates unidirectionally through 10 km of sorptive soil at rates of from 1 to 10 m/year, and that the underground water is discharged into a river used for drinking. The general result is that nuclides that are not sorbed by the soil, e. g., tritium, 14C, 1291, and possibly "Те, reach the river within a few thousand years. Other radionuclides that are sorbed by the soil are delayed for a much longer period, e. g., over a million years for plutonium, and are attenuated by radioactive decay and dispersion.

Procedures for evaluating the rate of migration in the event of intrusion of water into an underground repository are detailed in the foregoing references.

2.2 Batch Decay

Batch decay is concerned with the radioactive decay of a given amount of initially pure parent material. The decay products will build up and, if radioactive, will later die away as time progresses. An example is the decay chain resulting from the radioactive disintegration of 211 Pb, which is itself a member of the radioactive decay scheme of 23SU. Starting with 211Pb, the decay chain is

Nuclide: 2УРЬ -£-+ 2УВІ — 2^T1 ««и,

Half-life: 36.1 min 2.15 min 4.79 min stable

Denote by subscript: 12 3 4

Suppose that N° atoms of 211 Pb are freshly purified at time zero and there are no sources of 211 Pb present. The net rate of change of the number of 211 Pb atoms is

The net rate of change of the number of 211 Bi atoms is

and the corresponding equations for 207T1 and “’Pb are

Figure 2.6 shows the change with time of the number of atoms of each nuclide in the 211 Pb decay chain, per initial atom N° of 211 Pb. Figure 2.7 shows the variation with time of the activity, or disintegration rate XN, of each nuclide and the total activity of the mixture, relative to the initial activity XN° of 2UPb.

In the general case of a radioactive decay chain

N! -*■ jVj -*N3 -*•——- ► Nj -*■ ■ • ■ -+Nі->■ ■ ■

in which the parent material is present in an amount JV? at time zero, if none of the other members of the decay chain is initially present, and if there are no other sources of the parent material, the amount Nt of any nuclide present at time t can be written by analogy to Eq. (2.15):

t

N, =iV? XIX2 • ■ • X,_! 2 ——————— O’ > 1) (2.17)

/=1 П (Xfc-X/)

k=l

k*i

Equation (2.17) is known as the Bateman [Bl] equation. It is derived in Sec. 7.

By superposition, the batch-decay equation can be further generalized for the case of arbitrary initial amounts Nf of any of the radionuclides in the chain:

When a radionuclide decays to a daughter of half-life much shorter than that of its parent, the daughter builds up to an amount that remains in constant ratio to the amount of the parent, and the amount of the daughter then decreases at a rate controlled by the half-life of the parent. In this case, the daughter is said to be in equilibrium with the parent, even though the amount of the parent radionuclide may be changing with time. For example, for the batch decay scheme that led to Eq. (2.14), suppose that X2>Xi, and assume that for times of interest 2t > 1. Equation (2.14), written in terms of decay rates, then reduces to

N22 * e-t (2.19a)

лг Xj

In the limit of X2 > i the daughter builds up to a concentration such that its decay rate is identical to that of the parent. This is the condition of transient equilibrium, i. e., from Eq. (2.19a):

N22 «Л^Х, (2.19b)

Transient equilibrium is reached by 211 Bi from the batch decay of 211 Pb, as illustrated in Fig. 2.7. The time to reach this transient equilibrium is a few times the half-life of 211 Bi. The activities of 211 Bi and 2UPb would approach secular equilibrium, i. e., equal activities, if the ratio of the half-life of 211 Pb to that of 211 Bi were even greater. The second daughter, 207Tl, can also be said to be in transient equilibrium with 211 Pb, at times much greater than I/O* +X3), because both its half-life and that of its immediate precursor are both short compared with the 211 Pb half-life.

For this fuel-cycle analysis, it is convenient to specify a reactor design that satisfies a neutron balance and contains a charge of fresh fuel distributed uniformly throughout the reactor core.

Consider a portion of the reactor large enough to contain a representative sample of the entire contents of the reactor core. In a homogeneous reactor this might be any small amount of core volume; in a heterogeneous reactor it will be a region large enough to contain at least one set of repeating elements of core lattice structure.

We shall assume that a representative unit volume of this design contains Nm atoms of a single fissile species (e. g., 235 U) with absorption cross section and Ng atoms of a single fertile material (e. g., 238U) with absorption cross section Og. The unit volume is assumed also to contain steady-state amounts of 135Xe, 149 Sm, and other fission products with cross sections above 10,000 b, which build up to equilibrium concentration in a few days at the neutron fluxes typical of power reactors. It is assumed that no other fission products are present to an extent sufficient to affect the neutron balance. The items that affect the thermal-neutron
balance for this reference design condition are listed in Table 3.9. The cross sections are effective values averaged over the energy distribution of neutron flux.

The thermal-neutron balance equation for the reference design condition states that the rate of production of thermal neutrons equals the rate of consumption:

vtiePtbPNMoU = + Nt, oU + М*о*ф +

P

+ ЛМеФ + £ ^о*5ф + Niи%ф (3.24)

s

The amount of control absorber present in the reactor Ng is set by the reactor operator so as to keep the reactor just critical at the desired power level.

The reactivity of the fuel in the reference design reactor p* is defined as the ratio of the rate of absorption of thermal neutrons in control absorbers to the rate of production of thermal neutrons:

* _ Ме°еФ

vttePthpN^

ivitePthP-WMOH-Nfo;-lN? o}-NZeo$e — I N^-DB2* ______________ p s

PMePttiPNMaM