Chapter 11. Neutron Chain Reactions

The possibility in a mass of nuclear fuel such as uranium depends on (a) nuclear properties such as cross sections and neutrons produced per absorption (Section 6.3) and (b) the size, shape, and arrangement of the materials.

11.1. Criticality and Multiplication

To achieve a self-sustaining chain reaction, one needing no external neutron supply, a “critical mass” of uranium must be collected. To appreciate this requirement we visualize the simplest nuclear reactor, consisting of a metal sphere of uranium-235. Suppose that it consists of only one atom of U-235. If it absorbs a neutron and fissions, the resultant neutrons do nothing further, there being no more fuel. If instead we have a small chunk of uranium, say a few grams, the introduction of a neutron might set off a chain of several reactions, producing more neutrons, but most of them would escape through the surface of the body, a process called leakage. Such an amount of fuel is said to be “subcritical.” Now if we bring together approximately 50 kg of U-235 metal, the neutron production balances the leakage losses, and the system is self-sustaining or “critical.” The size is the critical volume and the amount of fuel is the critical mass. Neutrons had to be introduced to start the chain reaction, but the number is maintained without further additions. The term “critical mass” has become popular to describe any collection of entities large enough to operate independently.

Figure 11.1 shows the highly enriched metal assembly Lady Godiva, so named because it was “bare” (i.e., it had no surrounding materials). It was used for test purposes for a number of years at Los Alamos. If we add still more uranium to the 50 kg required for criticality, more neutrons are produced than are lost, the neutron population increases, and the reactor is “supercritical.” Early nuclear weapons involved the use of such masses, in which the rapid growth of neutrons and resulting fission heat caused a violent explosion.

Figure 11.1. Fast metal assembly “Lady Godiva.”

11.2. Multiplication Factors

The behavior of neutrons in a nuclear reactor can be understood through analogy with populations of living organisms; for example, of human beings. There are two ways to look at changes in numbers of people: as individuals and as a group. A person is born and throughout life has various chances of fatal illness or accident. On average the life expectancy at birth might be 75 years, according to statistical data. An individual may die without an heir, with one, or with many. If on average there is exactly 1, the population is constant. From the other viewpoint, if the rates of birth and death are the same in a group of people, the population is again steady. If there are more births than deaths by 1% per year, the population will grow accordingly. This approach emphasizes the competition of process rates.

The same ideas apply to neutrons in a multiplying assembly. We can focus attention on a typical neutron that was born in fission and has various chances of dropping out of the cycle because of leakage and absorption in other materials besides fuel. On the other hand we can compare the reaction rates for the processes of neutron absorption, fission, and leakage to see whether the number of neutrons is increasing, steady, or decreasing. Each of the methods has its merits for purposes of discussion, analysis, and calculation.

For any arrangement of nuclear fuel and other materials, a single number k tells the net number of neutrons per initial neutron, accounting for all losses and reproduction by fission. The state of the system can be summarized by:

The design and operation of all reactors is focused on k or on related quantities such as δk = k − 1, called delta-k, or δk/k, called reactivity, symbolized by ρ. The choice of materials and size is made to assure that k has the desired value. For safe storage of fissionable material k should be well below 1. In the critical experiment, a process of bringing materials together with a neutron source present, observations on neutron flux are made to yield estimates of k. During operation, variations in k are made as needed by adjustments of neutron-absorbing rods or dispersed chemicals. Eventually, in the operation of the reactor, enough fuel is consumed to bring k below 1 regardless of adjustments in control materials, and the reactor must shut down for refueling.

We can develop a formula for k for our uranium metal assembly with the statistical approach. As in Figure 11.2(A), a neutron may escape on first flight from the sphere, because mean free paths for fast neutrons are rather long. Another neutron (B) may make one scattering collision and then escape. Other neutrons may collide and be absorbed either (C) to form U-236 or (D) to cause fission, the latter yielding three new neutrons in this case. Still other neutrons may make several collisions before leakage or absorption takes place. The statistical nature of the process is revealed by the application of Computer Exercise 11.D, which involves the program SLOWINGS. Scattering, absorption, and escape are simulated with a Monte Carlo technique. A “flow diagram” as in Figure 11.3 is useful to describe the various fates. The boxes represent processes; the circles represent the numbers of neutrons at each stage.

Figure 11.2. Neutron histories.

Figure 11.3. Neutron cycle in metal assembly.

The fractions of absorbed neutrons that form U-236 and that cause fission, respectively, are the ratios of the cross section for capture σ^c and fission σ^f to that for absorption σ^a. The average number of neutrons produced by fission is ν. Now let η be the combination νσ^f/σ^a, and note that it is the number of neutrons per absorption in uranium. Thus letting be the fraction not escaping by leakage,

The system is critical if k = 1, or η = 1. Measurements show that η is approximately 2.2 for fast neutrons, thus must be 1/(2.2) = 0.45, which says that as many as 45% of the neutrons must remain in the sphere, whereas no more than 55% escape through its boundary.

Let us now examine more closely , the nonleakage factor, coming from the process of neutron loss through the surface of a reactor core without reflector. Leakage depends on scattering collisions and on the size and shape of the core. We would expect that the amount of neutron leakage depends on the ratio of surface to volume, because production occurs within the core and losses occur at the boundary. For a sphere, for example, the volume is V = (4/3)πR³ and the surface area is S = 4 πR², so the ratio is S/V = 3/R. As it turns out from the theory of neutron diffusion, the parameter that actually applies is B = π/R, the square of which, B², is called the “buckling”. It is also logical that leakage should be larger, the greater the transport mean free path (recall Section 4.6), and the smaller the absorption cross section (Section 4.3). This is indeed the case, involving the use of the diffusion length, as used in Section 4.6. The nonleakage factor for one neutron energy group in a bare reactor is thus = 1/(1 + B²L²).

Bucklings for three important shapes are as listed.

• Sphere, radius R: B² = (π/R)²
• Parallelepiped, L, W, H: B² = (π/L)² + (π/W)² + (π/H)²
• Cylinder, H, R: B² = (π/H)² + (j⁰/R)², where j⁰ = 2.40483.

Critical conditions for more complex situations including mixtures of fuels can be analyzed by use of program CRITICAL, discussed in Computer Exercise 11.A.

The effect of flux variation with position is illustrated by Computer Exercise 11.B, dealing with the program MPDQ.

The presence of large amounts of neutron-moderating material such as water in a reactor greatly changes the neutron distribution in energy. Fast neutrons slow down by means of collisions with light nuclei, with the result that most of the fissions are produced by low-energy (thermal) neutrons. Such a system is called a “thermal” reactor in contrast with a system without moderator, a “fast” reactor, operating principally with fast neutrons. The cross sections for the two energy ranges are widely different, as noted in Exercise 11.4. Also, the neutrons are subject to being removed from the multiplication cycle during the slowing process by strong resonance absorption in elements such as U-238. Finally, there is competition for the neutrons between fuel, coolant, structural materials, fission products, and control absorbers.

The description of the multiplication cycle for a thermal reactor is somewhat more complicated than that for a fast metal assembly, as seen in Figure 11.4. The set of reactor parameters are (a) the fast fission factor ϵ (epsilon), representing the immediate multiplication because of fission at high neutron energy, mainly in U-238; (b) the fast nonleakage probability , being the fraction remaining in the core during neutron slowing; (c) the resonance escape probability p, the fraction of neutrons not captured during slowing; (d) the thermal nonleakage probability , the fraction of neutrons remaining in the core during diffusion at thermal energy; (e) the thermal utilization f, the fraction of thermal neutrons absorbed in fuel; and (f) the reproduction factor η, as the number of new fission neutrons per absorption in fuel. At the end of the cycle starting with one fission neutron, the number of fast neutrons produced is seen to be ϵ pfη , which may be also labeled k, the effective multiplication factor. It is convenient to group four of the factors to form k^∞ = ϵ pfη, the so-called infinite multiplication factor that would be identical to k if the medium were infinite in extent, without leakage. If we form a composite nonleakage probability then we may write

Figure 11.4. Neutron cycle in thermal reactor.

For a reactor to be critical, k must equal 1, as before.

To provide some appreciation of the sizes of various factors, let us calculate the values of the composite quantities for a thermal reactor, for which ϵ = 1.03, p = 0.71, , , f = 0.79, and η = 1.8. Now k^∞ = (1.03) (0.71) (1.8) (0.79) = 1.04, (0.99) = 0.96, and k = (1.04) (0.96) = 1.00. For this example, the various parameters yield a critical system. In Section 11.4 we will describe the physical construction of typical thermal reactors.

11.3. Neutron Flux and Reactor Power

The power developed by a reactor is a quantity of great interest for practical reasons. Power is related to the neutron population and to the mass of fissile material present. First, let us look at a typical cubic centimeter of the reactor, containing N fuel nuclei, each with cross section for fission σ^f at the typical neutron energy of the reactor, corresponding to neutron speed υ. Suppose that there are n neutrons in the volume. The rate at which the fission reaction occurs is thus R^f = nυNσ^f fissions per second. If each fission produces an energy w, then the power per unit volume is p = w R^f. For the whole reactor, of volume V, the rate of production of thermal energy is P = pV. If we have an average flux and a total number of fuel atoms N^T = NV, the total reactor power is seen to be

Thus we see that the power depends on the product of the number of neutrons and the number of fuel atoms. A high flux is required if the reactor contains a small amount of fuel, and conversely. All other things equal, a reactor with a high fission cross section can produce a required power with less fuel than one with small σ^f. We recall that σ^f decreases with increasing neutron energy. Thus for given power P, a fast reactor, operating with neutron energies principally in the vicinity of 1 MeV, requires either a much larger flux or a larger fissile fuel mass than does the thermal reactor, with neutrons of energy less than 0.1 eV.

The power developed by most familiar devices is closely related to fuel consumption. For example, a large car generally has a higher gasoline consumption rate than a small car, and more gasoline is used in operation at high speed than at low speed. In a reactor, it is necessary to add fuel very infrequently because of the very large energy yield per unit mass, and the fuel content remains essentially constant. From the formula relating power, flux, and fuel, we see that the power can be readily raised or lowered by changing the flux. By manipulation of control rods, the neutron population is allowed to increase or decrease to the proper level.

Power reactors used to generate electricity produce approximately 3000 megawatts of thermal power (MWt), and with an efficiency of approximately 1/3, give 1000 MW of electrical power (MWe).

11.4. Reactor Types

Although the only requirement for a neutron chain reaction is a sufficient amount of a fissile element, many combinations of materials and arrangements can be used to construct an operable nuclear reactor. Several different types or concepts have been devised and tested over the period since 1942, when the first reactor started operation, just as various kinds of engines have been used—steam, internal combustion, reciprocating, rotary, jet, etc. Experience with individual reactor concepts has led to the selection of a few that are most suitable by use of criteria such as economy, reliability, and ability to meet performance demands.

In this Section we will identify these important reactor features, compare several concepts, and then focus attention on the components of one specific power reactor type. We will then examine the processes of fuel consumption and control in a power reactor.

A general classification scheme for reactors has evolved that is related to the distinguishing features of the reactor types. These features are listed in the following sections

(a). Purpose

Most reactors in operation or under construction have as purpose the generation of large blocks of commercial electric power. Others serve training or radiation research needs, and many provide propulsion power for submarines. At various stages of development of a new concept, such as the breeder reactor, there will be constructed both a prototype reactor, one that tests feasibility, and a demonstration reactor, one that evaluates commercial possibilities.

(b). Neutron Energy

A fast reactor is one in which most of the neutrons are in the energy range 0.1 to 1 MeV, below but near the energy of neutrons released in fission. The neutrons remain at high energy because there is relatively little material present to cause them to slow down. In contrast, the thermal reactor contains a good neutron-moderating material, and the bulk of the neutrons have energy less than 0.1 eV.

(c). Moderator and Coolant

In some reactors, one substance serves two functions—to assist in neutron slowing and to remove the fission heat. Others involve one material for moderator and another for coolant. The most frequently used materials are listed in the following:

 

Moderators	Coolants
Light water	Light water
Heavy water	Carbon dioxide
Graphite	Helium
Beryllium	Liquid sodium

The condition of the coolant serves as a further identification. The pressurized water reactor provides high-temperature water to a heat exchanger that generates steam, whereas the boiling water reactor supplies steam directly.

(d). Fuel

Uranium with U-235 content varying from natural uranium (≅0.7%) to slightly enriched (≅3% to 5%) to highly enriched (≅90%) is used in various reactors, with the enrichment depending on what other absorbing materials are present. The fissile isotope is produced and consumed in reactors containing significant amounts of U-238. Plutonium serves as fuel for fast breeder reactors and can be recycled as fuel for thermal reactors. A few reactors have been built with fertile Th-232, producing fissile U-233.

The fuel may have various physical forms—a metal, or an alloy with a metal such as aluminum, or a compound such as the oxide UO² or carbide UC.

(e). Arrangement

In most modern reactors, the fuel is isolated from the coolant in what is called a heterogeneous arrangement. The alternative is a homogeneous mixture of fuel and moderator or fuel and moderator-coolant.

(f). Structural Materials

The functions of support, retention of fission products, and heat conduction are provided by various metals. The main examples are aluminum, stainless steel, and zircaloy, an alloy of zirconium.

By placing emphasis on one or more of the preceding features of reactors, reactor concepts are identified. Some of the more widely used or promising power reactor types are the following:

• Pressurized water reactor (PWR), a thermal reactor with light water at high pressure (2200 psi) and temperature (600 °F) serving as moderator-coolant, and a heterogeneous arrangement of slightly enriched uranium fuel.
• Boiling water reactor (BWR), similar to the PWR except that the pressure and temperature are lower (1000 psi and 550 °F).
• High temperature gas-cooled reactor (HTGR) that uses graphite moderator, highly enriched uranium, and helium coolant (1430 °F and 600 psi).
• Canadian deuterium uranium (CANDU) that uses heavy water moderator and natural uranium fuel that can be loaded and discharged during operation.
• Liquid metal fast breeder reactor (LMFBR), with no moderator, liquid sodium coolant, and plutonium fuel, surrounded by natural or depleted uranium.

Table 11.1 amplifies on the principal features of the preceding five main power reactor concepts. A description of the RBMK, exemplified by the ill-fated Chernobyl-4 reactor, appears in Section 19.6. Other reactors include the Magnox and AGR of the United Kingdom and several concepts that were tested but abandoned (see encyclopedia article in References).

Table 11.1. Power Reactor Materials

	Pressurized Water (PWR)	Boiling Water (BWR)	Natural U Heavy Water (CANDU)	High Temp. Gas-cooled (HTGR)	Liquid Metal Fast Breeder (LMFBR)
Fuel form	UO2	UO2	UO2	UC,ThC2	PuO2, UO2
Enrichment	3% U-235	2.5 % U-235	0.7 % U-235	93 % U-235	15 wt. % Pu-239
Moderator	water	water	heavy water	graphite	none
Coolant	water	water	heavy water	helium gas	liquid sodium
Cladding	zircaloy	zircaloy	zircaloy	graphite	stainless steel
Control	B4C or Ag-In-Cd rods	B4C crosses	moderator level	B4C rods	tantalum or B4C rods
Vessel	steel	steel	steel	prestressed concrete	steel

The large-scale reactors used for the production of thermal energy that is converted to electrical energy are much more complex than the fast assembly described in Section 11.1. To illustrate, we can identify the components and their functions in a modern PWR. Figure 11.5 gives some indication of the sizes of the various parts. To gain some appreciation of the physical arrangement of fuel in power reactors, try out the graphics programs in Computer Exercises 11.E (ASSEMBLY) and 11.F (BWRASEM).

Figure 11.5. Reactor construction.

The fresh fuel installed in a typical PWR consists of cylindrical pellets of slightly enriched (3% U-235) uranium oxide (UO²) of diameter approximately 3/8 in (~1 cm) and length approximately 0.6 in (~1.5 cm). A zircaloy tube of wall thickness 0.025 in (~0.6 mm) is filled with the pellets to an “active length” of 12 ft (365 cm) and sealed to form a fuel rod (or pin). The metal tube serves to provide support for the column of pellets, to provide cladding that retains radioactive fission products, and to protect the fuel from interaction with the coolant. Approximately 200 of the fuel pins are grouped in a bundle called a fuel element of approximately 8 in (~20 cm) on a side, and approximately 180 elements are assembled in an approximately cylindrical array to form the reactor core. This structure is mounted on supports in a steel pressure vessel of outside diameter approximately 16 ft (~5 m), height 40 ft (~12 m), and walls up to 12 in (~30 cm) thick. Control rods, consisting of boron carbide or an alloy of cadmium, silver, and indium, provide the ability to change the amount of neutron absorption. The rods are inserted in some vacant fuel pin spaces and magnetically connected to drive mechanisms. On interruption of magnet current, the rods enter the core through the force of gravity. The pressure vessel is filled with light water, which serves as neutron moderator, as coolant to remove fission heat, and as reflector, the layer of material surrounding the core that helps prevent neutron escape. The water also contains in solution the compound boric acid, H³BO³, which strongly absorbs neutrons in proportion to the number of boron atoms and thus inhibits neutron multiplication (i.e., “poisons” the reactor). The term “soluble poison” is often used to identify this material, the concentration of which can be adjusted during reactor operation. To keep the reactor critical as fuel is consumed, the boron content is gradually reduced. A shield of concrete surrounds the pressure vessel and other equipment to provide protection against neutrons and gamma rays from the nuclear reactions. The shield also serves as an additional barrier to the release of radioactive materials.

We have mentioned only the main components, which distinguish a nuclear reactor from other heat sources such as one burning coal. An actual system is much more complex than described earlier. It contains equipment such as spacers to hold the many fuel rods apart; core support structures; baffles to direct coolant flow effectively; guides, seals, and motors for the control rods; guide tubes and electrical leads for neutron-detecting instruments, brought through the bottom of the pressure vessel and up into certain fuel assemblies; and bolts to hold down the vessel head and maintain the high operating pressure.

The power reactor is designed to withstand the effects of high temperature, erosion by moving coolant, and nuclear radiation. The materials of construction are chosen for their favorable properties. Fabrication, testing, and operation are governed by strict procedures.

11.5. Reactor Operation

The generation of energy from nuclear fuels is unique in that a rather large amount of fuel must be present at all times for the chain reaction to continue. In contrast, an automobile will operate even though its gasoline tank is practically empty. There is a subtle relationship between reactor fuel and other quantities such as consumption, power, neutron flux, criticality, and control.

The first and most important consideration is the energy production, which is directly related to fuel consumption. Let us simplify the situation by assuming that the only fuel consumed is U-235 and that the reactor operates continuously and steadily at a definite power level. Because each atom “burned” (i.e., converted into either U-236 or fission products by neutron absorption) has an accompanying energy release, we can find the amount of fuel that must be consumed in a given period.

Let us examine the fuel use in a simplified PWR that uses 20 w/o fuel and operates at 100 MWe or 300 MWt, as in a test reactor or a propulsion reactor. The initial fuel loading into a single zone is 1000 kg U. We apply the rule of thumb that 1.3 grams of U-235 is consumed for each megawatt-day of thermal energy, assuming that all fissions are due to U-235. In 1 year, the amount of U-235 consumed is (300 MWt) (1.3 g/MWt-day) (365 days) = 1.42 × l0⁵ g or 142 kg. We see that a great deal of the original 200 kg of U-235 has been consumed, with a final enrichment of 6.8 w/o. Let us assume that a completely new core is installed at the end of a year's operation. If we carry out the calculations as in Section 9.2, the fuel cost excluding fabrication and transport is $4.59 million. Most of that is for enrichment. The electricity produced is making the unit cost of fuel ($4.59 × 10⁶)/(8.76 × 10⁸) = $0.0052 or approximately half a cent per kWh. In Chapter 19 we will analyze fuel cycles in a large power reactor that has several zones with different enrichments and shuts down periodically to remove, rearrange, and install fuel.

We continue studying the operating features of our small PWR. Because no fuel is added during the operating cycle of the order of a year, the amount to be burned must be installed at the beginning. First, the amount of uranium needed to achieve criticality is loaded into the reactor. If then the “excess” is added, it is clear that the reactor would be supercritical unless some compensating action were taken. In the PWR, the excess fuel reactivity is reduced by the inclusion of control rods and boron solution.

The reactor is brought to full power, operating temperature, and pressure by means of rod position adjustments. Then, as the reactor operates and fuel begins to burn out, the concentration of boron is reduced. By the end of the cycle, the extra fuel is gone, all of the available control absorption has been removed, and the reactor is shut down for refueling. The trends in fuel and boron are shown in Figure 11.6, neglecting the effects of certain fission product absorption and plutonium production. The graph represents a case in which the power is kept constant. The fuel content thus linearly decreases with time. Such operation characterizes a reactor that provides a “base load” in an electrical generating system that also includes fossil fuel plants and hydroelectric stations.

Figure 11.6. Reactor control during fuel consumption in power reactor.

The power level in a reactor was shown in Section 11.3 to be proportional to neutron flux. However, in a reactor that experiences fuel consumption the flux must increase in time, because the power is proportional also to the fuel content.

The amount of control absorber required at the beginning of the cycle is proportional to the amount of excess fuel added to permit burn up for power production. For example, if the fuel is expected to go from 3% to 1.5% U-235, an initial boron atom number density in the moderator is approximately 1.0 × 10⁻⁴ (in units of 10²⁴). For comparison, the number of water molecules per cubic centimeter is 0.0334. The boron content is usually expressed in parts per million (i.e., micrograms of an additive per gram of diluent). For our example, with 10.8 and 18.0 as the molecular weights of boron and water, there are 10⁶(10⁻⁴) (10.8)/(0.0334)(18.0) = 1800 ppm.

The description of the reactor process just completed is somewhat idealized. Several other phenomena must be accounted for in design and operation.

At the periodic shutdowns for fuel replacement and rearrangement, a great deal of maintenance work is done. There is an economic premium for careful and thorough outage management to minimize the time the reactor is not producing power. Down times as low as 3 weeks have been achieved. Replacement of a steam generator requires several months. The capacity factor (CF) is an important parameter in the performance of a power reactor and the nuclear industry as a whole. It is the percent of output compared with the maximum. Included in its calculation are the outage for refueling and any other shutdown time. The median CF for 3-year periods in the United States has risen from 59% in 1974–1976 to more than 90% in 2004–2006. For data on individual reactors and analysis of trends, see References.

If a reactor is fueled with natural uranium or slightly enriched uranium, the generation of plutonium tends to extend the cycle time. The fissile Pu-239 helps maintain criticality and provides part of the power. Small amounts of higher plutonium isotopes are also formed: Pu-240, fissile Pu-241 (14.4 year half-life), and Pu-242. These isotopes and those of elements farther up the periodic table are called transuranic materials or actinides. They are important as fuels, poisons, or nuclear wastes. (See Exercise 11.14).

Neutron absorption in the fission products has an effect on control requirements. The most important of these is a radioactive isotope of xenon, Xe-135, which has a cross section at 0.0253 eV of 2.65 million barns. Its yield in fission is high, y = 0.06, meaning that for each fission, one obtains 6% as many atoms of Xe-135. In steady operation at high neutron flux, its rate of production is equal to its consumption by neutron absorption. Hence With the ratio σ^f/σ^a for U-235 of 0.86, we see that the absorption rate of Xe-135 is (0.86)(0.06) = 0.05 times that of the fuel itself. This factor is approximately 0.04 if the radioactive decay (t^H = 9.10 h) of xenon-135 is included (see Exercise 11.8). The time-dependent variation of neutron absorption in xenon-135 is the subject of Computer Exercise 11.C, which describes the program XETR.

It might appear from Figure 11.6 that the reactor cycle could be increased to as long a time as desired merely by adding more U-235 at the beginning. There are limits to such additions, however. First, the more the excess fuel that is added, the greater must be the control by rods or soluble poison. Second, radiation and thermal effects on fuel and cladding materials increase with life. The amount of allowable total energy extracted from the uranium, including all fissionable isotopes, is expressed as the number of megawatt-days per metric ton (MWd/ton).^[†] We can calculate its value for 1 year's operation of a 3000 MWt power reactor with initial U-235 fuel loading of 2800 kg; with an enrichment of 0.03, the uranium content was 2800/0.03 = 93,000 kg or 93 tons. With the energy yield of (3000 MW)(365 days) ≅1,100,000 MWd, we find 12,000 MWd/ton. Taking account of plutonium and the management of fuel in the core, a typical average exposure is actually 30,000 MWd/ton. It is desirable to seek larger values of this quantity to prolong the cycle and thus minimize the costs of fuel, reprocessing, and fabrication.

^† The metric ton (ton) is 1000 kg.

11.6. The Natural Reactor

Until the 1970s, it had been assumed that the first nuclear reactor was put into operation by Enrico Fermi and his associates in 1942. It seems, however, that a natural chain reaction involving neutrons and uranium took place in the African state of Gabon, near Oklo, some 2 billion years ago (see References). At that time, the isotope concentration of U-235 in natural uranium was higher than it is now because of the differences in half-lives: U-235, 7.04 × 10⁸ years; U-238, 4.47 × 10⁹ years. The water content in a rich vein of ore was sufficient to moderate neutrons to thermal energy. It is believed that this “reactor” operated off and on for thousands of years at power levels of the order of kilowatts. The discovery of the Oklo phenomenon resulted from the observations of an unusually low U-235 abundance in the mined uranium. The effect was confirmed by the presence of fission products.

11.7. Summary

A self-sustaining chain reaction involving neutrons and fission is possible if a critical mass of fuel is accumulated. The value of the multiplication factor k indicates whether a reactor is subcritical (<1), critical (=1), or supercritical (>1). The reactor power, which is proportional to the product of flux and the number of fuel atoms, is readily adjustable. A thermal reactor contains a moderator and operates on slowed neutrons. Reactors are classified according to purpose, neutron energy, moderator and coolant, fuel, arrangement, and structural material. Principal types are the PWR, the BWR, the HTGR, and LMFBR. Excess fuel is added to a reactor initially to take care of burning during the operating cycle, with adjustable control absorbers maintaining criticality. Account must be taken of fission product absorbers such as Xe-135 and of limitations related to thermal and radiation effects. Approximately 2 billion years ago, deposits of uranium in Africa had a high enough concentration of U-235 to become natural chain reactors.

11.8. Exercises

1. Calculate the reproduction factor η for fast neutrons, with σ^f = 1.40, σ^a = 1.65, and ν = 2.60 (ANL-5800, p.581).
2. If the power developed by the Godiva-type reactor of mass 50 kg is 100 watts, what is the average flux? Note that the energy of fission is w = 3.04 × 10⁻¹¹ W−s.
3. Find the multiplication factors k^∞ and k for a thermal reactor with ϵ = 1.05 p = 0.75, , , f = 0.85, and η = 1.75. Evaluate the reactivity ρ.
4. The value of the reproduction factor η in uranium containing both U-235 (1) and U-238 (2), is given by Calculate η for three reactors (a) thermal, with 3% U-235, N¹/N² = 0.0315; (b) fast, with the same fuel; (c) fast, with pure U-235. Comment on the results. Note values of constants:

    

   	Thermal	Fast
   σf1	586	1.40
   σa1	681	1.65
   σf2	0	0.095
   σα2	2.70	0.255
   ν1	2.42	2.60
   ν2	0	2.60

5. By means of the formula and thermal neutron numbers from Exercise 11.4, find η, the number of neutrons per absorption in fuel, for uranium oxide in which the U-235 atom fraction is 0.2, regarded as a practical lower limit for nuclear weapons material. Would the fuel be suitable for a research reactor?
6. How many individual fuel pellets are there in the PWR reactor described in the text? Assuming a density of uranium oxide of 10 g/cm³, estimate the total mass of uranium and U-235 in the core in kilograms. What is the initial fuel cost?
7. The core of a PWR contains 180 fuel assemblies of length 4 m, width 0.2 m. (a) Find the core volume and radius of equivalent cylinder. (b) If there are 200 fuel rods per assembly with pellets of diameter 0.9 cm, what is the approximate UO² volume fraction of the core?

8. • Taking account of Xe-135 production, absorption, and decay, show that the balance equation is
   • Calculate λ^X and the ratio of absorption rates in Xe-135 and fuel if φ is 2 × 10¹³ cm⁻²−s⁻¹.
9. The initial concentration of boron in a 10,000-ft³ reactor coolant system is 1500 ppm, (the number of micrograms of additive per gram of diluent). What volume of solution of concentration 8000 ppm should be added to achieve a new value of 1600 ppm?
10. An adjustment of boron content from 1500 to 1400 ppm is made in the reactor described in Exercise 11.9. Pure water is pumped in and then mixed coolant and poison are pumped out in two separate steps. For how long should the 500 ft³/min pump operate in each of the operations?
11. Find the ratio of weight percentages of U-235 and U-238 at a time 1.9 billion years ago, assuming the present 0.711/99.3.
12. Constants for a spherical fast uranium-235 metal assembly are: diffusion coefficient D = 1.02 cm; macroscopic absorption cross section Σ^a = 0.0795 cm⁻¹; effective radius R = 10 cm. Calculate the diffusion length L, the buckling B², and the nonleakage factor .
13. The neutron flux in a reactor varies with position. In a simple core such as a bare uranium metal sphere of radius R, it varies as φ = φ^c (sin x)/x, where x = πr/R. At the center of the sphere the flux is φ^c. Calculate and plot the flux distribution for a core with radius 10 cm and central flux 5 × l0¹¹/cm²−s.
14. A reactor is loaded with 90,000 kg of U at 3 w/o U-235. It operates for a year at 75% of its rated 3000 MWt capacity. (a) Apply the rule of thumb 1.3 g/MWt-day to find the consumption of U-235. What is the final enrichment of the fuel? (b) If instead one third of the energy came from plutonium, what would the final U-235 enrichment be? Note thermal cross sections, all in barns: U-235 σ^f = 586, σ^c = 95; Pu-239 σ^f = 752, σ = 270.

Computer Exercises

1. The evaluation of critical conditions for a variety of spherical metal assemblies can be made with the program CRITICAL. It uses a one-neutron group model with cross sections deduced from early critical experiments related to weapons. CRITICAL can handle any combination of uranium and plutonium. Run the program, choosing U enrichment and Pu content. Suggested configurations:
   • Pure U-235.
   • Godiva (93.9% U-235, experimental U-235 mass 48.8 kg).
   • Jezebel (pure plutonium, experimental mass 16.28 kg).
   • Natural U (0.0072 atom fraction U-235, should not be possible to be made critical).
   • Depleted U (0.003 atom fraction U-235).
   • Elementary breeder reactor (Pu-239 volume 10%, depleted U).
2. A miniature version of a classic computer code PDQ is called MPDQ. It finds the amount of critical control absorber in a core of the form of an unreflected slab, by solution of difference equations.
   • Load the program and study the listing.
   • Run the program and study the displays, then compare the results of choosing a linear or sine trial fast flux function.
   • With the constants given in Exercise 11.1, modify the program to calculate the critical control for a metal assembly as a slab of width 5 cm.
3. The amount of xenon in a reactor varies with time, especially when large changes in neutron flux occur, as at startup or shutdown. Program XETR (Xenon Transient) solves differential equations for the content of iodine-135 and xenon-135.
   • Load the program and examine the input constants and conditions. Study the trend in the output as the reactivity ρ (see Section 11.2) vs. time.
   • Use the concentrations of I-135 and Xe-135 calculated for long times after start up as initial concentrations, and set the flux equal to zero to simulate a sudden shutdown of the reactor. Note and discuss the trend in xenon with time.
4. Competition among three neutron processes—scattering, absorption, and leakage—is illustrated by the program SLOWINGS. It simulates the release of a series of neutrons at the center of a carbon sphere, and by use of slowing theory and random numbers, finds the number of neutrons absorbed and escaping.
   • Run the program several times to note statistical variations.
   • Increase the absorption cross section by a factor of 200 as if considerable boron were added to the sphere, and note the effect.
5. The program ASSEMBLY displays a pressurized water reactor fuel assembly, an array of 14 ×14 fuel rods. Run the program.
6. The program BWRASEM displays four boiling water reactor fuel assemblies with a cross-shaped control rod between them. Run the program to inspect.

11.9 References

Nero, 1979 Anthony V. Nero Jr, A Guidebook to Nuclear Reactors 1979 University of California Press Berkeley

Knief, 1992 Ronald Allen Knief, Nuclear Engineering: Theory and Technology of Commercial Nuclear Power 1992 Taylor & Francis Bristol, PA

Glasstone and Sesonske, 1994 Samuel Glasstone, Alexander Sesonske, Nuclear Reactor Engineering 4th Ed. 1994 Chapman and Hall New York Vol. 1, Reactor Design Basics, Vol. 2, Reactor Systems Engineering

Peterson and Newby, 1956 R.E. Peterson, G.A. Newby, An Unreflected U-235 Critical Assembly Nuclear Science & Engineering 1 1956112- Description of Godiva

Murray, 2007 Raymond L. Murray, Nuclear Reactors Kirk-Othmer Encyclopedia of Chemical Technology 2007 John Wiley & Sons New York

WWW Virtual Library: Nuclear Engineering WWW Virtual Library: Nuclear Engineering

http://www.nuc.berkeley.edu/main/vir_library.html http://www.nuc.berkeley.edu/main/vir_library.html

Links to nearly everything Links to nearly everything.

Uranium Information Centre (Australia) Uranium Information Centre (Australia)

http://www.uic.com.au http://www.uic.com.au

Briefing papers on a variety of nuclear topics, and many links Briefing papers on a variety of nuclear topics, and many links.

International Nuclear Safety Center International Nuclear Safety Center

http://www.insc.anl.gov http://www.insc.anl.gov

Database on Reactor Material Properties. By Argonne National Laboratory Database on Reactor Material Properties. By Argonne National Laboratory.

Nuclear Power Reactors Nuclear Power Reactors

http://www.world-nuclear.org/ http://www.world-nuclear.org/

Select Information Papers/Nuclear Power Facts/Nuclear Power Reactors Select Information Papers/Nuclear Power Facts/Nuclear Power Reactors.

From World Nuclear Association From World Nuclear Association.

Issue on Outage Management, April 2007 Issue on Outage Management Nuclear News April 2007

Blake, May 2007 E. Michael Blake, U.S. Capacity Factors: A Small Gain to an Already Large Number Nuclear News May 200727-

Thorium and breeding of U-233 Thorium and breeding of U-233

http://www.world-nuclear.org/ http://www.world-nuclear.org/

Search on Thorium Search on Thorium.

The Virtual Nuclear Tourist The Virtual Nuclear Tourist

http://www.nucleartourist.com/ http://www.nucleartourist.com/

Comprehensive coverage of nuclear power Comprehensive coverage of nuclear power. Explore links in Table of Contents. By Joseph Gonyeau.

Cochran et al., 1993 Robert G. Cochran, Nicholas Tsoulfanidis, W.F. Miller, Nuclear Fuel Cycle: Analysis and Management 2nd Ed. 1993 American Nuclear Society La Grange Park, IL

Cowan, July 1976 George A. Cowan, A Natural Fission Reactor Scientific American July 197636-

The Natural Reactor The Natural Reactor

http://www.physics.isu.edu/radinf/Files/Okloreactor.pdf http://www.physics.isu.edu/radinf/Files/Okloreactor.pdf

The ancient reactor in Oklo, Gabon The ancient reactor in Oklo, Gabon. By Andrew Karam. Enter Google with Natural Reactor for DOE/OCRWM site.