In the context of nuclear reactors the most important reactions are those where the incident particle is a neutron. Also, neutrons are electrically neutral and are able to interact with nuclei much more readily than charged particles due to the absence of any electrostatic barrier.

In a nuclear reaction the neutron may undergo a scattering collision or it may be absorbed by the tar­get nucleus. These possible neutron reactions will now be discussed in some detail here and in the following Sections but the general principles apply also to re­actions induced by particles other than neutrons.

4.7 Introduction

We have seen from a study of nuclear physics that a nuclear reactor is capable of generating heat at very high rates. The principal limitation on this rate of heat generation is the rate at which the heat can be transferred to the core coolant whilst maintaining the core components, in particular the fuel elements, within their design temperature limits. It is impor­tant therefore to be able to predict the rate at which coolants can extract heat from the core under various conditions of flow rates, pressures, geometries, etc. Hence the reason for a study of heat transfer and fluid flow in order to determine the rate of heat production in the reactor and to predict the tempera­ture patterns.

We will discuss the basic methods by which heat can be transferred and then apply them to the specific area of nuclear heat generation.

There are three methods by which heat can be transferred:

• Conduction.

• Convection.

•

Radiation.

For heat to be transferred by conduction the material through which the heat flows is essentially stationary, there being no net flow’ of atoms in solids or liquids, or of molecules in the case of gases. In reactors this mode of heat transfer is confined principally to the conduction of heat through the fuel element to the element surface.

The extraction of heat from the fuel element sur­faces by means of a gaseous or liquid coolant is prin­cipally by convection. The flow of the coolant over the element is achieved either by pumping, giving forced convective cooling, or by buoyancy forces aris­ing from differences in density. The former is by far the more important for the majority of reactor designs and particularly for gas cooled reactors; the latter is important in reactors where the coolant changes state over the fuel element, e. g., Boiling Water Reactors (BWR) and Steam Generating Heavy Water Reactors (SGHWR).

Heat transfer by radiation requires no contact or intermediate media between the two heat transfer sur­faces but since any appreciable heat transfer by this means requires high temperatures, i. e., above about 800°C, it is relatively unimportant in magnox, ad­vanced gas-cooled reactor (AGR) and-pressurised water reactor (PWR) systems.

In order that we can appreciate the principal design and operating factors affecting the rate of heat extrac­tion from the core, it is necessary to examine the laws of heat transfer, particularly those relating to conduc­tion and convection.

It was shown in Section 1.3.3 of this chapter, that the fission of U-236 into two equal fission products of palladium 118 releases 236 MeV of energy. Section

3.3.1 and Fig 1.8 show that this symmetrical fission is rare; asymmetrical fission with the release of two or three neutrons is much more likely. Calculation similar to that in Section 1 of this chapter, but for the more typical asymmetric fissions, gives an energy release per fission of 200 MeV, most of which appears in the form of kinetic energy of the products of the fission process and ultimately as heat energy. An ap­proximate distribution of the fission energy is given in Table 1.5 from which several points of interest can be obtained:

Table 1.5

Liberation of energy due to fission Instantaneous

Kinetic energy of fission fragments 168 MeV

Kinetic energy of fission neutrons 5 MeV

Prompt gamma rays 7 MeV

180 MeV

Delayed

Beta particles from fission product decay 7 MeV

Gamma rays from fission product decay 6 MeV

Antineutrinos 10 MeV

23 MeV [4] [5]

3.3.5 Decay heat

As previously mentioned, some of the energy released in fission is delayed; it is not released at the time of fission. This energy mainly comes from the sub­sequent decay of the fission products and is released as an exponential decline over a long time. Thus a reactor will continue to be a source of heat — decay heat — even though the fission process has ceased.

Immediately after shutdown a reactor will be pro­ducing about 6,5% of full power output; this falls to about 1.5% after an hour but is still about 0.5% three days later. Considering that full power may be of the order of 1500 MW (thermal), each 1% represents 15 MW of heat that is being produced. This has obvious implications on reactor design and operation in that there must be provision to remove a substantial amount of heat from the reactor core even after it is shutdown.

3.3.6 Summary of the fission process

Asymmetric binary fission is the most likely, with one fragment between 90 and 101 and the other between 132 and 143. The average number of neutrons produced by thermal fission (y) of uranium 235 is 2.43 fast neutrons of average energy 2 MeV. A small fraction (0.68% for U-235) of the fission neutrons are delayed (mean neutron lifetime for U-235 fission :

12.9 s). The energy released by the fission is 200 MeV, most of which — but not all — appears in the fuel in the kinetic energy of the fission products. A sig­nificant amount of energy is due to the decay of the radioactive products; the resulting handling and shielding problems and the decay heat must be catered for in the reactor design.

1.1.1 Einstein’s equation

Energy can take many forms and may be transformed from one form to another. For example, a projectile has kinetic energy arising from its speed, a hot object has thermal energy due to its temperature and an object on a ledge of a high cliff has potential energy by virtue of its position.

At the beginning of the century, in his Theory of Relativity, Albert Einstein showed that mass is also a form of energy. Mass may therefore be regarded as analogous to the parameters’ speed, temperature and height in the foregoing examples. Increasing any of these parameters requires energy to be supplied where­as a decrease leads to the spontaneous and inevitable release of the equivalent amount of energy Table 1.1.

Referring to Table 1.1, take as an example an elec­tric kettle containing water at temperature ti say. To increase the temperature to a higher value іг electrical energy must be supplied [s ms(t2 — ti) where m is the mass of water and s specific heat]. But hot water at t2 will spontaneously cool down to a lower tem­perature 11, the heat energy in this case being released to the surrounding atmosphere. In his theory Einstein derived the equation ДЕ = С2Дт where Дт is the change in mass and ДЕ the equivalent energy. He then postulated that all mass could be converted into energy and wrote his now familiar equation E = m C2 where the constant of proportionality C2 is the square of the speed of light in vacuo. Now C = 3 x 108 m/s and hence the energy associated with a very small change in mass is very large indeed. This is why the mass changes which occur in, for example, chemical reactions — like burning a fuel — are not normally detectable.

This factor is a measure of how well the design uti­lises the thermal neutrons in the moderator in getting as many as possible to where it matters, i. e., into the fuel.

_ number of thermal neutrons absorbed in the fuel total, number of thermal neutrons absorbed

In Fig 1.13 it is supposed that at some instant there are ni thermal neutrons in the moderator. A fraction of the ni neutrons is absorbed in the fuel thus giving fni thermal neutrons in the fuel [870 thermal neutrons for f = 0.87 say]. There will be (1 — f)m neutrons lost by capture in reactor materials other than U-238 and U-235.

Clearly, the more fuel there is in the reactor rela­tive to the moderator the higher the value of f; ultimately f = 1 for 100% fuel. Conversely the more moderator there is the smaller the value of f; ulti­mately f = 0 for 100% moderator.

Hence we may conclude that the value of f decreases progressively from unity towards zero as more and more moderator is added to a given amount of fuel.

There are two kinds of scattering events:

• Elastic scattering In a sense, elastic scattering is not a nuclear reaction because the neutron does not enter the nucleus to form a compound nucleus but is merely deflected by the nuclear fields. Elastic scattering may be regarded as a neutron colliding with a nucleus and rebounding; the total kinetic energy is conserved and the energy exchange be­tween neutron and nucleus may be calculated using the ordinary Newtonian laws of motion. The energy lost by the neutron in the collision depends on the angle through which it is scattered. Elastic scattering is primarily important as a way in which neutrons can lose energy and is discussed further in this context in Section 6 of this chapter.

1.3.1 Absorption reactions

There are two kinds of absorption events, capture and fission reactions:

Capture reactions Here the neutron is retained in the nucleus which then emits a nuclear particle and/or 7 radiation. Examples of neutron capture events follow:

* (n, >) reaction. The emission of the binding energy of the neutron as a 7 ray, the most common neutron capture process, is known as radiative capture.

115[„ + In — H6ln + 7 49 0 49

or In-115 (n, 7) In-116

59Co + In —[1] 60Co + 7

27 0 27

or Co-59 (n, 7) Co-60

In the above, as is often the case in radiative capture, the product nuclei indium 116 and cobalt 60 are both radioactive and emit j3/7 radiations with half life of 54 minutes and 5.3 years respectively.

These are examples of stable materials being trans­formed by neutron capture into radioactive iso­topes. This is known as neutron activation. In a reactor the neutron activation of indium is some­times used to measure the neutron density. Because of the 7 ray that accompanies the 0 radiation, activation of the cobalt present in ferrous materials can cause difficulties with maintenance work and in the eventual decommissioning of the plant.

. • (n, a) reaction:

Юв + In -* ?Li + 4hc

5 0 3 2

or B-10 (n, a) Li-7

This reaction is used in instrumentation to detect neutrons indirectly. Because neutrons are electri­cally neutral they do not ionise atoms to any great extent. However, the alpha particles produced in the neutron reaction with boron cause ample ionisa­tion and are readily detected.

* • (n, p) reaction:

14n + In — 14c + Ip

7 0 6 1

or N-14 (n, p) C-14

Alternatively, as the proton is identical with the nucleus of hydrogen, the reaction may also be written:

]4n + In " 14c + 1h

7 0 6 1

The product nucleus, carbon 14, is radioactive emitting a negative /3 particle and regenerating ni­trogen 14. The reaction is the basis for the carbon dating technique in archaeology: the nitrogen tn the atmosphere is bombarded by neutrons originating in the extra-terrestrial cosmic rays producing, it is presumed, a historically constant concentration of carbon 14 in the atmosphere and hence in plants and other living things. At death, the carbon 14 intake from the air ceases and the concentration in the ‘body’ or plant material reduces through radio­active decay. Knowing the initial concentration and half life (5570 years) of carbon 14, the moment of death can be determined.

Fission reactions The neutron absorption reaction other than capture is fission. The neutron absorbed by the target nucleus induces the resulting compound nucleus to split into usually two parts with the si­multaneous release of some neutrons and considerable energy, primarily in the form of the kinetic energy of the fission products.

Fission is the most important nuclear reaction of all as it is the source of the energy that enables gen­eration of power by nuclear means. The fission process will be discussed in detail in Section 4 of this chapter — firstly it is advantageous to introduce the concept of cross-section.

These are basically applicable to steady state conduc­tion in one direction through a uniform cross-section, e. g., a flat plate as illustrated in Fig 1.17.

Qt = total heat passing per unit time

A = area of plate perpendicular to direction of

heat flow

x = plate thickness

11 = temperature of hot face

i; = temperature of cold face

к = coefficient of thermal conductivity of the

plate material.

If we assume that к remains constant and that the relationship between heat and temperature is linear:

Qj/A = k(t] — t2)/x (1.1)

= q known as the heat flux.

This expression is sometimes written as:

Qt — (ti — t2)/(x/kA)

where x/kA is known as the thermal resistance.

This term is analogous to electrical resistance, its value depending upon the material properties and the geo­metry. It can, be applied in a similar manner to the application of’Ohm’s Law to electrical circuits. Thus to obtain a high rate of heat transfer through a solid fuel element we would endeavour to have a large tem­perature differential and a low thermal resistance, although of course there may be other non-heat trans­fer limitations to achieving this objective. The simple Equation (1.1) can only be applied directly to a flat plate geometry. The majority of fuel elements used in reactors are cylindrical and it will be useful when we apply the laws of conduction to fuel elements if we have the equation in a form which can be directly related to a cylindrical shape, e. g., a hollow cylinder or pipe (Fig 1.18).

By rewriting Equation (1.1) in the form Qt = — 2x к dt/dx we can apply it to Fig 1.18 and arrive at an expression:

Qt = (tі — t2)/(I/2irk) loge (R2/Ri) (1.2)

and as in Equation (i. l), (1/2тгк) Ioge (R2/Rj) is the ‘thermal resistance’.

Equation (1.2) is used extensively in calculations to obtain the heat loss through the walls of steam pipes, etc., due to conduction.

4.1 The chain reaction

Although the binary fission of a nucleus by an im­pinging neutron would no doubt be of academic interest, it would in itself be of little significance in the context of large scale power production. It is the accompanying release of neutrons which makes power production possible in that the neutrons may induce further nuclei to fission and thus lead to a chain reaction. Figure l. tl shows a rapidly diverging chain reaction.

Figure 1.11 is unrealistic because it presumes 100% fissionable material with no scattering events and no loss of neutrons by capture events or by leakage of the neutrons out of the system. However the figure is useful to illustrate the principle of a chain re­action and also for the need to ‘measure’ what is happening.

4.2 The multiplication constant к

The development of a chain reaction may be mea­sured by the multiplication constant (or reproduction factor) which is defined to be к = (number of neu­trons arising from the chain reaction in one genera — tion)/(number of neutrons arising from the chain reaction in the previous generation).

For simplicity it is advantageous at this stage to retain the presumption implicit in Fig 1.11 that there is no loss of neutrons out of the system; that is, the system — the nuclear reactor — is assumed to be of infinite size. The symbol is now used for the multiplication constant, the subscript infinity (®) being a reminder of the imposed condition of an infinite sized nuclear reactor. From the definition of the multiplication constant it is clear that:

• If kcc>l the chain reaction is diverging and the nuclear reactor is said to be ‘supercritical’. As the power produced is related to the number of fission events taking place, this is the condition for power raising. The larger the value of k«, the quicker is the rate of power rise.

• If ke = 1 the chain reaction is self-sustaining and the reactor is said to be ‘critical’. A nuclear power station producing power at a steady rate for per­haps months and years — a base load power station —■ is therefore critical. The word is unfortunate but understandable when it is recalled that the term ‘critical’ dates back to 12 December 1942. On that day Enrico Fermi and his colleagues first achieved a self-sustained chain reaction, an achievement which had aspects of uncertainty and peril for the participants.

• If к»< 1 chain reaction is converging and the reactor power is decreasing. As before, the rate of power decrease is determined by how much less than unity is the value of k®.

Before applying these ideas to the nucleus we must first introduce the term unified mass unit, u. The mass of the nucleus is of the order of 10 ~27 kg. Clearly, a kilogram is too large a unit for mass in this context and it is necessary to define a more convenient unit. This is the unified mass unit u and is defined on the basis that the mass of the carbon nuclide C-12 is exactly 12 u [the unified mass unit supersedes the previous unit of atomic mass unit based on the mass of oxygen 0-16 being 16 amu]. From this definition:

lu = 1.6604 x 10-27 kg

Applying Einstein’s equation E = m C2 with m in kg and C in m/s (= 3 x 10s), giving E in joules, then:

lu = 1.6604 x 10’27 x 9 x 1016 = 1.4944 x 10“10 joules

As previously, the kilogram was found to be inade­quate as a unit for mass, the joule also is too large a unit for energy. A more convenient energy unit is the electron volt, eV, and is the energy acquired by an electron in accelerating through a potential difference of 1 volt. Using these new units and recalling that the charge on an electron is 1.6022 x 10“19 coulomb, we have:

1 eV = 1.6022 x 10“19 joules

Therefore 1 joule = 6.242 x 1012 MeV Since lu = 1.6604 x 10“27 kg

lu ■ 931 MeV

Also:

Mass of the neutron = 1.008665 u

Mass of the proton = 1.007277 u

Mass of the electron = 0.000548 u

Of the fn[ thermal neutrons absorbed in the fuel some will be captured in U-238, others captured in U-235 and the remainder cause fission of U-235. The fission events will be accompanied by release of fast fissions.

The thermal fission factor is a measure of the gain in the number of neutrons by thermal fission:

number of fast neutrons arising from thermal fission

TJ = ——————————————————-

number of thermal neutrons absorbed in the fuel

In Fig 1.13 the fn і thermal neutrons give rise to 7j fn і fast neutrons. [1.33 x 870 = 1160 fast neu­trons. Alternatively, of the 870 thermal neutron ab­sorbed in the fuel, 392 are captured and 478 cause thermal fission of U-235. 478 x v = 478 x 2.43 = 1160 fast neutrons.]

The value of 7] is determined by the enrichment of the fuel and, as calculated in Section 4.4.2 of this chapter, 17 = 1.33 for natural uranium. In Fig 1.14, values of 17 are plotted against the U-235 content in the fuel. The figure demonstrates that if the expense of enrichment is to be undertaken then this may as well be up to 3% or so because of the initial rela­tively large gains in 17; the subsequent modest in­creases will probably not justify higher enrichments.