It could be thought that burnable poisons or fission products could be suddenly released from the core, thus determining a reactivity increase. It has been proved by numerous experiments that this release rate is always very slow, even at very high temperature, and even without making use of coated particle. For the above reason this cannot be considered a real accident.

The effectiveness of control rods decreases with time as a result of the burn-up of the absorbers. The burn-up of a control rod can be easily calculated once the flux distribution inside the rod is known from a transport calculation. The flux distribution obtained from a transport calculation (S„ or collision probability method) must first be normalized to the value corresponding to full reactor power. In the transport calcula­tion the rod is subdivided in s concentrical shells (e. g. the regions of a collision probability method) and for each shell j and energy group і an unnormalized flux щ is obtained. The diffusion calculation performed over the complete core gives for each group і the diffusion theory flux ф, (asymptotic flux) at the outer boundary of the rod, normalized to full reactor power. The net current entering the rod for each group і is given by

J. =§<*.. (13.1)

This expression follows from Fick’s law and from the definition of the extrapolation length d.

The total number of neutrons absorbed in group і per unit length of control rod is

Пі = Jilrrr (13.2)

where r is the control-rod radius.

The normalization factor / for the fluxes <pti can be obtained considering that

Пі = І) X. iiVicpaf (13.3)

j-i

where V) = volume of shell j per unit control rod length,

2ац = macroscopic absorption cross-section of shell j, group i.

With the normalized fluxes ftpa the reaction rates and burn-up of each shell can be calculated.

Practically only the outermost sheet of absorber burns because the flux decreases very rapidly inside the rod, so that in black rods the effect of burn-up is practically equivalent to a reduction of the absorber diameter. As burn-up proceeds the transport calculation has to be repeated in order to obtain the flux distribution for the new burn-up time steps.

Another important problem is posed by the heat production in the control rods, which must be known in order to provide the necessary cooling. This heat production results from absorption of y-rays and of neutrons.

Usually the main absorbing material of a control rod is l0B. The absorption of a neutron in l0B takes place according to the following reactions:

,0B + n —— — 7Li + a + y(2.79 MeV)

—7Li* + a + 7(2.31 MeV) (13.4)

?Li*——————————— — 7Li + 7(0.48 MeV)

As these ys do not have a very high energy, one can assume that they are absorbed by the metal components of the rod, so that each neutron absorbed in Boron produces

2.79 MeV = 4.47 x КГ13 W sec.

Another important control-rod material is iron.

The ys produced by Fe(n, y) reactions are usually only a small fraction of the y-flux of the core. This y-flux is usually calculated with computer codes used for shielding purposes. Once this flux is known the heat production can be calculated using the у-absorption coefficients for each control-rod component. The fine structure of the y-flux in the control rod is often neglected in these calculations.

The accuracy of HTR calculations has been tested with the above discussed experiments. Starting from the informations available about the uncertainty of the fundamental data and calculational methods it is also possible to have an independent assessment of this accuracy.

Some calculations have been performed in the past in this direction/54’55’ The data used for these calculations were not the best sets now available, so that the errors obtained are somewhat pessimistic. Figure 15.1 gives the relative effects of different uncertainties obtained by a General Atomic study of 1968.<54) The error in space-time depletion effects is here assessed supposing the use of one — and two-dimensional burn-up codes instead of three-dimensional calculations, and the error on space — temperature effects supposes that in the data preparation for the burn-up calculations the effect of local temperature differences in the core have been neglected.

The conclusion of this report is that the total uncertainty arising from cross-sections is of the order of 3% of the fuel cycle cost, or 1% of the total power cost. The estimated cost uncertainties associated with physics data and calculational methods are less than 3% of the total power cost.

A similar study was performed in 1969 for the THTR reactor/55’ analysing the effect of the uncertainties in microscopic nuclear data and in calculational methods on the attainable burn-up. The microscopic nuclear data analysed in this investigation are: absorption and fission cross-sections (including resonance data) and fission product yields. The analysis has been performed with a zero-dimensional burn-up code (BO)<56> and refers to the THTR reference fuel cycle. The reactor is supposed to be in equilibrium so that the average core composition determining keft is the average over all burn-up stages of a fuel element. The quantity Xi (cross-section or yield) being investigated is changed by an amount AXt corresponding to its uncertainty. Using the same residence time as in the reference case the new average composition and are

calculated. In this way is obtained the AkefT due to the change ДХ. Then the burn-up calculation is repeated changing the fuel residence time until reaches again the value of the reference case. In this way Д fifa is obtained. The total inaccuracy in kcff or in fifa is then calculated as square root of the summation of the square of each contribution.

This supposes a normal Gaussian distribution for every single uncertainty. The assumed uncertainties and the single contributions to the total errors are given in Tables 15.1, 15.2, 15.3 and 15.4. It must be remembered that better data are now available in many cases.

The total effect of ke« and burn-up is shown in Fig. 15.2.

+An increase of the absorption cross-section of 232Th causes also an increase in conversion ratio, so that the loss in does not correspond to a full loss in fifa. The opposite is true for 2”Pa.

The uncertainties in the calculation methods include some points characteristic of pebble-bed reactors like burn-up measurement, effect of fuel element flow, streaming in pebble bed, empty space above the core. The errors in space-temperature effects are of little importance in this type of reactor because of the continuous fuel-element movement.

Other errors considered are the inaccuracies of the representation of the reactor geometry by the mesh structure of diffusion theory codes (Akefr = 1.5%c), and the uncertainty of the order of %c in the fresh fuel composition (A= 1.0%o).

A particular treatment is required by the uncertainty in the energy per fission e released in the reactor (Де ~ 3%). This uncertainty has no effect on the burn-up measured in fifa, but on the burn-up measured in MW d/t and then on the fuel cycle cost. For a given reactor power the flux level is affected by an uncertainty equal to Де. In this case of continuous refuelling the change in flux level causes a change in equilibrium composition (average over all burn-up stages) and then on kcfs (Aksf! = 4%o in the case of ref. 55).

This reactive material can be either moderator or fuel. The water leakage into the core has already been considered. An insertion of graphite can happen in pebble-bed reactors where the upper reflector is suspended about 1 m above the core. In this case a part of the reflector can fall on the core producing a reactivity increase. In reactors with prismatic fuel a fuel element can suddenly fall into the core during a reloading operation. The reactivity increase is due to the insertion of the fuel element and to the elimination of neutron streaming out of the hole.

High-temperature reactors are very safe compared with other reactor systems, and a secondary shut-down system is not always required.

In the Peach Bottom reactor safety fuses are built in the upper part of the reflector. From these fuses hang short absorber rods, which fall into the core if the melting temperature of these fuses is exceeded.

In the Fort St Vrain reactor it is possible to compensate up to 12 nile of reactivity introducing B4C in granular form in the core channels.

Other secondary shut-down systems can also be provided by poison gases. The introduction of 1 atm of N2 or Kr can produce a reactivity loss of the order of 1 — f — 3% depending on the voidage (Kr is absorbing approximatively 1.5 times as much as N2). In the THTR reactor up to 14 nile reactivity might be compensated injecting BF3 as poison gas for additional emergency shut-down. The difficulty posed by cleaning the primary circuit before restarting the reactor discourages the use of poison gases.

Reference

1. P. Marien, Research and Development Division First Semi-annual Report for the period 1st July 1960-31st December 1960, p. 63. Dragon Project Report 34.

The first operation, which is seldom directly performed by the reactor designer, consists in the preparation of the nuclear libraries for the codes for spectrum calculations, starting from the basic data of the ENDF/B type.

The proper reactor design usually starts with zero-dimensional optimization calcula­tions for the equilibrium condition. Later space-dependent calculations for the equilib­rium condition are performed including control rods, temperature coefficient, heat — transfer and dynamics calculations. The running-in strategy is then defined and the above calculations are repeated for different stages of the running-in period.

The type of approximation needed in order to obtain sufficient accuracy without waste of computing time will have to be chosen for each of these steps by the designer. This choice is based sometimes on comparison with experiments, but most frequently on numerical experiments. This means that in order to check whether an approximation is sufficient for a certain type of calculation, the designer tries a higher approximation in one typical case. If the difference is sufficiently small the approximation can be considered as sufficient. This can be done for most types of numerical approximations: number of energy groups, space dimensions, mesh points, order of an S„ approxima­tion, size of time steps etc. Obvious limits to this procedure are posed by computer capacity and costs, but it is often possible to simplify the problems in these tests (e. g. the number of energy groups necessary for a two-dimensional calculation is tested on a one-dimensional test case). It must be remembered that not always the next higher approximation gives better results, but it is sometimes necessary to test on even higher approximations (e. g. a two thermal group calculation is not necessarily better than a one thermal group calculation, so that the test may require four or more thermal groups). Another typical numerical experiment consists in checking the method being studied with a Monte Carlo calculation.

The advantage of numerical experiments is that they are cheaper than actual experiments, and enable the designer to check one single approximation at the time, without involving the global error due also to other approximations and inaccuracies in the data. Obviously these analyses are only thoroughly performed when high comput­ing costs are involved, whereas in simpler cases higher approximations are used. This may lead to what can be easily considered a waste of computing time, but overall economic considerations may indicate that it is often cheaper to perform a sophisti­cated calculation instead of spending days in analysing which simplification can be tolerated without loss of accuracy.

In order to illustrate the practical use of the methods and codes described in this book, we give here some schematic flow diagrams of the most frequently performed reactor calculations.

The Sequence of Reactor Design Calculations

Estimate of core composition

Spectrum-averaging cross section codes

Estimate of equilibrium fuel cycle composition

Attractive fuel cycle

♦_____ .

І-D codes to refine knowledge of equilibrium cycle

Final estimates of core composition and fuel cycle cost for a given fuel management strategy

Fig. 16.1. Information flow diagram for fuel-cycle calculations (to estimate composition and power cost for a given fuel management scheme).*0

Figure 16.1 gives the flow diagram of a typical sequence of fuel-cycle calculations for the equilibrium and the running-in period, as established by General Atomic. w

A similar scheme for pebble-bed reactors is described in Fig. 16.2. Possible flow paths for burn-up and kinetics calculations are shown in Figs. 16.3 and 16.4m while Fig. 16.5 gives a possible way of designing the control-rod system.

All these flow paths are only indicative and change from establishment to establish­ment according to the computer codes being used, and to the peculiarities of the reactor under consideration.

Various computer codes have been mentioned in the course of the preceding chapters. It is not within the scope of this book to give a complete list of the codes which can be used for HTR calculations. Abstracts of the codes for nuclear calcula­tions, whose distribution is not restricted, are published regularly by the NEA Computer Programme Library of Ispra, and by the Argonne Code Center for US users.

Estimate of core composition

Spectrum-averaging cross­section codes

2-D X-Y calculations

—- f—-

Reactivity and power distributions for use in radial zoning studies, radial peaking factor estimates, etc., as a function of time

Reactivity and power distributions for use in axial zoning designs, rod movement estimates, axial peaking.

factor estimates, etc., as a function of time

Fig. 16.3. Information flow diagram for conventional core depletion calculations.

Fig. 16.4. Possible flow diagram for control rod calculations.

Some particular attention must be paid to the recent development of modular systems which permit a considerable authomatization in the sequence of many reactor calculations without the need of a separate data manipulation for each single calcula­tion. These systems allow the execution of a sequence of codes (modules) with various paths which may be selected at run time, and provide for exchange of information between these modules. This exchange of information is obtained writing and reading from a collection of the data (interface) which are produced or required by the operation of a module and are available for use by other modules.® The use of modular systems results in a saving of both manpower and computer time.

In the HTR field efforts in this direction started in the past with the development of coupled burn-up and spectrum codes (e. g. MAFIA’41) to which later transport codes have been added (VSOP<5>), but these cannot be properly called modular systems. The peculiarity of modular systems lies in the possibility of changing the path of the calculation at the will of the user. A typical example is given by the WIMS-E scheme’6’ which was originally a code for spectrum calculations’7’ and has now evolved to a system including almost all codes needed for a complete core calculation (see Table 16.1). The modules available in WIMS-E are listed in Table 16.1 with a brief description of their functions. The names of these modules are usually those of previously existing codes with the letter “W” added at the beginning.

This scheme is now the basis of most HTR calculations in Britain. Another modular system developed by UKAEA is COSMOS which is mainly intended for fast reactors, but has already been used for HTR calculations’8’ because it can perform three­dimensional burn-up calculations using its three-dimensional diffusion module TIGAR.

While important in other reactor types, these accidents either cannot occur or have negligible effects in high-tempeature gas-cooled reactors. In pebble-bed reactors the voidage between the fuel elements can change locally, but the effects on reactivity are always very small.

(d) Loss of coolant and blower failure

These accidents do not change directly the neutron multiplication properties of the reactor, but affect the heat-removal mechanism. A failure of this mechanism can either be the result of a depressurization or of a blower failure.

The first of these accidents can be treated considering that the coolant pressure decreases with time according to a given law, which depends on the type of leak. In case of blower failure the mass flow will decrease following a curve of exponential type. It is usually possible to guarantee an emergency cooling with a mass flow reduced to 6-10% of the original value (e. g. one of the blowers continues to operate at reduced power). An emergency cooling for the removal of the decay heat after a scram has occurred is in any case necessary since extreme temperatures have to be avoided.

The peculiarities of HTR physics

The calculation of an HTR core presents problems which are rather different from those encountered in the calculation of other thermal reactors. We list here the most important ones, although they have already been partially discussed in the preceding chapters.

The use of graphite as a moderator presents the advantage (in comparison with light water) of a low absorption, but requires a detailed treatment of the scattering law in graphite.

The small parasitic absorption associated with non-fuel components allows a high burn-up with low fuel cycle costs, but this requires a heavy loading of fertile and fissile materials.

The combined effects of the graphite scattering properties, high fuel loading and high temperature produce a rather hard neutron spec’trum, which may strongly deviate from the classical fast 1/E and thermal Maxwellian distribution. However, the ratio of moderator atoms to fuel atoms tends to be considerably higher than in light water reactors (mainly because of the low absorption of graphite) and the fraction of thermal fissions over total fissions is in HTRs larger than in LWRs. It can be seen in Fig. 14.1 that approximatively 30% of the fuel absorptions take place above 1 eV in PWRs, while this figure is only about 15% in HTRs.<3) However, because of the high moderator temperature the peak of the thermal distribution lies at higher energy in HTRs than in LWRs. A comparison of the HTR spectrum with that of other graphite moderated reactors is shown in Fig. 14.2<4) where also 235U and Pu cross-sections are plotted.

The He coolant has no influence on the core reactivity: because of this reason and of the use of a solid moderator, reactivity effects due to changes in core density are negligible.

Because of the rather homogeneous dispersion of fuel in the moderator, the thermal heterogeneity effects are much less pronounced than in other reactors (the thermal self-shielding factors are always very near to unity).

The use of coated particle fuel poses the problem of the treatment of the double heterogeneity, at the macroscopic level of the fuel pins and at the microscopic level of the coated particles. This is particularly important in the calculation of resonance absorption.

Because of the homogeneous dispersion of fuel in the moderator the resonance integral is in HTRs 2 or 3 times higher than in most other reactors.

Because of the hard neutron spectrum the treatment of some low-lying resonances (233U, 135Xe, Pu) becomes particularly important in this reactor type.

The effect of graphite crystal binding and of the low-lying resonances is particularly important in temperature coefficient calculations.

Because of the hard neutron spectrum, particular care must be taken in the calculation of spectrum and power distribution at the boundary between different regions (e. g. core-reflector interface).

HTR fuel has a very high burn-up. The resulting high fission product absorption requires a very detailed treatment of fission product build-up, considering all

important decay and absorption chains. This high burn-up imposes also a detailed treatment of fuel management and its effect on power distribution.

Very characteristic of this reactor type is the treatment of fuel performance limits which are not as simple as fixed melting temperatures which should not be exceeded, but depend upon a combination of fast neutron dose, temperature level and temperature gradient (this last quantity being proportional to the power density).

Particular problems can be related to the different types of fuel cycles. More heterogeneous designs have been considered in the past for low enriched U cycles in order to lower the required enrichment, but recently the same homogeneous design appears to be best suited for both cycles. The presence of Pu in low enriched cycles (or even more in cycles with Pu make-up) may pose the problems of conveniently treating the low-lying Pu resonances.

In general comparing calculational methods for HTRs and for other reactor types one must notice the greater importance of detailed spectrum calculations (it is necessary to take a large number of groups) but the less pronounced thermal heterogeneity effects. With possibly the exception of cases with high Pu loadings, detailed thermal cell calculations are not essential to HTRs. This explains the large use of zero-dimensional spectrum calculation codes in which thermal heterogeneity is simply represented by self-shielding factors.

For the same reason diffusion theory is sufficient for the analysis of most HTR design problems, while transport theory needs only to be used for special problems like calculations of self-shielding, control rods, burnable poison rods, resonance self — shieldings, etc.

Because of the high burn-up, a considerable amount of work has been invested in the development of HTR burn-up codes.

The peculiarities of HTR core dynamics have been treated in Chapter 12; we add here a list of the most important features.

Because of the combination of solid moderator, completely ceramic core, chemically and neutronically inert coolant, high thermal capacity and good fission product retention by the coated particles, core kinetics play a minor role in HTRs. Steam generators can be more affected by transients than the reactor core.

The moderator temperature coefficient can, in some cases, be positive in HTRs because of the effects of the low-lying resonances and of the absence of significant density coefficients so that the overall coefficient, still remaining negative, is smaller than in most liquid moderator reactors. The overall temperature coefficient is in large HTRs of the order of 10”5/°C compared to 10 4/°C for water-moderated reactors and КГ6-КГ7/°С for fast reactors.

The peak temperatures during power excursions are rather insensitive to the value of the temperature coefficient, mostly because of the high thermal capacity of the core.

In HTRs, as in all solid moderated gas-cooled reactors, core dynamics calculations are much easier than in reactors with liquid coolant or moderator.

Because of the high thermal capacity, in the case of transients the higher modes of the flux distribution do not contribute appreciably to the core temperatures, so that all dynamics calculations can be performed in the fundamental mode.

In the case of accident analysis the treatment of space dependence can be more important in thermal calculations than in neutron flux calculations. Space — dependent neutron flux calculations are most important in the analysis of the load following capability of the reactor.

References

1. R. C. Dahlberg, Physics of gas cooled reactors. ANS topical meeting on new developments in reactor physics and shielding, 12-15 Sept. 1972, Kiamesha Lake, NY ANS CONF-720901.

2. J. G. Tyror, J. R. Askew and I. Johnstone, Some problems in the physics of high temperature reactors. ANS topical meeting on new developments in reactor physics and shielding, 12-15 Sept. 1972, Kiamesha Lake, NY ANS-CONF-720901.

3. H. B. Stewart, M. K. Drake and R. C. Traylor, GA-8571, 5 Mar. 1968.

4. J. A. Desoisa, DCPM 18/CEGB 1.

5. K. Friedrich, L. Massimo and E. Vincenti, Space dependent studies of transients in high temperature reactors. Specialist meeting on Reactivity Effects in Large Power Reactors, Ispra, 28-31 Oct. 1970, EUR 4731 f-e.

Local temperature increases can be due to wrong gagging, channel blocking, loading errors, spatial instability, incorrect spatial distribution of the inserted control rods. The main problem posed by these local temperature increases is the difficulty of their detection. Monitoring of local temperature is not always possible or reliable in HTRs and one must often rely on proper planning of the operational procedure in order to avoid these temperature disturbances.

In many cases an accident of this type will only be detected by the increase of the primary circuit activity due to the fuel damage.

(e) Start-up accidents

Any one of the above-mentioned accidents can happen during start-up, but the typical start-up accident is a continuous insertion of reactivity resulting in too short a reactor period. Provided a scram occurs if the period gets smaller than a given value, no start-up accidents can lead to dangerous temperatures. If the reactor is cold the high thermal capacity of graphite gives plenty of time for the control system to intervene.

In the treatment of start-up accidents it is important to consider that only the prompt heat contributes to the core temperatures because delayed heat is not yet present.

15.1. Comparisons between theory and experiments

Although as we have seen, modern HTR calculations do not make use of any experimental correlation, many experiments have been performed in order to test the accuracy of the calculational methods and data used for HTR design.

Zenith I

These experiments started in 1959 with the zero energy Zenith I reactor in Winfrith which reached temperatures up to 800°C.(1 3) The experiments included criticality, temperature coefficient, flux distributions, integral and differential (time of flight) spectrum measurements, fine structure, etc. These experiments have been performed on the U-Th systems, while later the Zenith reactor has been used for measurements on Pu-Th systems.

Dragon

Although not designed for physics experiments the Dragon reactor experiment provided, and still provides, a considerable amount of valuable data, first on U-Th systems and later on low enriched U systems.

Calculations on the Dragon reactor are difficult because of the small size and irregular nature of the core, but in spite of that it has been possible to analyse some zero energy experiments with a good accuracy. Reactivity was predicted to within 200mNile.<4)

Measurements of fast neutron damage have been performed and compared with the fast flux obtained from routine core calculations (see § 8.12). Physics measurements on highly irradiated fuel are under way.’5’