We have already seen that a considerable number of factors affect the rate of heat transfer and all or some of these factors must be considered during the design and during the subsequent operation of the reactor. The application of the basic heat transfer laws involves the correlation of a number of variables into empirical equations. These variables will include both the fundamental properties of the fluids and solids involved and the experimental data. A conven­ient method of grouping together the fundamental properties is known as ‘Dimensional Analysis’.

Although this method is useful in analysing a varie­ty of heat transfer problems we will apply it to the determination of the surface heat transfer coefficient (h) since this is important in assessing the thermal capacity of the reactor core.

When heat is being transferred from a solid to a fluid as in a reactor fuel channel the dimensional analysis technique shows that the fundamental quan­tities involved can be grouped into three dimensionless numbers:

where h = surface heat transfer coefficient

de = equivalent diameter (defined as 4 x area of flow/wetted perimeter)

к = fluid thermal conductivity

V = velocity

<5 = density

Д = viscosity

Cp = specific heat (at constant pressure)

Figure 1.23 shows the relationship between three groups, Nu, Re, Pr, being typical for most fluids with a low thermal conductivity, e. g., СОг — It can be seen that the relationship is affected by the type of flow con­ditions in the fuel channel. With gas cooled systems the flow will always be turbulent and the straight line in this zone is represented by the equation:

Nu = 0.023 Re0-8 Pr0-4 (1.9)

This is a useful and reasonably accurate relationship for many reactor heat transfer applications and will enable the surface heat transfer coefficient (and hence the rate of channel heat extraction) to be determined when the factors included in Equations (1.6), (1.7) and (1.8) are known. The value of these factors will vary with the design and operating conditions.

The application of Equation (1.9) can be illustrated by solving the equation in terms of h:

h = 0.023 Re0-8 Pr0-4 k/de

= 0.023 (Gde/д)0-8 x (Срд,/к)°4 x k/de h = J G0,8 de ~0-2 (1.10)

where J = (0.023k)°6 (Cp)0-4/^04

Thus h is a function of:

J — the properties of the fluid

G (бр) — mass velocity

de — a property of the geometry

If any of these factors are varied by design or by operating conditions the value of h will change and hence the thermal output of the fuel channels. At the design stage of the reactor for instance it may be necessary to investigate the effect of the use of different coolants on the heat transfer rates in the fuel channel, e. g., the use of helium in place of CO2 to overcome possible C02-graphite problems in the AGR. Changes in operating temperatures or pressures will produce changes in the gas properties contribu­ting to the value of J. Of particular importance is the relationship between pressure and h, the latter reducing in value as the gas pressure falls. Thus a depressurisation will result in a decrease in the rate of cooling of the fuel elements even if the mass flow rate is maintained at a constant level. The precise value of the changes in h requires detailed calculation but Equation (І.10) will give a useful indication of the effect of changing the three major factors:

• The properties of the coolant fluid.

• The mass velocity.

• The channel/fuel element geometry.

The application of these dimensionless numbers will depend upon many factors including the type of coolants, the process of heat transfer, the flow con­ditions, etc. Application of the numbers will be found in text books on heat transfer, the objective of in­troducing them at this stage being to assist in under­standing the principal design and operating factors affecting the rate at which heat can be extracted from the reactor core. The many numbers are used prin­cipally in computer models (ARGOT, HOTSPOT, SCANDAL) to obtain accurate temperature profiles and heat transfer rates under varying design, opera­tional and fault conditions.

4.8 Temperature drop across a fuel channel

The heat transfer concepts which have been discussed so far would enable the designer to determine the temperature distribution from the centre of the fuel element to the bulk coolant. Figure 1.24 summarises this distribution:

• The temperature drop in the fuel rod (tp — tf) is due to conduction and the heat generation within the rod, resulting in the parabolic temperature pattern.

• The temperature drop through the fuel sheath (tf — ts) results from heat transfer by conduction.

• The final temperature drop from the surface of the sheath to the bulk temperature of the coolant (ts — tc) is by convection.

4.9 Temperature variations along a fuel channel

The temperature pattern shown in Fig 1.24 applies to a cross-section at any plane along the fuel channel. Although the pattern of this temperature distribu­tion will be similar at each level of the channel the temperatures and rates of heat transfer will vary with

Fig. 1.24 Temperature drop across a fuel channel

position along the channel. For instance, the coolant inlet and outlet temperatures for a typical magnox reactor would be И0°С and 3S0°C respectively, and 300°C and 650°C for an AGR. Thus the value of at least one major factor affecting the rate of heat transfer is varying along the channel and it is impor­tant to be able to calculate the coolant, sheath and fuel temperature patterns resulting from this variation.

Another important factor which must be considered is the rate at which heat is generated within the fuel. In developing the expression for the temperature drop in the fuel rod Equation (1,4) we assumed that the rate of heat generation was constant and proportional to the volume. However, this assumption is only correct for a single position in the fuel rod, the rate of heat generation per volume of rod varying with the position along the channel and also with the position of the channel in the reactor core.

In order to determine the temperature patterns in the core it is necessary to consider in detail the heat dstribution within the core resulting from the nuclear reactions and also the absorption of this heat by the channel coolant.

The alternative to Section 4.4.1 of this chapter — fuel enrichment and fast neutrons — is to retain natural uranium as the fuel but combined with slow neutrons, and thus take advantage of the greatly increased likelihood of U-235 fission occurring. Figure

1.6 (c) shows that the value of ar is several hundred times greater for slow neutrons than for fast.

Instead of the П] fast neutrons of Section 4.3, let n ] slow neutrons be introduced into the infinite mass of natural uranium. The calculation in Section 4.3.4 of this chapter to obtain the next generation of П2 neutrons may now be repeated, but of course there is no possibility with slow neutrons of inelastic collisions occurring. The neutrons are simply either captured or cause fission.

The necessary microscopic cross-section values for natural uranium may be obtained by taking the weighted average of the values for U-235 and U-238:

a (natural uranium) =

1 138

_ a (U-235) + — a (U-238)

139 139

The values given in Table 1.2 are applicable to neutrons of the ‘standard’ thermal energy 0.025 eV. Thus:

П2 = Оf nj/(<7f + Oc)v

= 4.15 П|/(4.15 + 3.43) 2.43 = 1.33 П|

Therefore к® = 1.33 > 1.

Hence natural uranium fuel will give k® > 1, theo­retical maximum value 1.33, provided the fissioning neutrons are of thermal energy. This is the basis of the class of reactors known as thermal reactors, per­haps with some slight enrichment of the fuel.

The flaw in the foregoing is that, following the absorption of the ni thermal neutrons, the next generation of П2 neutrons are as a result of fission events and are therefore fast neutrons. These must be converted — moderated — into thermal neutrons if the chain reaction in the natural uranium is to be sustained. This is achieved by having a second material, a moderator, in the reactor in which the neutrons released by fission may lose their energy and become thermal neutrons by successive elastic collisions with the moderator nuclei.

The above radioactive decay processes often result in the changed nucleus, the daughter product, having an excess internal energy. Studies of the energy of radio­active emissions — and of the energy necessary to produce ‘artificial’ radioactive nuclei — have led to an understanding of the internal energy of the nucleus. It seems that if nuclei are stimulated in some way they cannot absorb just any quantity of energy. A nucleus can exist only with certain discrete internal energies. The state with the least internal energy is the most stable and is called the ground state of the nucleus.

If the nucleus is in some other discrete energy level above ground state it is said to be ‘excited’. The nucleus may then return to the ground state, or some other intermediate excited state, by giving off gamma (?) radiation. This is electromagnetic radiation occupying the very high frequency short wavelength end of the electromagnetic spectrum. It is therefore akin to X-radiation but even more penetrating. The energy of the emitted у ray will be exactly — and, for a given element, uniquely equal to the energy difference between the initial and final states of the excited nucleus; this is the basis of gamma spectro­scopy. There is no change, of course, in either the mass number or atomic number of the nucleus. (See also Section 1.5.4 of this chapter.)

To summarise: if in radioactive decay a daughter product is formed in an excited state, as is very often the case, the particulate radiation emissions are accompanied by у radiation as the nucleus drops to a lower or ground state.

T4 = logn(2)/X = 0.693/X = 0.693т

After five half lives only about 3% of the original radioactive nuclei remain, Fig 1.5. Values for half lives for particular isotopes range from fractions of a second (Be-8 : 3 x 10-16 s) to millions of years (U-238 : 4.5 x 109 years). The neutron is also an example of a radioactive substance; when it is freed from a nucleus it is a /3 emitter, and therefore trans­forming to a proton, with a half life of 10.8 minutes.

The unit of radioactivity used to be the Curie (Ci) and was originally defined in terms of the rate of activity of radium, the first radioactive substance studied. It was subsequently redefined to be 3.7 x 1010 nuclear disintegrations per second. The Curie has now been replaced by a new unit, the becquerel (Bq), which is exactly equal to 1 disintegration per second.

4.5 The effective multiplication constant, ketf

In the foregoing the simplifying assumption has been made that the reactor has no boundary — an infinite reactor. This is not so for any real reactor, of course, and it is now necessary to include in the overall neutron balance of the fission chain reaction the possibility that some neutrons will ‘leak’ irreversibly out of the reactor — Fig 1.12. Thus, when considering finite sized reactors, the value of the infinite multi­plication constant will need to be modified to take into account those neutrons that are lost from the neutron life cycle through leakage. This is known as the ‘effective multiplication constant’ keff.

4.6 Non-leakage probability, PNL

By studying the neutron life cycle it was shown in Section 6.3 of this chapter that k® may be expressed as the product of four factors:

k® = рет/f

where the ‘infinity’ suffix is a reminder that the re­actor is presumed to be of infinite size. The possibility of neutrons diffusing out of the reactor or, more importantly, the proportion of neutrons that do not leak out may be incorporated in the multiplication constant by having a fifth factor: the non-leakage probability Pnl-

Thus keff = pei)f x Pnl = k® x Pnl

where keff is the effective multiplication constant for the finite reactor.

k® is the infinite multiplication constant for the equivalent reactor in terms of fuel and moderator materials and their basic incre­mental geometric arrangement within the core (i. e., the fundamental lattice cell ar­rangement) but of infinite size.

PNL is the non-leakage probability; that is the probability that a neutron remains in the reactor and is absorbed there.

number of neutrons absorbed

„ in the reactor

PNL = ———————————— IS < 1

number of neutrons absorbed

+ leaking out

Thus, keff is the product of four factors petjf (k*,) which describe the neutron behaviour inside the re­actor and a fifth factor Pnl which expresses the pro­bability that the neutrons remain in the reactor.

The probability that a neutron does leak out of a reactor, or the fraction of all neutrons that do leak, is:

Pl = 1 — Pnl

Consider the compound nucleus uranium 239 formed by the absorption of a neutron by U-238. For the neutron to induce fission the sum of the binding and kinetic energy transferred to the U-239 compound nucleus must exceed its fission activation energy. From the liquid drop model the activation energy of U-239 is 7 MeV; the difference between the binding energies of U-238 and U-239 is 5.5 MeV, Thus the kinetic energy of the incoming neutron must be at least 1.5 MeV. In practice, arising from imperfections in the liquid drop model, it is found that the threshold value for the neutron energy is 1.1 MeV.

Materials, such as U-238, which may undergo fission following absorption of fast neutrons of a few MeV kinetic energy are called fissionable materials.