Figure 3.14 shows the total steady-state fuel-cycle cost for an interval of 1.0 year between refuelings as a function of feed enrichment for batch fractions, /, of |, 5, J, and j. The batch fraction is defined as 1 jn, where n is the number of fuel zones. Also plotted in this figure are levels of constant energy production (E) or capacity factor (L1) and lines of constant burnup (B). The unit costs of fuel-cycle materials and services are those anticipated for the year 1980, to be described in more detail in Sec. 5.

To illustrate use of Fig. 3.14, the example of the line Ґ =0.9 will be discussed. Suppose that this 1060-MWe reactor is expected to operate at an availability-based capacity factor L’ = 0.9 with a 1-year interval between refuelings. The minimum fuel-cycle cost of $41 million will occur at a batch fraction /= j and a feed enrichment of 3.75 w/o MSU. This will require fuel to sustain an average burnup В of slightly over 40,000 MWd/MT. If average burnup should be limited for mechanical reasons to slightly over 30,000 MWd/MT, the minimum fuel cycle cost of $42 million will occur at /= f and a feed enrichment of 3.2 w/o, the combination suggested by the manufacturer for this reactor.

Figure 3.15 shows the unit fuel-cycle cost in mills per kilowatt-hour as a function of the same variables. This unit cost is obtained by dividing the total cost in dollars by the electric energy in megawatt-hours. For example, the unit cost at L’= 0.9 and /=5 is $41,000,000/7317 X 103 MWh = 5.6 $/MWh or 5.6 mills/kWh. Because of the overlap of lines,

Figure 3.14 Effect of enrichment and batch fraction on total fuel cycle cost _l per steady-state cycle, electric energy per 5 cycle (£), availability-based capacity factor (£’), and bumup (В).

Figure 3.15 Effect of enrichment and batch fraction on steady-state unit fuel-cycle cost.

representation of unit costs in Fig. 3.15 does not bring out the effect of the several variables on costs as well as representation of total costs in Fig. 3.14.

2.4 Capture Reactions

In fission reactors the transmutation reactions of principal importance involving neutrons are capture and fission. All nuclides (except 4 He) take part in the radiative capture reaction (n, y), an example of which is

235ir 4. 1 „ 236it 1

ю U + 0П-* 92U + o7

This reaction produces an isotope of the reacting nuclide with mass number increased by unity and one or more gamma rays, which carry off most of the energy of the reaction. Other capture reactions, possible for a few nuclides (mostly those of low mass number), result in emission of an alpha particle (n, a):

*?B + In -* ^Li + tHe

or a proton (и, p):

ІО + in ■* ‘?N + ІН

2.5 Fission Reactions

The fission reaction is responsible for the sustained production of neutrons in a nuclear reactor and for most of the energy released. In this reaction, one neutron is absorbed by a heavy nuclide, which then splits into two nuclides each in the middle third of the periodic table, and several neutrons, which are available for initiating additional fissions. All elements beyond lead undergo fission with neutrons of sufficiently high energy; the only readily available long-lived nuclides that undergo fission with thermal neutrons are 233U, 23SU, 239Pu, and 241 Pu.

An example of the fission of 235 U into 144 Xe and 89 Sr has already been given. The fission reaction may take place in a number of alternative ways. Light fragments have been observed to have mass numbers from 72 to 118, heavy fragments from 118 to 162. At a given mass number fragments have also been observed with atomic numbers varying over a range of three or more. For example, 133Te, 133I, and 133Xe have all been observed as primary fission fragments. Finally, the number of neutrons produced in an individual fission event may be anywhere from zero to four or more. As a result, a large number of alternative fission reactions take place, of the general form

+ o" z,’ L + z’H + x’on

where L and H denote the light and heavy fission fragments, respectively, , A,, and X all vary between limits, and Z2 and A2 are determined by the conditions

Zt +Z 2 = 92

A, +A2 + X = 236

To illustrate the characteristics of different methods of fuel management, the example of the 1054-MWe PWR designed by the Westinghouse Electric Company for the Donald C. Cook Nuclear Plant of the American Electric Power System [A1 ] has been used.

We shall be interested in estimating the fuel-cycle performance of this reactor during a steady-state cycle, in which fresh fuel has a 235 U enrichment of 3.2 w/o 235 U and spent fuel has a 23SU enrichment of around 1.0 w/o and also contains around 0.6 w/o fissile plutonium. The reference design condition used to evaluate effective neutron cross sections and other reactor physics parameters during irradiation is taken to be 2.7 w/o 235 U. This value, slightly higher than the arithmetic mean of the fissile content of fuel at the beginning and end of irradiation, is intended to reflect the higher cross section of fissile plutonium compared with 235 U.

Table 3.11 gives the properties of each region of the core lattice for the reference design of this reactor. Information to be used directly in neutron balances and other fuel-cycle calculations are the volume fraction of each region v, the concentration N of each molecular or nuclear species in that region, and the ratio of the thermal flux in that region to the thermal flux in the fuel region ф. The rate of reaction of thermal neutrons with species ; in region j per unit lattice volume is Ща^ф/ф, where

Ny is the number of atoms or molecules of species / per unit volume of region /

Oj is the effective cross section of species і for thermal neutrons Vj is the volume fraction of region j in the lattice

ф/ is the ratio of thermal-neutron flux in region / to thermal flux in the fuel ф is the thermal-neutron flux in the feel region

Characteristics assumed for this reactor in the reference design condition are listed in Table 3.12.

Table 3.13 gives effective cross sections for thermal neutrons and other nuclear properties of the materials in the core of this reactor. These effective cross sections have been calculated by the procedure recommended by Westcott, which has been outlined in Chap. 2, from data provided by Westcott [W3] and Critoph [С1]. To obtain appropriate nuclear reaction rates, these effective cross sections are to be multiplied by the thermal-neutron flux, ямвіі, where Имв is the density of neutrons in the Maxwell-Boltzmann part of the spectrum and 0 is the average speed of the Maxwell-Boltzmann neutrons.

(3.37)

The value of 80 b given for fission-product pairs is an approximate, constant value to be used independent of the fuel from which the fission products are formed and independent of the flux time to which the fission products are exposed after formation. For the present PWR, the effect of these variables on the cross sections of fission-product pairs, evaluated by extrapolation of Walker’s [Wl] tables, is as follows:

As irradiation progresses, the effective cross section of fission products from each fuel nuclide decreases. This decrease is partially offset by greater production of fission products from plutonium, which have a higher cross section than those from 235 U. The constant value of aF = 80 given in Table 3.13 takes these two effects approximately into account.

Table 3.14 gives the neutron balance for this reactor in the reference design condition, charged with U02 fuel containing 2.7 w/o 435 U uniformly distributed, operating at full power of 3250 MW, and with equilibrium xenon and samarium. Items 1 through 5 deal with events experienced by neutrons with energies high enough to cause fission in 338 U, above 1 MeV. Items 6 through 10 deal with events experienced by neutrons while being slowed down from fission to thermal energies. Items 11 through 23 give the production and consumption of thermal neutrons.

One significant feature of Table 3.14 is that the production of thermal neutrons, item 11, equals the total consumption, item 23. A second point to be noted is that the reactivity is given by the ratio of item 22 to item 23.

0 2112

p = ud = 0A349 (338)

A third point is that the constant term К in Eq. (3.33) is the sum of items 12, 15, 16, 17, 18, 19, and 20.

К = 0.0008 + 0.1431 + 0.0390 + 0.0043 + 0.0216 + 0.0724 + 0.0143 = 0.2955 (3.39)

Finally, the initial conversion ratio ICR, the ratio of the atoms of 239 Pu produced per atom of 235 U consumed, is the sum of items 2, 8, and 15.

ICR = 0.0111 + 0.4619 + 0.1431 = 0.6161 (3.40)

These terms determine the rate at which plutonium builds up during irradiation. One reason for giving the neutron balance in such detail is to be able to evaluate these terms.

1 INTRODUCTION

The production of power from controlled nuclear fission of heavy elements is the most important technical application of nuclear reactions at the present time. This is so because the world’s reserves of energy in the nuclear fuels uranium and thorium greatly exceed the energy reserves in all the coal, oil, and gas in the world [HI], because the energy of nuclear fuels is in a form far more intense and concentrated than in conventional fuels, and because in many parts of the world power can be produced as economically from nuclear fission as from the combustion of conventional fuels.

The establishment of a nuclear power industry based on fission reactors involves the production of a number of materials that have only recently acquired commercial importance, notably uranium, thorium, zirconium, and heavy water, and on the operation of a number of novel chemical engineering processes, including isotope separation, separation of metals by solvent extraction, and the separation and purification of intensely radioactive materials on a large scale. This text is concerned primarily with methods for producing the special materials used in nuclear fission reactors and with processes for separating isotopes and reclaiming radioactive fuel discharged from nuclear reactors.

This chapter gives a brief account of the nuclear fission reaction and the most important fissile fuels. It continues with a short description of a typical nuclear power plant and outlines the characteristics of the principal reactor types proposed for nuclear power generation. It sketches the principal fuel cycles for nuclear power plants and points out the chemical engineering processes needed to make these fuel cycles feasible and economical. The chapter concludes with an outline of another process that may some day become of practical importance for the production of power: the controlled fusion of light elements. The fusion process makes use of rare isotopes of hydrogen and lithium, which may be produced by isotope separation methods analogous to those used for materials for fission reactors. As isotope separation processes are of such importance in nuclear chemical engineering, they are discussed briefly in this chapter and in some detail in the last three chapters of this book.

Figure 1.1 Fission of 535 U nucleus by neutron.

The equations of Sec. 6.2 give the number of atoms of each fission product after a reactor has been run at stated conditions for a specified time. If the reactor is then shut down, the fission products build up and decay in accordance with the laws of simple radioactive decay, which were outlined in Sec. 3. If the nuclides in the decay chain are removed only by radioactive decay during reactor operations, the equations of Sec. 3 describe the changes with time of the number of atoms of any nuclide in the decay chain. If a member of a fission-product decay chain or its precursors in the decay chain are removed by neutron absorption, equations for the amount of each nuclide present at time t after shutdown may be obtained by applying the equations of radioactive decay to the amount present at shutdown.

We may illustrate by calculating the number of atoms of 1351 and 133 Xe present in a reactor that had been operated at a flux 0 long enough to build up a steady-state content of
1351 and 135Xe, and then shut down for a time t. No removal by processing is assumed, so that fi and /хе are equal to zero. The steady-state contents of 1351 and 13sXe have already been obtained as Eqs. (2.117) and (2.118), respectively. The number of N of 135I atoms remaining at time t is obtained by applying Eq. (2.13), with N = N*:

Af,=JV, Vxi’ (2.120)

Similarly, the number of iVXe of 135 Xe atoms present at time t is obtained by applying Eq. (2.18), with jVi° = N* and 7V&e = N£e:

e-*it — e-*xet

Nxe = Kh Ц———— Ц—— + Л&е-**.’ (2.121)

лХе ~ ЛІ

Substituting Eqs. (2.117) and (2.118) into (2.121):

where ф is the neutron flux that existed prior to shutdown.

The transient poisoning ratio, which is the ratio of the neutron absorption in 135 Xe to fission absorption if the reactor is to be started up again after shutdown time t, is obtained from Eq. (2.122):

Figure 2.16 Xenon poison ratio during reactor operation at constant flux and after shutdown.

The increase is caused by the sudden reduction in the overall removal rate constant for xenon when the reactor is shut down, whereas the rate of production of xenon from its main source, the decay of 1351, decreases only slowly with time as the iodine decays. For low neutron fluxes (0< 1013) prior to shutdown the xenon buildup after shutdown is less important because the xenon burnout by neutron capture is then small relative to xenon removal by radioactive decay.

For a metallic element to be extractable by an organic solvent immiscible with water, it appears to be necessary that the element be capable of forming an organic-soluble, electrically neutral complex compound with the solvent or with an added complexing agent. The compound

between uranyl nitrate and tri-n-butyl phosphate (TBP) is an example of the first kind, the compound between thorium nitrate and salicylic acid, of the second.

Formation of these extractable complexes involves coordination bonds with the metal cation, i. e., the sharing of electrons from the complexing agent to complete previously unfilled orbits of the cation. The alkalies and alkaline earths are not easily capable of forming such compounds because they have no empty electron orbits, and hence cannot be readily extracted with organic solvents immiscible with water. On the other hand, elements of the transition groups, such as the rare earths, uranium and the other actinides, iron, nickel, and cobalt, form coordination compounds with ease and are readily extracted by organic solvents immiscible with water.

The complex compounds formed by the metal cations in solvent extraction systems are illustrated first by the formation of complex compounds between cations and neutral molecules in aqueous solution. An example of an equilibrium reaction involving such formation is [P2]

Ag+ + 2NH3 * Ag(NH3V

The complex compound Ag(NH3)2+ is easily destroyed, i. e., the reaction is reversed, by adding acid to remove dissolved NH3 or by adding a halide ion to precipitate silver halide.

Examples of stronger complexes formed with anions are [P2]

Fe2+ + 6CN" — Fe(CN)64′

Pt2+ + 4СГ ^ PtCU2*

Cations that form such complexes are characterized by a coordination number, i. e., the number of complexing groups that are attached to the cation. Cations in the above reactions exhibit coordination numbers of 2 for Ag+, 6 for Fe2+, and 4 for Pt2+.

Compounds with less than full coordination are also formed, such as

Th4+ + N03" ^ ThN033+

and Th4+ + 2N03 ‘ — Th(N03 )2 2+

with the amount of Th(N03)22+ complex increasing with N03” concentration. In a cation — anion complex with less than full coordination bonding from the anion, the cation may become fully coordinated by adding water molecules to the complex. Complexes formed by anions of weak acids are usually more stable than complexes formed by anions of strong acids.

When an organic ion or molecule is able to form coordination bonds with a metallic cation in more than one place in the organic molecule, an especially stable complex called a chelate compound is formed. Among the effective chelating agents are molecules that contain two ketone structures, such as the 1,3-diketones with the structure

-C-CH2-C-

II II

0 0

A molecule containing the diketone group forms a heterocyclic ring coordination compound with a metal cation by losing a proton and attaching itself to the cation through both oxygen atoms. The number of chelating molecules added is thus half the coordination number of the cation in the complex. A chelating agent of common use in laboratory extraction is the 1,3-diketone thenoyltrifluoracetone (TTA), with the enol form

HC-CH

II II

НС С—C—CH=C-CF3

/ II I

SO OH

Representing the organic chelating compound by HK, the overall reaction involved in the chelate extraction of a metal in the ionic form Mn+ is

Mn+(aq) + nHK(o) ^ MK(o) + nH*(qq)

with an equilibrium constant К given by

= [MK(o)] [H>?)]n (4 n

[Mn+(aq)] [HK(o)J« ^ ‘

In these equations aq denotes the aqueous phase and о the organic. Quantities in brackets are the activities at equilibrium, i. e., the molar concentration of the indicated species times its activity coefficient. For example:

Th4+ + 4HK ^ TT1K4 + 4H+

and U02 2+ + 2HK — U02K2 + 2H+

As indicated by Eq. (4.1), the relative concentration of the metal chelate in the organic phase at equilibrium decreases with increasing concentration of hydrogen ion in the aqueous phase. When an aqueous solution containing extractable metal ions is contacted with an immiscible organic carrier, such as benzene, containing dissolved chelating agent, the chelating compound must dissolve in the aqueous phase, ionize, and react with the metal ion, and the metal chelate then dissolves in the organic phase. The low solubility of the chelates and their slow rates of formation limit the industrial-scale application of chelate separation [C8, S4].

More rapid extracting reactions result from the formation of relatively loose nonchelating complexes with organic molecules. A widely used organic complexing agent for the extraction of the actinide elements thorium, uranium, neptunium, and plutonium is TBP, which probably forms bonds by the electron from the phosphoryl oxygen atom in the structure [S4]

(BuO)3P+-Q-

where BuO denotes the butoxy group. Examples of overall extracting reactions involving covalent bonds with TBP are

U022+(a?) + 2N03′(a<?) + 2TBP(o) * U02(N03)2 -2TBP(o)

Ри4+И) + 4N03’H) + 2TBP(o) — Pu(N03)4-2TBP(o)

The above reactions are shifted to the right, thereby increasing the relative amount of metal cation in the organic phase, by increasing the concentration of uncombined TBP in the organic phase and by increasing the concentration of aqueous nitrate ion. The latter is accomplished by adding a salting agent such as HN03 or A1(N03)3. Nitric acid also forms a hydrogen bonded complex [S2] with TBP and extracts according to the overall reaction

Haq) + N03′(a?) + TBP(o) =* HN03-TBP(o)

Similarly, TBP promotes solubility of water in the organic phase.

Subsequent chapters include discussion of TBP solvent extraction to purify uranium (Chap. 5) and thorium (Chap. 6) and to separate and recover actinides in irradiated reactor fuel (Chap. 10). Also discussed in Chap. 5 is a third type of organic extractant, consisting of an organic-soluble acid or base, of moderately high molecular weight, which extracts metals as simple or complex organic-soluble salts [P2], appropriately characterized as a “liquid ion exchanger.” Examples are di(2-ethylhexyl) phosphoric acid and trioctylamine, both used in extracting uranium from ore leach liquors (cf. Table 5.14).

It is characteristic of nuclear reactions of the type occurring in nuclear reactors that the sum of the number of neutrons and protons in the reactants equals the sum in the products. The same is true of the charge of the reactants and products. Consequently, in balancing nuclear reactions, the sum of the A’s of the reactants must equal the sum of the A’s of the products; and the sum of the Z’s of the reactants must equal the sum of the Z’s of the products. As an example of a balanced equation for a nuclear reaction, we may consider one of the fission reactions that occurs when 235U absorbs a neutron:

+ In — *JJXe + fiSr + 3in

The neutron is represented by Jn, a nuclide with nuclear charge 0 and mass number 1.

2 RADIOACTIVITY

2.1 Types

Radioactive nuclides break down spontaneously in six principal ways, illustrated by the following examples:

f|Sr -* “Rb + ?e (positron)

e + _?e -*• 2q7 (0.51 MeV photons)

5. Electron capture:

“Sr + _?e -*• fjRb + x-rays

6. Spontaneous fission:

HSCf -*■ fission products + neutrons

Some nuclides may decay alternatively in more than one way. For example, 14 percent of “““Kr decays by emission of a gamma ray, according to the above equation, and 86 percent decays by emission of a beta particle to form 85 Rb.

Sections 3.4 and 3.5 have dealt with an idealized situation in which a PWR is operating in the steady state with an exact fraction (e. g., one-third) of the fuel replaced at each refueling. A real reactor seldom reaches a steady-state condition and may have a number of fuel assemblies that cannot be divided evenly into fuel zones containing equal numbers of assemblies. The purpose of this section is to describe briefly a real reactor and the results of a computer-based analysis of the fuel-cycle performance of this reactor through a succession of cycles.

2.2 Reactor Construction

The reactor to be discussed is the large PWR manufactured by the Westinghouse Electric Company, which has been built for the Diablo Canyon station of the Pacific Gas & Electric Company, the Donald C. Cook station of American Electric Power Corporation, and the Zion station of Commonwealth Edison Company. Rated capacities of 3250 MW (thermal) and 1060 MW (electric) have been used. The following brief description of this reactor was abstracted from the Safety Analysis Report of the Donald C. Cook station [А1].

Figure 3.16 is a cutaway view of this reactor. The reactor vessel is a cylinder 13 ft in diameter with an ellipsoidal bottom. The top of the vessel is closed with a flanged and bolted ellipsoidal head, which is removed for refueling. When in operation the reactor is filled with water at a pressure of 155 bar (15.5 MPa). The water enters the inlet nozzle at the left at a temperature of 282°C and leaves the outlet nozzle at the right at 317°C. The effective average temperature of the water is 301.6°C, which will be taken as the temperature of the Maxwell-Boltzmann component of the neutron flux.

There are 193 fuel assemblies held between the upper and the lower core plates. Figure

3.17 is a horizontal cross section through the portion of the reactor containing the assemblies. Inlet water flows down in the two annular spaces between the reactor vessel and the core barrel, turns at the bottom of the vessel, and flows upward through the fuel assemblies inside the core baffle.

Figure 3.18 is a dimensioned horizontal cross section of one fuel assembly. The assembly consists of a 15 X 15 square array of zircaloy-4 tubes set on 0.563-in square pitch. Two — hundred four of these tubes are filled with U02 pellets, pressurized with helium and closed with welded zircaloy end plugs. The zircaloy cladding for these fuel tubes is 0.422 in outside diameter, with a 0.0243-in wall. The overall length of tubing filled with U02 is 12 ft. At 20 points in the fuel assembly, zircaloy-4 guide tubes are provided for control rods. During normal operation these tubes are filled with water, burnable poison rods, or movable control rods. The central position in the fuel assembly is occupied by a zircaloy thimble for in-core instrumenta­tion. It is sealed off from the water that surrounds the fuel assembly. The 225 zircaloy tubes of the assembly are held in place over their length by nine evenly spaced spring clip grids made of Inconel-718.

The mass of zircaloy in guide tubes and instrument thimble is 9.5 kg/assembly, and the mass of Inconel is 8.6 kg.

The reactor core consists of 193 fuel assemblies mounted on 8.466-in-square pitch. The initial loading of fuel and control poison in the core of this reactor is shown in Fig. 3.19. Fuel assemblies marked M are provided with movable control rods that can be inserted or withdrawn by control rod drives that enter through the head of the reactor vessel (Fig. 3.16). The numbers

193 FUEL ASSEMBLIES

(8, 9, 12, 16, or 20) placed at other fuel positions give the number of fixed burnable poison rods containing boron carbide placed in the indicated assembly during the first fuel cycle. During normal operation at full power, the movable control rods are fully withdrawn. Long-term reactivity changes are controlled by depletion of the burnable poison and by adjusting the concentration of boric acid dissolved in the cooling water.

The number of nuclei reacting in a specified way with neutrons in unit time is proportional to the number of nuclei present and to the concentration of neutrons. In the language of chemical kinetics, neutron reactions are first-order with respect to concentration of nuclei and neutrons, and it is because neutron reactions are simple first-order irreversible processes that a very detailed quantitative treatment of the rate processes in a nuclear reactor can be given.

The expression for the rate of reaction of neutrons with reacting nuclei N is

where n is the concentration of neutrons, in number per unit volume, and KR is the specific rate constant. It has become customary to express KR as the product of another constant a, called the cross section, and the neutron speed v, so that Eq. (2.41) becomes

The product vn is termed the neutron flux ф and is the measure most commonly used to describe the neutron intensity in a reactor.

For a given neutron density n and speed u, the product оф is the first-order rate constant and is the fraction of the reacting nuclei consumed by the reaction per unit time. It plays the same role in rate equations as the radioactive decay constant X.

We are now in position to derive equations that will give the degree of bumup nuclear fuel can experience before it ceases to be critical. First, we must determine how the concentration of each nuclide that affects the neutron balance changes with time. We consider fuel that at time zero contains N$s atoms of 235 U per cubic centimeter, atoms of 238 U, and no other uranium isotopes, plutonium, or fission products. This fuel is then exposed to a thermal — neutron flux <p(t), which may be a function of time. The variation in concentration of each nuclide in this fuel with time is obtained as follows.

*To be used with Westcott [W3, Cl) s factor s2.

+To designate isotopes of uranium, plutonium, and other actinide elements, it has become conventional to use two-digit subscripts, such as 49 for 239 Pu, in which the first digit is the atomic number minus 90 and the second digit is the last digit of the mass number.

* These values of v are from [ С1 ]. They are used here because effective cross sections are from [Cl ] also. These values of v differ slightly from App. C.

® In fast fission.

‘ Pairs of fission products with cross sections less than 10,000 b.

235 U. The rate at which 235 U is consumed is

^25 = N? s Є’”»»

The flux time is the fundamental variable used to express the extent of exposure to irradiation. Even when the flux varies with time, Eq. (3.44) is strictly correct. The flux time thus defined is in units of neutrons per square centimeter. Expressed in these units its magnitude is around 1021 in a typical reactor, as when fuel has been irradiated in a flux of 1014 «/(cm2 — s) for 107 s. Consequently, it has become customary to express flux time in units of 1021 njcm2, termed neutrons per kilobarn and written «/kb. The flux time is often called the fluence.

236 U. 236 U is produced by capture of neutrons in 235 U at the rate per unit volume of ■^25^250^25/(1 + a2s)> where a25 is the ratio of neutrons captured by 233U to neutrons producing fission in 23SU. The rate of consumption of 236 U by neutron capture is TV26a260. Hence the net rate of change of 236 U concentration is

^26 _ A25 а25а25 0 лт n j.

—~T————— . ■■■,’“——— f*26°26 0

dt 1 + a2S

Plutonium isotopes. The net rate of formation of 239Pu is

Absorption of thermal neutrons in 239 Pu

4~ . ;28————— Г (V2S^2S°2S + T)49^49049 + T)4N4i 04l )ф (3.47)

1 + «2S 728 — 1 (Cmt )

Absorption of fast neutrons from 23SU, 239 Pu, and 241 Pu in 238 U

This equation may be written as

^g9 = Л^28028 4- «25^25025 — 7^49CT49 4" K41/V41O41 (3.48)

where кт = т? шеЛ (1 — p) 4- nm —3®——————- —

1 4- a28 728 — 1

7 = 1 — K49

To solve this equation exactly, the dependence of 241 Pu concentration, TV4i, on flux time must be derived. This is obtained by considering the rate of change of concentration of 240 Pu,

dNiO _ <*49^49 °49 _ „ _

j/j——— і "Vі,——— JV40O40

do 1 + «49

and of 241 Pu,’1′

To obtain an exact solution, Eqs. (3.48), (3.49), and (3.50) should be solved simultaneously, with the dependence of N25 on flux time given by (3.44). The solution is of the form

jV49 = Л&(/449 + B49e’s’e + C49e-S’e + D49e-S>e)

+ NtsiE^e-0»9 + F49e~s’e + Gne-W +H49e-S>e) (3.51)

where Su S2, and 53 are the three roots of the cubic equation

In this section, an approximate solution is derived that does not require finding the roots of the cubic equation (3.52).

^Equation (3.50) neglects the term —Х41Л^4і ІФ representing the decay of 13.2-yr 241 Pu to 241 Am. To have included this term would have made it necessary to use the time t as independent variable rather than the flux time 6, which would have greatly increased the complexity of integrating these equations. The magnitude of X*! /ф in bams for 241 Pu (half-life 13.2 year) in a typical PWR (ф— 5 X 1013 n/(cm2 "s) is

1024 x______________ 0-693_____________ = ЗО h

(13.2)(3.15 X 107)(5 X 1013)

This is small compared with the neutron-absorption cross section for 241 Pu, which is o41 = 1377 b.

The approximation consists in neglecting the formation of 239Pu by absorption of resonance neutrons from 141 Pu, a procedure justified as long as k4iJV4i04i <укА9а^. With this approximation Eq. (3.48) reduces to

=^SS°2S + K2sNif02S — yNnqOy) (3.53)

With 5 given by (3.44), the solution of this equation, subject to iV49 = 0 at t = 0, is ЛГ49 = C, + C2e-a»e — (С, + С2)е-а‘^

jn — А’мОді 1 0497

C = K2s^2sa2s 0497 ~ °25

With jV4, given by (3.54), the solution of Eq. (3.49), subject to Nw = 0 at t = 0, is

242Pu. Consumption of 242Pu by absorption of neutrons will be neglected. The rate of accumulation of 242 Pu then is given by

dNn _ tt4iM)i Q4i0

dt 1 + a41

With iV4] given by (3.61), the solution of Eq. (3.57), subject to N42 = 0 at t = 0, is

«41041 Le + C7(l — е~°1ів) + C,( 1 — e-W»)

1+“41 |_ * 025 0497

+ C9(l — e~ff»9) _ (C« + C7 + C8 + C9)(l — e-°«’9)

040 041

Fission products. Burnout of fission products by neutron absorption will be neglected. The rate of formation of fission-product pairs from 235 U is

dNp{25) _ .’V^s 07s Ф

dt 1 + «25

With N25 given by (3.44), the solution of Eq. (3.68), subject to Np(25) = 0 at t = 0, is

v® n — е~°кв}

Np(25) = .g—— ^

1 + а25

The rate of formation of fission-product pairs from 239 Pu is

dNpi.49) Л49 CJ49 ф

Л 1 + «49

With N#) given by (3.54), the solution of this equation, subject to Np(49) = 0 at t — 0, is

The rate of formation of fission-product pairs from 241 Pu is

dNp(41) = A^4i o4i Ф

dt 1 + «49

By comparing this equation with Eq. (3.66) for 242Pu, we see that

The term for capture of resonance and fast neutrons from fission of 241 Pu has been omitted to be consistent with the approximation used in Eq. (3.53).

Checks of numerical woik. Two equations useful in checking numerical calculation of nuclide concentrations are

and TV2°8 — TV28 = Nf(28) + TV4, + AV(49) + TV«o + Ntl + TV42 + 7VF(41) (3.78)

Burnup. For this chapter we shall assume that the heat of fission is 0.95 MWd/g fissioned, or

9.5 X 10s MWd/MT. This corresponds roughly to a heat of fission of 200 MeV for *®*U. Fuel burnup B, in megawatt-days per metric ton, is thus related to the weight fraction w of fuel fissioned by

В = 9.5 X 10stv (3.79)

The weight fraction fissioned is

… _ 235TVF(25) + 238TVF(28) + 239TVF(49) + 241TVH41)

235jV°5 + 238Л& (3’80)

Bumup can be measured experimentally, either from the amount of heat liberated in a particular fuel element or from the amount of fission products found in it. Flux time, on the other hand, is much less readily determined experimentally and is subject to more ambiguity because of the various ways in which neutron flux may be defined. Whereas flux time is the natural independent variable to characterize fuel exposure in calculations, it is preferable to use burnup in describing practical situations.