Physics of fast reactor control rods

Progres~ in Nuclear Enerf4y, Vol. 16, No. 3, pp. 287 321, 1985. Printed in Great Britain. All rights reserved.

Copyright I

0149 1970/85 $0.1)0+.50 1985 Pergamon Press Ltd.

PHYSICS OF FAST REACTOR CONTROL RODS J. L. ROWLANDS Atomic Energy Establishment, Winfrith, Dorchester, Dorset, U.K. (Received 1 July 1985)

1. INTRODUCTION In most designs of fast reactor, reactivity control is achieved by moving neutron absorbing control rods. The control rods are usuaUy separated into two groups, the shut-down rods and the regulating rods. Thus to raise power, the shut-down rods are completely withdrawn from the reactor core and the regulating rods are partly withdrawn. Their position is adjusted to achieve the required power level. An increase in power results in increases in the temperatures of the fuel and the coolant (relative to the coolant inlet temperature), These temperature increases cause a net reduction in the reactivity of the reactor which is balanced by a partial withdrawal of the regulating rods. Other changes, such as the reduction in fuel reactivity with burn-up, are compensated by gradual withdrawal of the regulating rods. Regulating (or operating) rods can be moved either as a curtain or individually controlled in order to shape the reactor power. Shut-down rods can be further subdivided into primary and secondary shutdown rods. These have different mechanical and operational characteristics in order to reduce the probability of common mode failure. The designer usually allows some flexibility in the way in which the function is divided between regulating and shut-down rods. For the control of a large commercial fast reactor, comprising some 360 fuelled subassemblies, typically 24 control rods are required. Figure 1 shows the core plan of SUPER-PHENIX. Since the control rods occupy core positions created by omitting fuelled subassemblies, the control rod channels are geometrically similar in size and shape to the subassemblies. In current designs the subassemblies are clusters of fuel pins in hexagonal wrappers. A typical geometrical arrangement of a control rod is illustrated in Fig. 2. An outer steel guide-tube, which has the dimensions of a core subassembly wrapper, is located on the reactor lattice. The cluster of 19 natural boron carbide absorber pins is enclosed in a steel wrapper box which

0 Inner core subossernblies C) Outer core subossernblies • Control rods Secondery shut-down rods Q Breeder subossemblies ~) Steel reflector ossemblies Fig. L SUPER-PHENIX core plan.

moves within the guide-tube. Below the absorber section is a spike which moves within the spike guidetube. This acts as a shock absorber when the rod is dropped. The guide-tube, absorber region wrapper box, spike and spike guide-tube must be designed to accommodate irradiation induced swelling and bowing whilst, at the same time, being free from vibration. In some designs (for example, the SUPER-PHENIX complementary shut-down rods) the absorber section is articulated to enable it to move down through a deformed guide-tube. Regulating rods have an irradiation lifetime of a few years. The use of boron enriched in its a°B isotopic content, in place of natural boron, enhances the absorption of neutrons, but increases the

287

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J.L. ROWLANDS

Fig. 2. Cross-sectional view of a boron carbide control rod within its guide-tube. The rod comprises a cluster of 19 pins within a steel wrapper.

cost and reduces the irradiation lifetime of the control rod. In this case the rod usually consists of a larger number of smaller diameter pins because of the higher heat deposition rate in enriched boron carbide. The accuracies to be aimed for both in the prediction of control rod reactivity requirements and in the reactivity worths of the arrays of control rods which must be provided to match these needs are decided by economic considerations. To cover uncertainties it is necessary either to provide extra control rod positions or to enhance the effectiveness of the rods adopted for the design. Either course is expensive and consequently a high precision is demanded for predictions of requirements and rod effectiveness. Extra control rod positions carry the penalty of an increase in the number of control rod mechanisms and control rods, and also in reactor size, whereas increasing rod effectiveness (by boron enrichment, for example) increases the cost of the rods and reduces their lifetime in the reactor. The reactivity changes, 6p, which must be compensated for by control rods in a typical 300 MW(e) sodium-cooled prototype fast reactor are as follows: (1) The loss of reactivity with fuel burn-up. This is caused by the reduction in the fissile material content of the fuel and the build-up of fission products. Fuel restructuring and elongation effects due to irradiation, and possible fuel pin movements, must also be allowed for. The total requirement depends on the maximum fuel burnup and the fuel management strategy. For a six-

batch fuel cycle (in which, on average, one sixth of the core is removed and replaced at each refuelling) and a maximum burn-up of 10~ of fuel atoms, the reactivity falls by about 1.5~o in each stage (or run) in the cycle. An additional 0.5 ~0 6p is typically required to cover differences in the patterns of reloads. Thus the total fuel burn-up control requirement is about 2~o 6p. (2) The reactivity change which occurs when the reactor is taken from the normal shut-down temperature up to power. In the normal shutdown state of a sodium-cooled fast reactor the sodium is at a temperature of about 250°C. To raise the reactor to power, reactivity must be added to compensate for the reductions in reactivity resulting from the increases in the sodium inlet temperature from about 250°C to 400°C and in the temperatures of the fuel, cladding and coolant in the core (above the coolant inlet temperature) consequent upon the increase in power. These effects depend on the isothermal temperature coefficient and the power coefficient of reactivity, respectively. Reactivity effects due to changes in temperature of the control rod mechanisms themselves must also be allowed for. The combined effect is about 1 ~o 6p. (3) The variations of reactivity with time following shut-down from power. These are small in a fast reactor and the main effect arises from decay of 139Np (produced as a result of neutron capture in 238U)to 23 9 p u ' This typically adds about 0.2 % 6p. (4) A shut-down reactivity margin is required for refuelling and replacing control rods. The margin must be sufficient to cover any possible mishandling, such as incorrect fuel loading or unintentional removal of control rods. A margin is also required so that the reactor is shut-down even when one or more rods fail to drop into the core (due to electrical failures, mechanism failures or a rod sticking in a guide-tube). A shut-down reactivity margin of 3 ~ 6p is typical. (5) At the end of the irradiation period the control rods are in their most withdrawn position. They must be inserted sufficiently far, however, to provide any radial power shaping which may be required and to provide a sufficiently high rate of change of reactivity with rod movement to give the necessary transient control. The loss of control rod reactivity due to rod absorber burn-up must also be taken into account. A typical minimum requirement for these purposes is 0.2~ 6p. In this example, the total reactivity which must be invested in the control rods is about 6~o. Larger, commercial-sized, fast reactors will have somewhat

Physics of fast reactor control rods different requirements for several reasons. The variation of reactivity with burn-up is less but the number of batches in a fuel cycle is likely to be fewer than six and the maximum fuel burn-up might be greater than 10%. The variation in reactivity with temperature and power changes can be greater because of the more negative Doppler and larger sodium density coefficients in the larger reactor. The rods must be designed to allow adequate cooling and to accommodate irradiation induced swelling and deformation. This requires prediction of heat deposition rates, gas production rates (in particular helium production by the (n, ~) reaction in 1°B) and neutron irradiation induced atomic displacement swelling rates. The temperature limits determine the diameters of the absorber pins. Accommodation of swelling requires the provision of gaps between the absorber pin and its cladding and between the wrapper enclosing the pin cluster and the guide-tube in which it moves. These considerations determine the volume of absorber which can be included in the rod and in consequence the reactivity the rod can control. The irradiation lifetime of the rod is usually determined by swelling rather than loss of absorber effectiveness. Another factor which must be estimated is the radioactivity induced in the rods, such as the tritium produced in boron carbide (by the l°B (n,T, 2~I reaction), or the activation products produced in other potential control rod materials such as europium and tantalum. This has to be predicted for planning the handling, heat removal, storage and disposal of used control rods. The accurate prediction of the effects of the insertion of control rods on the reactor power distribution is another important requirement, It is necessary to ensure that the reactor can provide the design power output at all stages of the fuel cycle. Control rod insertion also influences the irradiation dose gradients which, in turn, affect the irradiation induced distortions of fuelled subassemblies, the control rods themselves and associated guide-tubes. Taking these factors into consideration, the broad target accuracies (1 SD! associated with control rod predictions can be summarized as follows: Rod reactivity ± 5'I,~ Absorption rates and heat deposition rates within control rods +__5% Ratio of peak to average total power in core fuel assemblies with different patterns of control rod insertion + 1O/o Irradiation doses and dose gradients in control rods and in subassemblies adjacent to control rods. + 10% The reactivity effects of the control rod followers (or

289

the channels from which the absorber rods have been removed) must also be predicted accurately because it is relative to the rod follower that rod reactivity worth must be estimated. The rod follower reactivity effects must also be calculated when the required reactor fuel enrichments are being estimated. Extensive programmes of measurements of control rod reactivity worths have been made in zero power criticality facilities. Reaction rates within the absorber regions and fission rate distributions within cores having different configurations of control rods, including partly inserted rods, have been measured. These measurements provide a comprehensive validation of calculational methods to accuracies approaching these requirements. Measurements made on operating power reactors have further supplemented this validation. Having summarized the role of control rods in a fast reactor and some of the problems facing the designer which have led to the choice of control assembly now frequently adopted, some of the design aspects of control rods will be treated in more detail. The neutron physics characteristics of control rods will be discussed, the calculational methods presently used will be summarized and the experimental information obtained to validate these calculational methods will be reviewed.

2. NEUTRON PHYSICS CHARACTERISTICS OF CONTROL RODS

The reactivity worth of a neutron absorbing control rod depends on the type of absorber material, the diameter of the absorber region and the density of the absorber. In cases where the absorber section is not cylindrical (for example a pin cluster) an effective diameter is taken which is either the diameter of the circle enclosing the pin cluster, or an equivalent mean chord. The effective diameter can also be defined as the diameter of a 'black' rod (i.e. one which absorbs all incident neutrons) which has the same reactivity worth. However, this latter definition is not usual in fast reactor applications. The reactivity worth of a rod is not directly proportional to the density of absorber material because there is a significant neutron flux dip into the rod. Typically, the average absorber reaction rate and reactivity effect per absorber atom are only about one half of the values for a thin isolated sample of the absorber material. The variation of rod reactivity worth with absorber density is, typically, approximately proportional to the square root of the density. The relative reactivity worths of different absorber

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J.L. ROWLANDS

materials depend on the macroscopic neutron absorption cross-section and the volume of the absorber which can be contained within a rod. It has already been noted that the maximum volume fraction is determined by temperature limits and the need to accommodate irradiation induced swelling and gaseous reaction products. Tantalum has a good thermal conductivity and no significant gaseous reaction products and so a tantalum control rod can be made up from plates assembled into a stack through which pass a few coolant channels. This gives a high volume fraction of absorber. Such a high volume fraction is not achieved for boron for which the rod comprises a cluster of pins containing boron carbide pellets. In the case of natural boron carbide (19.8% I°B) a 19 pin cluster in a hexagonal array (1 + 6 + 1 2 ) is typical, whereas for boron carbide rods enriched in the absorber isotope ~°B, a larger number of pins can be required (the number depending on the enrichment). A 37 pin design ( 1 + 6 + 1 2 + 1 8 ) i s typical for a 40~o enriched rod. Europia is being evaluated as a possible control absorber. However, for europia an even larger number of pins is required, typically 127 pins for 'monoclinic'europia or 85 pins for 'cubic' europia. The volume fraction is then even smaller. For europium boride the number of pins can be fewer, 19 or 37, although when the boron is enriched a larger number of pins is again required. Taking into account the relative volume fractions, the reactivity worth ratios relative to natural boron carbide rods are about 0.8 for tantalum, 1.0 for europia and 1.3 for europium boride. An advantage of tantalum and europium over t°B is that the absorption reaction products are also good absorbers. A disadvantage is that they are highly radioactive, thus requiring special provisions for out-of-reactor cooling, handling and long term disposal. The main source of heat generation in boron carbide control rods is the kinetic energy of the reaction products from the (n, ct) reaction. Neutron scattering recoil and gamma energy deposition also contribute. Irradiation induced swelling occurs in boron carbide because of the helium produced in the (n, ct) reaction and this can be the factor limiting control rod lifetime. Irradiation induced atomic displacements cause swelling of steel and this can result in diametral growth of control rods and bowing of rods and their guide-tubes. This can also limit control rod lifetime. The helium pressure in boron carbide pins can be relieved by venting them, but ingress of sodium can then present post-irradiation handling problems. In the following subsections selected neutron physics aspects of fast reactor control rods are discussed in more detail.

2.1. Potential fast reactor absorber materials An indication of the effectiveness of different absorber materials when used in control rods can be obtained from small sample reactivity worth measurements. Measurements have been made for samples of absorber materials at the centre of ZEBRA Core 14 (the PFR mock-up assembly) (Ingram et al., 1975). The results, given in Table 1, have been expressed in terms of the reactivity per unit volume normalized to a value of unity for natural boron carbide. On this basis, it can be seen that europium is better than any of the other rare earth elements and that rhenium and irridium would also make effective control rod absorber materials, although their high cost makes their use unlikely. Moderating materials have the potential to enhance the reactivity worth of a rod by scattering neutrons from energies at which the absorption cross-section is small to energies at which it is large. The 11B and carbon in a boron carbide control rod have this effect. The effect can be markedly enhanced by introducing hydrogen compounds into the rod and the use of zirconium hydride and gadolinium hydride in control rods has been studied for this reason. Measurements have been made in ZEBRA, comparing rods containing different materials arranged as a 19 pin cluster (Ingram et al., 1975). The largest reactivity effect was

Table 1. Relative reactivity worths of different absorbers

Absorber B Rh Pd Ag In Sm Eu Gd Tb Dy Ho Er Tm Yb Lu Hf Ta Re lr Pt Au

Sample Thickness Material (mm) B4C Metal Metal Metal Metal Metal

EU203 Metal Tb,~O~ Metal Ho20 3 ErzO3 Tm20 3 Yb20 3

tu203 Metal Metal Metal Metal Metal Metal

5.5 0.5 1.0 1.0 1.0 2.1 2.4 1.1 5.5 2.1 5.5 5.5 5.5 5.5 5.5 0.8 1.0 0.5 0.5 1.0 1.0

Normalized reactivity per unit volume 1.00 0.72 0.42 0.63 0.35 0.36 1.43 0.41 0.69 0.41 0.57 0.27 0.40 0.25 0.57 0.44 0.72 1.37 1.29 0.45 0.55

Physics of fast reactor control rods found for a rod consisting of an outer annulus of boron carbide and a central zone of zirconium hydride. This composite rod had a worth 1.5 times that of the natural boron carbide reference rod, equivalent to enriching this to about 50~o ~°B. Thus the introduction of hydrogen can considerably reduce the quantity of absorber required for a given worth but a consequence is a more rapid fractional burn-up of the absorber isotope. Measurements of the effect on the 239pu fission rate distribution showed that only when the hydride was in the outer region of the rod did the power increase near the rod. This increase was most marked for gadolinium hydride but even in this case the average 23'~pu fission rate in the fuel region adjacent to the rod reached a value only 4}~, above the value with no absorber rods present. In a composite rod in which the moderator region was enclosed by a region of absorbing material the 239pu fission rate pattern was similar to that obtained with an unmoderated rod.

291

25

2O "13 15

._>

t 0

cr 05--

I

L

I

I

q

2

3

4

M o s s of

lab in control rod (kg)

Fig. 3. Variation of reactivity worth with l°B content of control rod.

2.2. Dependence ql'control rod reactivity worth on

absorber density and rod diameter In the ZEBRA Mozart programme measurements were made of the variation of control rod worth with I°B content (Broomfield et al., 1973). The assembly was a mock-up of the Japanese reactor MONJU. The effective diameter of the control rod absorber pin cluster was about 9 cm. The variation of the rod reactivity worth with l°B content is shown in Fig. 3. The mass of I°B ranged from 0.57 to 4.88 kg. For this size of rod and range of l°B content the reactivity worth varies approximately in proportion to the square root of the ~°B content, reflecting the increase in flux dip into the rod with increasing I°B content. For a natural boron carbide P F R /C F R rod, which has an absorber pin cluster effective diameter of about 12 cm, the rod worth is 15~0 higher than for a M O N J U rod (effective diameter 9 cm) with the same ~°B content. For arrays of rods the variation in rod worth with boron enrichment and rod diameter is different from the variation found for single rods. This is because interaction effects increase with rod worth. These are discussed further in Section 2.3. The reactivity worths of rods containing europium oxide have been compared with natural boron carbide rods in the EMC programme in Z P R O (Dobbin and Daughtry, 1976). Measurements were made with different volume fractions of absorber in the rods, the fractions being 4, 8, 16 and 32~o of the theoretical density (6.99 g/ml for E U 2 0 3 and 1.64 g/ml for natural B4C), the diluent materials being sodium and steel.

Table 2. Relative reactivity worths of rods containing Eu203 and B4C Volume fraction (°o) 4 8 16 32

Relative reactivity worths Eu203 B4C 5.6 9.3 14.9 22.6

4.0 7.2 12.3 19.8

Worth ratio EuzO3/B4C

1.40 1.29 1.21 1.14

The relative reactivity worths are presented in Table 2. For the smallest volume fraction the worth of Eu203 is 1.4 times the worth of B4C. As the volume fraction increases, the worth of europia becomes closer to the worth of natural boron carbide. Resonance shielding in the two natural isotopes of europium, '51Eu and t 5aEu ' is a factor contributing to this relative reduction in the worth of Eu20 a as the volume fraction increases. Supercell calculations of the effects of small variations in the boron content and enrichment of a 19 pin cluster boron carbide rod have been made and the results are given in Table 3. The reference case, Case 1, is a natural boron carbide rod having a pin cluster diameter of 15 cm. In Case 2 the boron carbide content is reduced by 10°~ keeping the dimensions fixed and retaining the same steel and sodium content. In Case 3 the area of the pin cluster is reduced by 10~o so that the boron content is the same as Case 2, but the boron density is the same as Case 1. This reduces the rod worth by about 1.2~,, relative to Case 2. However, the

292

J.L. ROWLANDS Table 3. Variation of rod worths and reaction rates with aspects of control rod design

Case 1. Reference case: Natural B4C 2. 10~0 reduction in B4C content 3. 10~o reduction in B4C content and tighter pin packing to reduce the area occupied by the pin cluster by 10~o (~5~o reduction in cluster radius) 4, 10% average burn-up 5. 30% enriched l°B 6. 40c% enriched l°B 7. 50% enriched m°B

% change '°B reaction rates in rod worth Centre pin Ring of 6 Ring of 12

-5.2

3.51 3.34

3.70 3.50

4.33 4.03

-6.4

3.26

3.42

3.98

- 3.8 15.5 26.9 35.8

3.43 4.00 4.32 4.55

3.58 4.28 4.70 5.01

4.07 5.31 6.09 6.75

reduction is sensitive to associated changes in the sodium and steel fractions. Case 4 gives the effect of a 10% average burn-up of the I°B in the reference rod. The reduction in m°B content is 10.6% in the outer ring of 12, 9.1% in the ring of 6 and 8.6 % in the central pin. Although there is a larger reduction in ~°B in the outer ring of pins than in Case 2 the higher moderator fraction (carbon, 7Li and 4He) in Case 4 results in an enhanced rod worth. The remaining cases show the effects of boron enrichment on rod worth and relative pin reaction rates. The reaction rate in the outer ring is about 6% higher than the average for the pin cluster, in the reference case. The difference increases to 11% for the 50% enriched boron rod. For the normalisation used to produce the relative reaction rates (constant total fission rate in the supercell) the ratio of the reaction rate in the outer ring of pins to the control rod reactivity worth increases by 5% on going from natural boron to 50~o enriched boron, the ratio of average rod reaction rate to reactivity worth remaining approximately constant.

2.3. The dependence of control rod reactivity worths and interaction effects on core size A control rod perturbs the neutron flux both locally and over the reactor as a whole. The local flux dip towards the rod is about 10% in the adjacent fuel region and increases to about 40% at the centre of the rod. An off-centre control rod can produce a flux tilt across the reactor core of a factor 2 or more. The flux tilt increases with the reactivity worth of the rod and with the size of the reactor core. The flux tilt is particularly large in annular geometry cores which have a large central breeder zone. The flux dip and flux tilt effects result in interactions between control rods.

The reactivity worth of a pair of adjacent control rods is less than the sum ofworths of the rods when inserted separately. The worth of a pair of rods inserted on opposite sides of the reactor, and well separated, is greater than the sum of the individual rod worths. These interaction effects are, typically, + 1 0 % in 250 MW(e) sized fast reactors and a factor of 2 in commercial sized last reactors. Calculations have been made to study the effect of reactor size on the reactivity worths and interaction effects for control rod arrays. Three-dimensional diffusion theory calculations were performed, using both natural and 90% enriched boron carbide rods, for three core diameters, 2.4, 3 and 3.6 m. The reactor models contained 19 control rod positions, comprising a central rod, a ring of 6 in the inner core and a ring of 12 in the outer core of the two enrichment zone core. The diameters of the control rods were taken to vary in proportion to the core diameter and the relative radial positions of the rods remained constant. The boron carbide density in the control rods remained the same in each case and so the mass increased in proportion to the square of the core diameter. The rod reactivity worth calculations are summarized in Table 4. The worths of all control rod patterns studied decrease with increasing reactor size, the worth of the outer control rod ring varying least rapidly. An increase in I°B enrichment from 19.8~o (natural) to 90% leads to an increase in array worths of up to about a factor of 2. Interaction effects between rings of rods and the central rod are summarized in Table 5. In this table the interaction effect is defined as the ~o difference between the reactivity worths of a combination of central rod, inner ring or outer ring and the sum of the worths when these are inserted separately. The central rod and the inner ring are sufficiently close for their interaction to

293

Physics of fast reactor control rods Table 4. Control rod worths for different boron enrichments and core sizes

Case

1 2 3 4 5 6 7 8 9 10

Worth 6(l/keft)(~o) 2.4 m core 3.0 m core 3.6 m core

Control rods which are inserted Natural boron carbide control rods Central control rod Inner control rod ring Outer control rod ring All control rods Central control rod +inner control rod ring Central control rod +outer control rod ring Inner+outer control rod rings 90% enriched (X°B) control rods Enriched inner control rod ring Enriched outer control rod ring Enriched inner + outer control rod ring

0.6 3.1 3.3 8.0 3.4 4.2 7.5

0.4 2.3 3.1 7.0 2.4 3.9 6.7

0.3 1.5 3.1 6.2 1.6 3.8 5.9

5.3 6.2 15.9

3.4 5.2 12.8

2.0 4.7 10.3

Table 5. Control rod interaction effects for different boron enrichments and cole sizes

Case

1 2 3 4 5

Description (rod rings which are inserted) Natural boron carbide rods Central rod + inner ring + outer ring Central rod + inner ring Central rod + outer ring Inner ring+outer ring Enriched boron carbide rods Inner ring + outer ring

reduce the reactivity worths, the central rod being in the range of the inner ring's flux depression, whereas the central rod a n d outer ring, and inner and outer rings interact so as to increase the reactivity worths of the rods. Interaction effects involving the central rod and inner or outer rings are small because of the small central rod worth. I n t e r a c t i o n effects are greater for enriched b o r o n r o d s although, for the case studied here (inner + outer ring), they vary less rapidly with core size t h a n the n a t u r a l b o r o n rod interaction effects. The interaction effects increase approximately in proportion to the core diameter. The interaction effects between individual rods of a ring can be m u c h larger than these values between whole rings.

2.4. Effect of rod removal on the worth o f a control rod array in a large f a s t reactor It is necessary to carry out m a i n t e n a n c e on control rods a n d their m e c h a n i s m s which involves the removal of some control rods from the s h u t - d o w n reactor. Moreover, allowance has to be m a d e for the

Interaction effect (~o) 2.4 m core 3.0 m core 3.6 m core

+ 13.6 - 7.8 + 6.6 + 16.6

+ 20.3 - 9.6 + 9.2 +23.8

+ 25.3 - 11.3 + 9.8 +28.3

+ 38.5

+48.3

+ 53.6

possibility of failure of a rod to d r o p when the reactor is s h u t - d o w n from power. It is necessary to predict the m a x i m u m change in s h u t - d o w n reactivity margin which can result from the removal of a few rods. Studies of the reactivity addition which results when a small group of rods is removed from the s h u t - d o w n core of a large fast reactor have been m a d e by Austin (1973). As expected, interaction effects can have a strong influence o n this reactivity addition. W h e n the rods removed are symmetrically positioned a b o u t the core centre their average worth is close to the average value for the rods in the rings from which they are removed. W h e n the rods are removed from one side of the core flux tilts are induced which reduce the average reactivity worth of the remaining rods. The calculations show that, for the reactor model used in this study, the reactivity w o r t h of a n array of rods can be reduced by up to 25~o by removing 12~o of the rods. Thus, for the cases studied, the m e a n reactivity worth of the rods w i t h d r a w n can be as large as two times the average worths of the rods in a full array. F o r a larger reactor the interaction effects can be even greater.

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J.L. ROWLAND$

2.5. Reaction rates within and near to the control rods Fission rate distributions through control rods for 235U and 23su have been measured in ZEBRA Assembly 13 (Marshall et al., 1976) which was a mockup of the U.K. Prototype Fast Reactor PFR. The rods comprised 19 pin clusters. The fission rates were measured in the central pin and the two rings of pins, the measurements being normalized at a point 16 cm from the centres of the rods. The fission rates measured for a natural boron carbide control rod and a europium control rod are given in Table 6. The fission rate dip for 23su is similar in both rods, being a dip of 35% in the central pin. The 235U fission rate dip is larger, being 39% in the boron carbide rod and 43% in the europia rod. The fission rate dip in the fuel neighbouring the rods was about 10%. Table 6. Fission rates measured within control rods

Position of measurement Normalization point (15 cm from centre of rod) Outer ring of pins Inner ring of pins Central pin

Boron rod Fission rates

Europia rod Fission rates

235U

23su

235U 23su

1.0

1.0

1.0

1.0

0.73 0.64 0.61

0.75 0.67 0.65

0.73 0.60 0.57

0.78 0.67 0.67

A 239pu fission rate scan in the region neighbouring the rods, normalized at a point 23 cm from the centre of the rod, showed a dip of about 11% at a point 9 cm from the centre of the rod. The fuel extended to within about 8 cm of the centre of the rod and the flux dip at this point would be larger. A control rod follower, which consists of sodium and the steel of the guide-tubes, also perturbs the fission rates. The 23su fission rate dips by about 10% at the centre of the channel, whereas the 239pu fission rate peaks by about 15%.

the neutron interaction point energy deposition rates (associated with fission, scattering recoil and charged particle emission) and the gamma energy yield rates (arising from fission, inelastic scattering and (n, ~) reactions) in a typical core flux. Gamma energy deposition cross-sections have also been condensed with a PFR core gamma spectrum. The energy deposition rates have been calculated using a total gamma flux chosen to give equivalence between gamma energy yield and deposition within a subassembly. Gamma energy deposition rates in this gamma flux have been calculated for fuel, sodium, steel and B4C and combined with the neutron point energy deposition rates. Simplified composition data have been used. The fuel is taken to consist of 238U, 239pu and oxygen and the steel is represented by iron. Spectrum averaged neutron interaction point energy deposition and gamma energy yield cross-sections are given in Table 7. The point energy deposition crosssections are subdivided into elastic scattering and inelastic scattering nuclear recoil, charged particle recoil and fission product recoil. (Charged particle recoil energy yield has only been treated for substances with large (n,p) and (n, ct) cross-sections, such as 6Li and l°B.) The gamma sources are inelastic scattering, capture and fission. The gamma energy deposition cross-sections averaged over the PFR core gamma spectrum have been converted to units of W/g in Table 8. The values range from about 4.5 W/g for light elements to 11.5 W/g for heavy elements. The energy deposition rates in the peak PFR neutron flux (8.5 x 1015n/cm 2 sec) are given in Table 9 for an inner core subassembly. The total corresponding gamma flux which gives a balance between the gamma energy yield and deposition rate within a subassembly is 1.8 x 105 7/cm ~ sec. Fission product recoil is the main source of energy in a subassembly. In light isotopes, such as sodium, elastic scattering nuclear recoil is of a comparable magnitude to gamma energy deposition. In steel, however, gamma energy deposition is the main source of heating. In boron carbide the particle recoil resulting from the (n,ct) reaction is the main source of heating. (These values for B4C do not allow for the flux dip in a control rod but relate to the core fuel flux.)

2.6. Energy deposition rates in fast reactor materials The energy yield in a neutron-nucleus reaction is not all deposited at the point of the reaction (in the form of kinetic energy). Some of it is carried away as kinetic energy of emitted neutrons and gamma rays, to be deposited over a wider volume. To illustrate the relative contributions to the heating of reactor components energy yield cross-sections (averaged over a PFR core neutron spectrum) have been used to derive

3. CALCULATIONAL METHODS Different levels of sophistication can be adopted in the calculational methods used for different applications. The amount of detail in the representation of the whole reactor, and of the control rods, and the method used to calculate the reactivity and power distribution are chosen with regard to the required

Physics of fast reactor control rods

295

Table 7. Spectrum averaged neutron interaction energy yield cross-sections (MeV x barns) Point energy deposition Substance

Elastic scattering

Inelastic scattering

l°B 11B C O Na Fe Eu Ta z38U 239pu

0.163 0.182 0.166 0.137 0.156 0.030 0.020 0.013 0.013 0.011

0.002 0.002 0.001

Capture

Fission

6.059 -0.002

0.012 0.007 0.009 0.010 0.006 0.004

--9.657 320.04

Table 8. Gamma energy deposition in a reference total. ;, flux of 1015/cm2 sec

Material B C Na Fe Ta UO 2 Stainless steel PuOz (Pu/UO2)

BaC

W/g 4.414 4.781 4.643 4.823 8.459 11.523 (4.823) (11.523) (11.523) 4.488

Density (g/ml)

W/ml

0.8394

3.897

10.96 7.97 11.40 11.03 2.52

126.3 38.44 131.4 127.1 11.31

accuracy, the speed of execution of the method and the extent to which the method has been validated. For example, in planning the fuel reloading of a reactor a two-dimensional plan representation of the reactor is usually considered satisfactory whereas, for calculating the variation of the fuel composition and the reactivity of the core with burn-up, during operation, a three-dimensional model in which the insertion positions of the control rods are represented, is usually adopted. Few group diffusion theory whole reactor calculations are generally considered to be sufficiently accurate to calculate power distributions and variations in reactivity and fuel composition with burn-up for reactor operational purposes. F o r some reactor operational calculations even simpler methods are used. Influence function and interaction model methods calculate rod array reactivity worths and reactor power distributions as simple polynomials in terms of control rod positions (using coefficients which are precalculated or determined experimentally). However, for the design of control rods, and for

Gamma energy yield Total

Inelastic scattering

6.224 0.184 0.167 0.135 0.168 0.037 0.029 0.023 9.676 320.05

0.007 0.007 0.006 0.002 0.078 0.163 0.440 0.622 0.554 0.261

Capture

Fission

1.063

0.013 0.085 15.034 5.594 1.590 2.928

0.870 24.57

Total 1.070 0.007 0.006 0.002 0.091 0.248 15.474 6.216 3.014 27.759

assessing the reactivity worths of rods in relation to the requirements, the most accurate methods which have been validated are used. Validation is required for all methods and this is discussed in Section 4. In the course of a design study and during the operation of a reactor it is inconvenient to change calculation methods and consequently new improved methods are only adopted when they offer a significant improvement and when it is convenient. Meeting quality assurance requirements involves carrying out a programme of verification and validation work before a new method can be introduced. Consequently, simple approximate methods can continue in use even when more accurate methods could be used with comparable computing efficiency.

3.1. General features oJ the calculational methods The detailed structure of control rods is not represented explicitly in the usual deterministic whole reactor diffusion theory or transport theory calculations. The calculations are carried out at two levels: the fine structure level for a subregion of the reactor containing the control rod and the whole reactor level using appropriately homogenized cross-sections. A calculation is first made for the subregion, or control rod supercell, in which the fine structure of the control rod and a surrounding region of core material is represented. The neutron cross-sections for the control rod and surrounding region are then separately homogenized over the hexagonal lattice areas of the subassembly and control rod. These homogenized cross-sections are then used in the whole reactor calculation. This is typically three-dimensional diffusion theory supplemented by an evaluation of the approximations arising from the use of diffusion theory and from the finite spatial mesh. More accurate whole reactor calculation methods than diffusion

5.03 4.33 2462

212.47

5.99 0.54 222.8

84.31

2.01 1.00 689.36 692.37

The corresponding peak gamma flux is 1.8 × 1015.

Na Fels.s.) U/PuO 2 Subassembly total/ml Natural B4C

Point energy deposition W/subassembly W/g W/ml ml

14.49

3.25 3.63 24.38 36.53

2.73 28.91 268.41

1.09 6.65 75.15 82.90

G a m m a energy yield W/subassembly W/g W/ml ml

8.13

8.40 8.74 20.70 20.48

7.05 69.65 228.79

2.82 16.02 64.06 82.90

G a m m a energy deposition W/subassembly W/g W/ml ml

Table 9. Energy deposition at the peak neutron flux of 8.5 x 10~5/cm 2 sec

92.4

14.4 9.3 243.5

233

12.1 74.0 2691

4.83 17.02 753.42 775.27

Total energy deposition W/subassembly W/g W/ml ml

Z

O

Physics of fast reactor control rods theory are being brought into use including threedimensional transport theory, such as Pn methods (Fletcher, 1981) and diffusion synthetic transport theory methods (Alcouffe, 1977; Michelini, 1978). Nodal transport theory methods offer the prospect of more accurate whole reactor calculations in three dimensions in which the finite mesh effects are negligibly small (Lawrence, 1984). Simpler, but more approximate modified diffusion theory methods are used in diffusion theory calculations to treat the sodium filled channel region from which a control rod has been withdrawn (the rod follower channel). Sodium is a low density material and diffusion theory overestimates neutron transport in low density regions when the diffusion coefficient is defined in the standard way (D = 1/3 Ztr I. Following the whole reactor calculation, detailed reaction rate distributions must be obtained by combining the control rod fine structure flux calculation with the whole reactor flux calculation, or alternatively by repeating the fine structure calculation with boundary conditions obtained from the whole reactor calculation. Monte Carlo methods can model all the details of a reactor. They are not used routinely at present for calculating the characteristics of large fast reactors because of the high computing costs required to achieve the desired accuracies and the lower cost of modified diffusion theory which achieves acceptable accuracies in most applications. However, Monte Carlo methods are used in validation studies. For the small fast reactor KNK, in which transport theory effects are important, the Monte Carlo code MOCA is used as the standard method. An example of the more approximate methods used for fuel management planning studies and core following is the flux synthesis method of Pipaud et al. (1980). The flux is represented in the form ~0g(x, y, zl = ~ ~ (z)' q~' (x, y) i-1

where the q~ (x, y) are flux distributions calculated for different planes of the reactor (for example, a 'rod inserted' plane and a 'rod withdrawn' plane within the core and planes within the upper and lower axial breeder regions). Another simplified method, which is used for PFR reload planning and subcriticality monitoring, is a two-dimensional centre plane model with axial leakage represented by an axial buckling. In this model partly inserted control rods can only be represented as fully inserted or fully withdrawn, although part insertion is sometimes treated by an appropriate fraction of the macroscopic absorption cross-section.

297

A much simpler approximate method is the use of influence functions to calculate the effect of control rod movement on reactor power distributions and on rod interaction effects (i.e. the difference between the reactivity worths of an array of rods and the sum of the worths of the individual rods). The power distribution with N rods present consists of the reference power distribution multiplied by N factorsf,(Z,) where Z, is the degree of insertion of rod n. These factors give the effect of the individual rods on the power distribution and the total effect is approximated as a product (or, in some formulations, as a sum) of the individual rod effects (the factor depending on the degree of insertion of the rodl. Similar tabulated factors can be used to calculate control rod interaction effects, or the reactivity worth of an array of partly inserted rods. Approximations in the calculation methods, such as those arising from the use of diffusion theory, the effects of neutron energy group condensation and the finite mesh, must be investigated for each reactor model and control rod design. The relative importance of different approximations can be estimated from studies reported in the literature and this section summarizes some of these studies. Taking the target accuracy for the prediction of control rod reactivity worths and absorption reaction rates to be _+5~, (1 SD) an approximation in one aspect of the methods which introduces an error of less than about _+2 °/o can be regarded as acceptable, although it is necessary to take note of such errors and the way that they combine.

3.2. Geometrical modelling in reactor calculations A plan view of a power reactor has core and breeder region subassemblies and control rods arranged on a hexagonal lattice. A typical lattice is illustrated in Fig. 1. In whole reactor calculations homogenized cross-sections are usually used in each of the hexagons although the radial variation of composition with burn-up is sometimes treated, using a finer subdivision of the model (such as 24 triangles per hexagon). The macroscopic cross-sections for core and breeder subassemblies are usually derived for the volume averaged composition of each subassembly. However, for accurate calculations of Doppler effects and sodium voiding reactivity, account must be taken of the fuel pin and subassembly wrapper box flux heterogeneity to derive appropriate cell averaged cross-sections. In the usual reactor design, a control rod and its guidetube occupy one hexagonal lattice cell, and homogenized cross-sections are derived for the hexagon using supplementary calculations as described in the next subsection. Alternatively, with a fine spatial mesh, the

298

J.L. ROWLANDS

subregions of the control rods can be represented explictly in the whole reactor calculation but such a fine mesh is usually only practicable for the analysis of specially designed experiments, such as a central control rod in a uniform cylindrical core. A design of control rod which does not occupy the whole area of the hexagonal lattice cell has been proposed in a gascooled fast reactor study. In this design a cluster of absorber pins moves in a channel at the centre of a fuelled subassembly. Such a design requires a fine spatial mesh for an adequate representation. However, for the usual design, whole reactor calculations can be made for a lattice of uniform hexagonal cells (in which the macroscopic cross-sections are equivalent homogeneous values) supplemented by calculations which derive the reaction rates in subregions of a control rod from the whole reactor fluxes plus the fine structure flux calculations. Different geometrical representations are possible for whole reactor calculations: H E X × Z, with each hexagonal lattice position represented by one mesh point; T R I x Z with six equilateral triangles per hexagonal lattice position (or subdivisions by factors of 4 to give 24 or 96, etc. triangles per hexagonal lattice position) or X Y Z , with the hexagonal elements replaced by rectangles having the same area and centres. (These rectangles are divided into four to enable the whole lattice to be represented by a mesh which is uniform in the X and Y directions, the mesh spacing in one direction being equal to half the hexagonal lattice pitch and in the other direction being 3/2 times this. This rectangular representation can then be further subdivided if required.) Many zero power criticial assemblies are composed of square elements and this forms the basic unit for the geometrical modelling of these (in X Y Z geometry). Control rod reactivity worth calculations and fuel management studies are often made in few energy groups. To derive these condensed group crosssections simpler geometrical models are frequently used. Typical of these are R Z cylindrical geometry whole reactor and supercell models. In the R Z geometry whole reactor model a control rod at the centre of the reactor can be represented by a region having the same cross-sectional area. However, offcentral rings of control rods must be homogenized with core material to produce appropriate data for the annular region containing the rods in an R Z model. An R Z geometry supercell model can be used for homogenization calculations. In this model a central control rod region is surrounded by a region of core material in proportion to the ratio of core subassemblies to control rods. A reflecting (or white) boundary condition is used at the outer radial boundary. These R Z

geometry models can also be used to derive axial bucklings for use in two-dimensional plan model calculations. Homogenization calculations are also made using one-dimensional cylindrical geometry models, with axial leakage treated by means of an equivalent absorption (or axial buckling). For more refined homogenization calculations pin cluster geometry collision probability codes are used to calculate reaction rates in individual pins and to obtain homogenized cross-sections. Monte Carlo calculations have been made to validate the more approximate methods. These have used more detailed representations of the control rods in the reactor.

3.3. Control rod homogenization In whole reactor calculations it is usual to homogenize the control rod cross-sections over the lattice area containing the rod. A typical design of control rod for an LMFBR consists of a cluster of absorber pins (e.g. boron carbide) clad in steel, contained in a steel wrapper which moves inside a steel guide-tube, the whole assembly occupying one hexagonal lattice cell. The absorber pin cluster (and the associated sodium and steel) occupies about one half of the lattice area, the remaining area of the cell containing sodium and steel. In some studies a simple volume averaging has been used to obtain the homogenized lattice area cross-sections. More accurate homogenization methods require calculations for a supercell model with a detailed representation of the control rod surrounded by a region of core material. Another homogenization procedure is to use the neutron flux calculated for a supercell model to average the reaction cross-sections for the regions of the control rod with flux × volume and to average the transport cross-sections with neutron current x volume. This method is a significant improvement over simple volume averaging but it does not reproduce the average neutronics properties of the rod precisely. Improved homogenization methods are being developed in many countries, mainly for thermal reactor applications. Generally it is not possible to reproduce all the properties of the control rod region in the whole reactor calculations. Obtaining an improved equivalence can involve an iterative sequence of calculations between the whole reactor and supercell models. The definition of homogenized parameters depends upon whether they are to be used in diffusion theory or transport theory calculations and whether the whole reactor finite mesh is to be allowed for. Thermal reactor homogenization usually produces parameters to be used in a finite mesh diffusion theory whole reactor calculation.

Physics of fast reactor control rods The aim of homogenization is to reproduce the region integrated reaction rates and surface neutron currents for each energy group and each homogenized region. However, in practice these are usually reproduced only for a reference case and only the average net neutron currents are preserved. Homogenization methods were the subject of an IAEA Sponsored Specialist's Meeting held in Lugano in November 1978 (IAEA-TECDOC-231) and, more recently, have been discussed by Smith (1980) and Koebke (1981). The questions discussed in this subsection are (1) The detail required in the control rod representation in the homogenization calculation (pin cluster geometry or an approximate cylindricalization). (2) The accuracy of simple volume averaging and of flux x volume averaging. (3) Improved homogenization methods. The effects of the approximations depend on the rod geometry and the strength of the absorber. They must be viewed in the light of other approximations, such as the possible use of diffusion theory and the effect of the size of mesh used in the whole reactor calculations. In the analysis of the ZEBRA Mozart experiments (Broomfield et al., 1973) the pin cluster geometry collision probability code PIJ (Anderson, 1973) was used to calculate the flux distribution and this flux was then used to weight the cross-sections for each region to produce macroscopic cross-sections for use in the homogenized reactor control rod region. Calculations using PlJ for the 19 pin cluster geometry were compared with calculations made using a cylindrical representation (with the compositions homogenized in annular regionst. For boron carbide control rods the pin cluster geometry calculations gave a reactivity worth about 1 ,~o/ lower for natural boron and 3'~, lower for highly enriched boron (80 and 90~o enriched ) than the cylindrical geometry representation. For tantalum control rods a further complication is the treatment of resonance shielding. This effect is difficult to treat in the complex pin cluster geometry but estimates made using simpler geometries indicate a difference between the two geometrical representations (pin cluster and cylindricalizedl of about 45~. The magnitude of the approximation involved in simple volume averaging has been estimated for SUPER-PHENIX control rods by Bouget et al. (1979). They compared R Z geometry calculations for a central control rod using compositions volume averaged over the lattice area containing the control rod with calculations in which the control rod region is divided into an inner absorber region and an outer region of sodium and steel. This is a cylindricalized representation of a SUPER-PHENIX control rod which consists of an array of 37 boron carbide pins "~

/o

299

enclosed in a steel box which moves inside a hexagonal steel tube. The differences in rod worths calculated for the two types of representations are calculated for different boron enrichments. Rod worth is about 6~,, larger for a highly enriched boron rod and about 8~,, larger for a 25 I~ enriched rod, when the composition is averaged over the whole subassembly lattice area. The approximation involved in flux xvolume averaging has been examined (Rowlands and Eaton, 1978; Rowlands et al., 1979) for a cylindrical representation of a PFR control rod. This rod consists of a cluster of 19 natural boron carbide pins which has been cylindricalized in a similar way to that described above for the SUPER-PHENIX control rod. The calculations were made for a supercell consisting of the two regions of the control rod (inner absorber region with an outer sodium/steel region) surrounded by a region of core material, with a white outer boundary condition. Axial leakage was treated by a buckling mode and the calculations were made using the DSN code WDSN ST (Brissenden and Green, 1968). Calculations made using homogenized cross-sections for the control rod cell area were compared with those for the two region control rod model (in the three region supercell model). These showed that volume averaging overestimates reactivity worth for this cylindrical model by 12.8~o and flux x volume averaging overestimates worth by 4.0~o. The calculated rod worth would be even smaller if the individual pins of the cluster were represented. Methods of homogenization which reproduce the reactivity worth of the rod in a transport theory supercell calculation were also investigated. The methods involve averaging the flux x volume weighted cross-sections with the adjoint flux calculated for the homogenized model. The condition for reactivity equivalence is of the form

,[ ~OslX~(Z - E(x)) ~o*(x) d r = 0 Z = j~0s(X)Ztx )~0~(x i d v/,f~os(X) ~o*Ixl d V where ~os(x) denotes the flux calculated for the fine structure supercell model and q~'(x) the adjoint flux calculated using the homogenized cross-sections and the integration is over the homogenization volume. For moderation cross-sections the adjoint flux relates to the secondary energy. For transport cross-sections the correlation between the angular variation of the flux and adjoint flux (relative to the angular averages or scalar values) is involved in the averaging. An iterative procedure must be followed in which the homogenized cross-sections used to obtain the

300

J.L. ROWLANDS

adjoint flux solutions for the homogenized model are improved in successive cycles. In the first cycle the adjoint flux is calculated using the flux × volume averaged homogenized cross-sections. In the second cycle the homogenized cross-sections are obtained by weighting with the f l u x x a d j o i n t flux (first cycle) x volume. It is found that two cycles are sufficient to obtain rod reactivity worths and reaction rates within 1~o of the heterogeneous model values. The axial and radial transport corss-sections were separately averaged with the neutron current and the homogenized model adjoint current calculated for the two directions. There are many ways in which reactivity equivalence can be obtained. In the first study (Rowlands and Eaton, 1978) equivalence was

obtained separately for capture, moderation and transport in each energy group. In later work (Rowlands et al., 1979) the same group scaling factors were applied to the flux x volume averaged group capture and moderation cross-sections and different factors were applied to the current averaged axial and radial transport cross-sections. The factors calculated for the homogenization of a natural boron carbide control rod are given in Table 10. It was found that, in addition to reproducing the rod reactivity worth and absorption rate accurately, the method reproduces the flux dip at the surface of the rod to a satisfactory accuracy. The overall spectrum shape was also well predicted, although the detailed shape in the 3 keV sodium resonance was not reproduced accurately.

Table 10. Factors applied to the flux averaged cross-sections to give reactivity equivalence Energy group 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37

Lower energy 10 MeV 6.065 3.679 2.231 1.353 820.8 keV 497.9 302.0 183.2 111.1 67.38 40.87 24.79 15.03 9.119 5.531 3.355 2.035 1.234 748.5 eV 454.0 275.4 167.0 101.3 61.44 37.27 22.60 13.71 8.315 5.044 3.059 1.855 1.125 0.683 0.414 0.100 Thermal

Capture moderation

Radial transport

Axial transport

0.9900 0.9906 1.0026 0.9924 0.9786 1.0100 1.0069 1.0083 0.9956 0.9894 0.9757 0.9692 0.9611 0.9266 0.9032 0.8602 0.7932 0.7218 0.8185 0.8059 0.7790 0.7276 0.7339 0.7323 0.7139 0.7447 0.6951 0.7071 0.6956 0.6929 0.7266 0.7620 0.7778 0.7430 0.8759 0.7878 0.8688

0.8744 0.8629 0.8348 0.8078 0.7929 0.8391 0.8807 0.8529 0.8906 0.8559 0.8579 0.8929 0.8879 0.8893 0.9830 0.9516 0.9829 1.0077 0.9860 0.9573 0.9410 0.9591 0.9844 1.0179 1.0275 1.0600 1.0486 1.3644 1.1857 1.5278 1.2512 1.2574 1.4137 3.2528 0.8037 0.7917 0.9032

0.9940 0.9950 0.9941 0,9936 0.9931 0.9960 0.9940 1.0033 1.0210 0.9798 0.9776 0.9750 0.9731 0.9749 0.9778 0.9713 0.9906 1.0026 0.9569 0.9398 0.9251 0.9292 0.9171 0.9179 0.9058 0.8871 0.9011 0.9068 0.8950 0.9091 0.9310 0.9472 0.9530 0.8966 0.9659 0.9360 0.9629

301

Physics of fast reactor control rods Methods which reproduce the group absorption rates and surface currents have also been investigated. It should be emphasized that the above methods produce homogenized cross-sections for use in transport theory calculations and further adjustments would be required to produce constants for use in diffusion theory calculations. It should also be noted that this method has not been tested for a control rod situated in a flux gradient. An alternative approach to homogenization used in some calculation schemes is to derive cross-sections which reproduce the control rod surface currents in diffusion theory calculations in the mesh used for whole reactor calculations. The cross-sections are thus corrected for the heterogeneity, for the approximation involved in using diffusion theory and for the finite mesh effect. However, there are other effects in fast reactors which require the use of transport theory, such as core-breeder boundary effects. With the development of three-dimensional nodal transport theory codes the homogenization method need not allow for the use of diffusion theory nor the finite mesh effect.

There are some fundamental differences between the procedures used by different authors: Method 1. The axial diffusion coefficient in the channel is defined as the ratio of the net axial neutron current in the channel to the flux gradient when there is a linear axial flux gradient and the channel is of infinite axial length and surrounded by an infinite uniform medium. Method 2. The axial diffusion coefficient in the channel is chosen to reproduce also the enhanced axial leakage in the region neighbouring the channel. An alternative approach would be to modify the axial diffusion coefficient in a region neighbouring the channel. Method 3. An additional correction is made for the finite channel length and for the non-linear axial variation of the flux (sinusoidal or exponential). In addition to these different methods there are differences in the approximations made to the transport theory solution, with analytical formulae being used to approximate the spatial and angular integrals,

3.4. Effective d([fi~sion coefficients./or a control rod channel

The fast reactor spectrum averaged value of the macroscopic transport cross-section for sodium is about 0.07 barns (the value being about 0.03 barns at MeV energies). This is much smaller than the average value for fast reactor core material, which is typically 0.2 barns. Diffusion theory calculations made using the standard definition of the diffusion coefficient (1/3 Z,,) in control rod channels from which the rods are withdrawn overestimate the axial leakage. This causes the calculated reactivity worths of the control rods (relative to sodium filled channels) to be underestimated. Improved estimates of the axial leakage associated with the control rod channels (or control rod follower regions) can be obtained by using modified diffusion coefficients within the channel. The procedure used to derive these is to make transport theory calculations for a simple geometrical representation of the channel in a uniform medium or supercell environment. The modified diffusion coefficients are defined as the ratios of the net neutron currents to the gradients of the scalar flux and it is usual to derive different values for different directions (Dx, D r and D_ for example). For cylindrical channels approximate methods and formulae have been derived for the axial and radial diffusion coefficients for both voided and low density channels. These have been reviewed by Rowlands and Eaton (1980).

3.4.1. Effective axial diffusion coefficients.[or voided channels. The differences can be illustrated by formulae derived for a voided cylindrical channel. Laletin (1960) has calculated the axial current for a voided channel of radius a extending axially through a reactor medium of height H and mean free path 2 = 1/Y. The net current per unit area within the channel can be written in the form J..= - ( i ' q ~ / P z ) . (2/3)-(1 + ( a / 2 ) " ( 2 - q ) t

where (~3q~/?z) is the axial gradient of the scalar flux in the surrounding medium and q is a term which depends on the axial flux shape, the channel length and the axial position within the channel. For a linear axial flux variation, and a channel of infinite length, q=0. For a sinusoidal axial flux variation (with buckling B21 and a channel through a medium of infinite extent q=qs=(3n/4)"

B ~ a (to first order in B~).

The net axial current outside the channel is larger than the value given by the usual definition of the diffusion coefficient by the amount A.I ,.h = - ~ta2 " (?~o/t~zJ" (2/3 ~.

The channel axial diffusion coefficient which reproduces both the current in the channel and the increase in current outside the channel above the value given by

302

J.L. ROWLANDS

the standard definition of the diffusion coefficient in the surrounding region is O~ = (2/3). (2 + (a/2) • (2 - q)). 3.4.2. Effective diffusion coefficients for low density channels. The approach adopted by Rowlands and Eaton (1980) is to derive a value for the channel from Benoist's approximate formula for the axial diffusion coefficient for a cell containing a channel (Benoist, 1964) (a Method 2 approach) f

O k = (1/3)'~{(V6pi/E;)'P,;.~}/{Z(VI) } i,j

i

where Pi~,k is the directional first flight collision probability for the direction k, and ~i is the transport cross-section for region i. V~ denotes the volume of region i and ¢pl the scalar flux in the region. Assuming a flat flux, and volume averaging of axial diffusion coefficients, results in an expression for the channel diffusion coefficients DzRM= (1/3E, ).{(2 - (Zo/E t )) + ((Z,/Z o ) - 2 +

(Zo/Z,))- Poo.~} where E o relates to the channel and E~ to the surrounding region. The approximate formula for Poo,~ derived by Bonalumi (1971) has been used in the calculations, but with channel radius a, replaced by

a* = a ( l - q/2) so as to give a correction for the non-linear axial flux variation (or finite core height) which is consistent with the limiting value for the voided channel (and hence a Method 3 expression). This generalized form is called the buckling modified formula, BM1. An alternative formula for a* has also been studied. The values of q used are:

qc=(3n/4).B~a (for a channel through a core region) and q8 = (3/2)' (L/H 2 )" a (for a channel through an axial blanket region) where L is the exponential decay length of the flux in the axial blanket ~o(z)= q~(O) • exp( - z/L) and H is the axial blanket thickness. Bonalumi's approximate formulae for Poo and P .... are: Poo(q) = 1 - 1/{ 1 + 2q - 2#q/( 1 + 2/tq)} where ~/= aZ o p = 4 / 3 - exp(-0.3r/) and Poo,z(q)=Poo(q). {1 + 1/(2(1 +41/))}. For the radial direction Bonalumi's formula for the

channel diffusion coefficient has been used

1 ~4aZl+(3-s)(l+s)], D,a=3ZI( 4aEo+(l+s) 2 j w h e r e s = ~ .

3.4.3. Effective axial diffusion coefficients calculated using the W D S N - S T code. Another method which is used to obtain axial diffusion coefficients for a low density channel involves the DSN code WDSN-ST. This one-dimensional transport theory code treats leakage in the perpendicular direction by means of a buckling mode. The flux is taken to have the two component form

N(r,z,~)) = No(r,~)cos B~z + N 1(r,~)sin B~z. The axial leakage, L, is calculated for a supercell model which includes the channel and the channel diffusion coefficient Dc is defined by r = (Of,pore + Os~osV,)B~

where Ds is the axial diffusion coefficient for the surrounding region in the absence of the channel and is the B, value D,={l-(l./b~).tan

~b~}/{X.b~.tan

~b~}

where b z = Bz/'£,. This method treats the effect of the axial buckling explicitly and avoids the flat flux approximation (although the diffusion theory scalar flux should be used, rather than the WDSN-ST flux, to obtain parameters for use in diffusion theory calculations). The WDSN-ST method therefore involves fewer approximations than the buckling modified formula and can be used for multi-annular geometry (whereas the simpler formulae apply to a homogeneous channel in a single uniform region). However, it is often more convenient to use the approximate formula, rather than to derive values for Dz using the WDSN-ST procedure and the values are of comparable accuracy. Furthermore this code does not provide a method for calculating effective diffusion coefficients for channels in axial blanket regions. 3.4.4. Supercell calculations of channel reactivity effects. Calculations have been made of the reactivity effect of a cylindrical channel in a two-dimensional cylindrical supercell consisting of a core region and an axial blanket region. Two types of channel have been calculated, Model A, having a channel extending through the core region only, and Model B, with the channel extending through both the core and axial blanket regions. Calculations have been made in 37 energy groups for three channel radii (5, 7 and 9 cm)

303

Physics of fast reactor control rods for a channel containing 85~o sodium and 15Vo steel (representing a control rod channel containing a steel follower) in a fast reactor core and axial blanket environment. The transport cross-sections are similar in the core and blanket regions. Transport theory calculations made using the code D O T (Mynatt, 1969) are compared with diffusion theory calculations made using the code S N A P (McCallien, 1975) in Table 11. Both unmodified and modified diffusion coefficients are used. The results are also related to calculations made for the homogenized supercell in which the channel composition is averaged with the surrounding core or axial blanket composition. The final converged k value for all the diffusion theory calculations was obtained using the neutron balance edit in the S N A P code. For the small spatial mesh used in these calculations the value is sensitive to the method used to compute k from the final converged flux. This is because of the effect of computer rounding errors. In the transport calculations a vacuum boundary condition has been used on the outer boundary of the axial blanket and in diffusion theory the corresponding boundary condition has been taken to be D(O~o/c~z)/~p = - 0.469. The main reactivity effect of the sodium filled channel arises from the section of the channel in the core region (the effect for Model B being only a little greater than for Model A). For the channel in the core region only (Model A) the change in reactivity is largely due to the change in supercell material content. The effect of the separation of the sodium into a low density channel is small, the values obtained for the homogenized model being close to the D O T values. The axial blanket channel reactivity effects (the

differences between Model B and Model A) are larger than the effects found for the homogenized models. In this case channel streaming is the dominant effect and it is well predicted by the buckling modified formula. However, the differences between the Model A and the homogenized model reactivity effects are not well predicted using the buckling modified formula. Calculations made for a one-dimensional cylindrical model, with axial leakage represented by an equivalent absorption, show a similar difference between diffusion theory and transport theory. The effect is probably associated with the difference between the transport theory and diffusion theory value of the channel flux dip, with this being underestimated by diffusion theory. The D O T calculations give a large channel flux dip at high energies, twice as large as do the diffusion theory calculations. The larger high energy channel flux obtained in the diffusion theory calculations results in a larger high energy moderation and axial leakage and hence a larger reduction in reactivity associated with the channel. Results have also been obtained using channel axial bucklings derived from the diffusion theory calculations in the buckling modified diffusion coefficient formula. The axial bucklings vary strongly with energy, being negative at low energies and increasing from 5 m -2 at 10keV to 15m 2 at 10MeV. However, using these values had little effect on the channel reactivity worth for this sodium/steel filled channel, compared with using an average value at all energies. Clearly the use of modified diffusion coefficients results in improved values for the channel reactivity effect but there are aspects of the effect which are not treated using the present formula.

Table I 1. Channel reactivity effect calculated for a fast reactor control rod follower (ilk in ~°4,j(37 energy group calculations for a supercell model}

Channel radius (cm)

DOT

Diffusion theory Unmodified formula BM formula Difference Difference Value (°o) Value (i~,)

Homogenized cell Transport theory Diffusion theory

Model A

a= 5 a=7 a =9

4.16 8.58 15.35

4.44 9.08 16.12

6.7 5.8 5.0

4.26 8.77 15.69

2.4 2.2 2.2

4.23 8.82 15.86

4.25 8.84 15.92

Model B

a=5 4.35 4.85 11.4 4.45 2.3 4.27 a =7 9.00 9.89 9.8 9.18 2.0 8.91 a =9 16.10 17.47 8.5 16.38 1.7 16.12 Calculations for a one-dimensional cylindrical model with axial leakage represented by absorption a=5 5.61 6.46 15.2 5.77 2.9 5.50 a= 7 11.49 13.09 13.9 11.89 3.5 11.48 a=9 20.24 22.85 12.9 21.05 4.0 20.70 Channel composition: 85~o sodium and 15~/osteel (guide-tube and rod follower}

4.29 8.95 16.19

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J.L. ROWLANDS

3.4.5. Fischer's method for calculating modified diffusion coefficients. Fischer (1981) has developed a procedure for treating the buckling dependence explicitly, rather than by simply including a term which reduces to the term in the formula derived by Laletin (1960) for the case of a voided channel, as is done in the BM method described above. For the case of zero axial buckling the expression derived using Fischer's method is in agreement with the above method. When the axial buckling is non-zero the expression for the diffusion coefficient is expanded in the buckling and the individual terms in the expansion (which involve integrals containing the K i functions) are calculated by numerical integration, or by series expansion. Fischer's expression involves terms in B2, however, whereas the BM formula contains a term proportional to B. Fischer's formula for the diffusion coefficient for a voided channel thus differs from the formula derived by Laletin.

3.5. Dependence of calculated control rod reactivity worths on the finite difference mesh The usual linear finite difference approximations used to solve the neutron diffusion equation and the discrete ordinates transport theory equation are accurate to first order in the mesh interval. However, terms proportional to odd powers of the mesh interval cancel for a homogeneous problem with uniform mesh spacing. Consequently, for a sufficiently small mesh, and provided that boundary (or inhomogeneity) effects are small, calculated k values should vary linearly with mesh interval squared, when the mesh is varied uniformly in all directions over the whole reactor model. In X Y Z geometry the finite difference error has the following form for small mesh intervals e=~(h2 + h,2 ) + ~h~2 = ~;ax, +/3h~ where axr is the area of the mesh element in the xy plane, and h~ denotes the mesh interval in the x direction. The finite difference error is larger for a reactor model containing absorbing control rods than for the reactor model with control rods withdrawn, resulting in a mesh dependent error in the calculated rod reactivity worths. There are errors in the power distribution, both over the reactor as a whole and adjacent to control rods, associated with the finite mesh approximation. The sign and the magnitude of the error are different for mesh centre (or mesh interval) finite difference diffusion theory codes (which overestimate k) and for mesh edge finite difference diffusion theory and discrete ordinate transport theory codes (which underestimate k). There is also an error associated with the

number of angular ordinates used in discrete ordinate transport theory calculations. By making calculations for two uniform mesh sizes in the range for which the dependence is of the quadratic form an extrapolated value which approximates the zero mesh interval value can be obtained. Sakagami and Takeda (1978) have investigated the mesh dependence for three-dimensional diffusion theory calculations in H E Z x Z and T R I × Z geometry. They concluded that the quadratic dependence is a good approximation for the mesh range they have studied and that the finite difference approximation error in the whole reactor reactivity can be written in the form 6p "" (Ar)2(Clp + C2) -~-(AZ)2(C3~ -~-C4)

where p is the reactivity worth of the control rods. (This form applies both for partly and fully inserted rods.) In the mesh centre diffusion theory calculations for a 300 MWe LMFBR with one mesh point per hexagonal subassembly and an axial mesh interval, AZ, of 11.5 cm k is overestimated by 0.8~o with the control rods withdrawn and by 1.9~o with the rods inserted. The calculated reactivity worth of the control rods is about 13~o lower for the hexagonal mesh than for a triangular mesh of six triangles per subassembly and AZ = 3.38 cm. For the triangular mesh the error in k is 0.11~ with the rods withdrawn and 0.23~o with the rods inserted. An investigation of mesh dependence has been made using a two-dimensional X Y geometry plan representation of the ZEBRA Assembly MZC by Sugawara (see Broomfield et al., 1973). This was an assembly containing arrays of control rods representative of the 300 MWe MONJU reactor. The elements of ZEBRA are approximately 5 × 5 cm square and the MONJU Mock-up control rods occupy a region of four elements. Calculations using mesh centre diffusion theory and a mesh of one point per element (the standard mesh) underestimate rod reactivity worth by 8~o whereas a mesh of four points per element (with a spacing of 2.5 cm) underestimate rod worth by 2~o. Calculations made using TWOTRAN (Lathrop and Brinkley, 19701 and the equal weight diamond difference scheme show a much smaller variation with mesh area, the control rod worth discrepancy being less than about 10/o for the standard mesh (i.e. a factor of 8 less than the diffusion theory method). The variable weight diamond difference scheme gives a reactivity error which varies quadratically with mesh area. Negative flux fix-up schemes produce non-linear variations, but these are sometimes required to give convergence. Sweet (see Rowlands et al., 1979) has found that by first

Physics of fast reactor control rods obtaining convergence with negative flux fix-up, and then a further stage of convergence without fix-up, convergence can be obtained and a linear variation of k with mesh area results (for the mesh range studied). Sweet found a linear dependence on mesh interval squared for mesh centre diffusion theory and DOT DSN transport theory calculations for mesh intervals of less than 4 cm. He also found that for very small mesh intervals (less than about 1 cm) the diffusion theory codes could give erroneous k values when calculated from the ratio of successive fission source distributions. However, a neutron balance calculation using the same flux solution gave accurate values. The dependence of calculated properties on S, order in general depends on a number of factors; the choice of quadrature scheme, the finite difference scheme and the geometry (e.g. cylindrical or X Y ) . In onedimensional cylindrical supercell model calculations made using the code WDSN-ST, control rod reactivity worth was found to increase by about 1~o when the S, order was increased from S~ to $16. The quadrature scheme used by this code divides the unit sphere into segments of equal area. The increase was slightly different for a two region representation of the control rod and a homogenized representation, being 1.0~o for the heterogeneous (two region) representation and 0.7~o for the homogenized representation. In whole reactor calculations for a 1200 MWe design of fast reactor (the U.K. Commercial Demonstration Fast Reactor, CDFR) made using the code DOT in X Y geometry, Sweet found that the change in control rod reactivity worth (for a full array of operating rods) on going from S4 to Ss was within the uncertainty in the calculations (about +0.1 ~). This change from S4 to Ss produced a slight decrease in Keff of about 4 × 10- 5. Sweet has made calculations with a standard mesh of 7.5 × 6.5 cm (four rectangles per subassembly). Diffusion theory calculations using this mesh underestimate rod reactivity worths, compared with the extrapolated zero mesh values, by amounts varying from 2 to 8~,,, depending on rod position and the number of rods in a ring. Doubling the number of mesh points in each direction reduces the error by about a factor of 4. Sweet estimated that for a triangular geometry representation with six triangles per subassembly the underestimate of rod worth due to the finite mesh size is about equal in magnitude to the overestimate due to the use of diffusion theory. Thus the two effects almost cancel, to give a net deviation of less than about +2~o between worths from diffusion theory with this triangular mesh and DOT transport theory calculations extrapolated to zero mesh interval. The mesh errors in the DOT transport theory values of control rod worth, calculated using the standard mesh,

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are smaller than those in the diffusion theory values in most cases, the average underestimate of the value being 2~o for the standard mesh and 0.5~o for the doubled mesh. Coarse mesh diffusion theory and transport theory methods are being developed and evaluated and these can reduce the finite mesh corrections to negligibly small values. There are different approaches to the derivation of the coarse mesh equations, including polynomial expansions within the node and local fine mesh diffusion theory or transport theory solutions. Applications of the methods to fast reactor control rod calculations have been evaluated by Suzuki (1975) and Takeda et al. (1979). 3.6. Effect of nurnber of energy groups The dependence of the reactivity worth and reaction rate distribution on the number of energy groups used in the calculation is a function of the energy group structure, the method used to calculate the condensing spectrum and the number of zones in which different condensing spectra are used. Kato et al. (1976) found that the errors in k are approximately inversely proportional to the square of the number of energy groups for their particular strategy of group condensation. With two energy groups the ~o error in control rod worth was found to be 2.2~o (rod worth being underestimated in the two group calculation) and with more groups the error was smaller, although it did not vary monotonically with number of groups. Hammer et al. (1974) used six energy groups condensed using fundamental mode spectra for the core media, and using a standard flux for blanket regions and for the control rod. A comparison between 6 and 25 group calculations in R Z geometry showed differences of 1.6~o or less in the reactivity worths of a B4C absorber rod and a sodium channel. Sugawara made calculations in X Y geometry for two off-central rods containing highly enriched boron carbide. Calculations made in 4, 6, 9 and 37 groups and using two different sets of condensing spectra were compared. The first set consisted of separate zone integrated condensing spectra for inner core, outer core and blanket regions (with rods withdrawn), and also for sodium filled control rod channels and for the control rods themselves. In the second method the inner core spectrum was used for condensing the control rod region cross-sections. For the first method the error in control rod worth was an overestimate of about 2~o, or less, for 4, 6 and 9 groups. Using the second method (the inner core weighting spectrum in all regionsl the overestimate was larger: 2.7~ for nine groups, 3.2'~, for six groups and 4.9~ for four groups.

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3.7. The approximations involved in the use of diffusion theory Comparisons between diffusion theory and transport theory calculations, when extrapolated to zero mesh size and high order S, approximation, show that diffusion theory overestimates reactivity worth for a central control rod (or a control rod in a supercell calculation). For off-central rods, differences in the overall flux shape between diffusion theory and transport theory can affect the differences in control rod reactivity worths. In his X Y geometry calculations (with a single axial buckling) for CDFR natural boron carbide control rods Sweet found that diffusion theory overestimates reactivity worth by 7 ~ for a central rod and by 4Yo for the full array of operating control rods, the overestimate for individual rods and rings of rods ranging from 2 to 7~o. Sugawara in his calculations for the ZEBRA Mozart experiments, found that R Z geometry diffusion theory calculations for a central rod (compared with S4 TWOTRAN calculations) overestimate rod worth. The overestimate was found to be 6.5 ~o for a natural boron carbide MONJU sized rod and 9.6~o for an 80Yo enriched MONJU sized rod. For the larger PFR sized natural boron rod in the same core the overestimate was 7.7Yo. The overestimate was found to be 4~o smaller for the off-central rods than for the central rod. The overestimates were different for the R Z and X Y geometry calculations. Differences in the treatment of axial leakage associated with the rod follower could contribute to this difference, as well as mesh approximations and discrete ordinate, S, quadrature schemes. Monte Carlo calculations have been made using the MOCA code in the analysis of control rod measurements made in the Karlsruhe facility SNEAK. Although the uncertainties in the Monte Carlo calculations are comparable with the differences, the worths from diffusion theory calculations are found to be larger than those from Monte Carlo calculations by about 4~o. Comparisons in R Z geometry between diffusion theory and $4 transport theory show differences, typically, of 10~. Calculations made by Hammer et al. using a onedimensional cylindrical geometry model show diffusion theory to overestimate boron carbide control rod worths typically by 53/0 compared with transport theory calculations. The effect of scattering anisotropy was also examined. Reactivity worth increases of up to 4~o were found for a small rod (r = 3 cm) compared with the values obtained when the transport approximation was made. A comparison, made by Sweet, of reaction rate

distributions calculated in X Y geometry using diffusion theory and transport theory shows that diffusion theory overestimates reaction rates in control rods, the overestimate being about 10~o in the 238U fission rate and about 5~o in the 239pu fission rate. The discrepancies for the control rod follower are smaller (although still significant for the 238U fission rate). The differences are affected by differences in the overall reactor flux distribution, particularly for the outer core relative to the inner core, which results in the differences between diffusion theory and transport theory being a function of radial position. The differences outside the rod are smaller, diffusion theory being typically 5~, higher for the 238U fission rate adjacent to the rods and about 1~,,, higher for the 239pu fission rate. An energy group breakdown of the absorption rate in a natural boron carbide rod in a supercell model shows that the absorption rate is overestimated by diffusion theory by up to 20~o at MeV energies decreasing to an overestimate by about 3 ~ below 50 keV.

3.8. Geometrical modellin 9 approximations 3.8.1. Two-dimensional plan model calculations and axial bucklinos. Two-dimensional plan model calculations are of two types, centre plane representations and axial average representations. Axial leakage is represented by means of an equivalent absorption which is usually written in the form DgB29 (although it is not necessary to derive the group dependent bucklings, B0z, as an equivalent group dependent absorption, L~0,can be derived directly). The more accurate calculations use zone and group dependent bucklings, with different bucklings in the control rod zones, the control rod follower zones and other reactor zones. The bucklings can be obtained from threedimensional calculations or two-dimensional reactor or supercell calculations in R Z cylindrical geometry in which a central control rod or control rod follower is present. Mid-plane bucklings are obtained from the axial flux curvature (or flux second difference) in the mesh elements adjacent to the mid-plane. For control rod studies and the calculations of whole reactor properties axial average bucklings are more appropriate. These are defined by the axial leakage equivalent absorption cross-section, L~,y, for zone r and group g

L~ o =Do., " (Bo.,)2 = ~SJ.,o(x, y, zc) dx dy/ S~Dy,,(x, y, z)" q~(x, y, z) dx dy dz

Physics of fast reactor control rods where the axial current at the top of the core (axial position zc) is

J~,g(x, y, z~)= -Do(x, y, zc) " (~o/~z)zc (assuming a symmetrical, half-height model) and the two integrations are over the cross-sectional area and volume of the zone r respectively. The use of separate, group dependent bucklings (or axial leakage equivalent absorption cross-sections) in the inner core, outer core, control rod and control rod follower regions results in a significant improvement in accuracy. The effect has been studied in R Z geometry by Sugawara who showed that the reactivity effect of a control rod follower (relative to fuel) was overestimated by about 20~o when a single valued core region buckling was used. The reactivity worth of an 80~o enriched boron control rod (relative to fuel) was overestimated by about 4Vo when core region multi-bucklings were used in the rod regions (although the single valued buckling gave agreement within 1~o in this case). The use of group and region dependent multi-bucklings reduced the discrepancies to about 1~o or less. The reactivity worths of edge fuel elements (which were used to determine the reactivity effects of control rods by the critical balance method in the ZEBRA Mozart experiments) were not as well predicted using multibucklings as with single bucklings, however, the discrepancies being -2.6~o and 1.1 °/o respectively. Axial bucklings for control rod follower regions should be calculated using effective axial diffusion coefficients appropriate to a low density region (or axial leakage equivalent absorption cross-sections obtained directly from transport supercell calculations).

3.8.2. Comparison of cylindrical and rectanyular 9eometry models. A comparison of one-dimensional cylindrical geometry and X Y geometry rod worth calculations for a central 80~o enriched boron carbide control rod (with an equal area representation for the rod in the two models) has been made by Sugawara. The agreement was good, the difference being only 0.1 ~o. This is less than the uncertainties associated with the convergence and mesh corrections.

3.8.3. Use (?[flux synthesis methods. The program KASY (Buckel et al., 1977) has been used for X Y / Z geometry diffusion theory synthesis calculations in the analysis of control rod measurements made in the SNEAK criticial facility and in reactor design calculations. This code has been used successfully in the analysis of reaction rate distributions and reactivity

307

worths for arrays of control rods. The success depends on the selection of appropriate trial functions. The method is particularly valuable for a series of similar calculations for a reactor, such as fuel management calculations.

3.9. Potential accura~T qf control rod calculational

methods Current calculational procedures have the potential of achieving the target accuracy of _+5 ~o in control rod reactivity worth and rod reaction rates. However, nuclear data errors which affect either control rod absorber worth relative to fuel worth or the calculation of the overall reactor flux distribution could result in larger discrepancies. Consequently it is necessary to test the calculational methods by analysing experiments. A calculational procedure which is representative of currently used methods is outlined below. In this scheme whole reactor diffusion theory calculations are made, with corrections derived from two-dimensional transport theory calculations made for representative cases. (i) Supercell calculations with a detailed representation of the control rod to obtain homogenized cross-sections in the form required for whole reactor calculations and for deriving the reaction rate fine structure. These could be made using pin cluster geometry collision probability codes or, more approximately, DSN calculations for a simplified cylindricalized representation. (2) Calculations of effective diffusion coefficients for control rod channels. (3) Diffusion theory calculations to obtain spectra for condensing to few energy groups (5 to 10) using different weighting spectra in different reactor regions, including control rod absorber regions and control rod channels. These could be obtained using a reference three-dimensional reactor model or a simpler model (RZ with a central control rod or plan model with axial bucklings). This stage is only required when a large number of calculations are to be made and the computing costs involved in using about 25 groups are unacceptably large. (4) Derivation of group dependent axial bucklings for each region using the above models. There are two reasons why this stage could be required. The first is to enable the diffusion theory methods to be compared with two-dimensional plan geometry DSN calculations and the second is to enable a wider range of diffusion theory calculations to be made with less computing costs.

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J.L. ROWLANDS

(5) Three-dimensional diffusion theory calculations for the primary control rod configurations and two-dimensional plan model calculations, with group and region dependent axial bucklings, for the wider range of fully inserted control rod configurations of possible interest. (6) Comparisons of the standard mesh diffusion theory calculations with extrapolated zero mesh transport theory calculations, in both R Z models and plan models with axial bucklings, to evaluate corrections to rod worths, rod absorption rates and core fission rate distributions (or to confirm that these corrections are acceptably small) for representative reference models. Corrections to irradiation damage doses and dose gradients should also be derived. The effect of varying the S, order and spatial mesh should be examined. (7) Calculations of the energy deposition taking into account gamma migration and the sources of energy: fission, charged particle emission recoil, capture gammas, scattering recoil and inelastic scattering gammas. Hence calculations of temperature distributions for different control rod positions. (8) Calculations of neutron irradiation dose-rates and dose-rate gradients and hence the swelling and bowing of components. (9) Calculations of the variation of control rod composition and reactivity with irradiation. Several developments are enabling simpler methods to be introduced. The improvements in speed and reductions in costs of computing form one important factor. Three-dimensional multi-group calculations can now be made in diffusion theory routinely. Twodimensional plan geometry calculations are still being made, either because there is a need for continuity in the methods being used in a particular application or to validate diffusion theory methods using twodimensional DSN methods. However, threedimensional Pn transport theory methods are available (Fletcher, 1981) and fast three-dimensional transport theory methods such as the transverse leakage coupled one-dimensional nodal methods (Lawrence, 1984) are being developed and introduced into use. Automatic energy group rebalance methods also speed up solutions. Adoption of these multi-group threedimensional transport theory methods will eliminate the need for energy group condensation and for supplementary calculations to validate diffusion theory methods and to derive modified diffusion coefficients. The development of Monte Carlo methods which provide the required accuracies with acceptable computing costs are in prospect and these will eliminate the need for the homogenization stage.

3.10. International intercomparison calculations for a large L M F B R Calculations 1/ave been made for a specified calculational model using different nuclear data libraries in an international intercomparison of calculations for a large LMFBR (LeSage et al., 1978). These included calculations of the reactivity worth of a central natural boron carbide control rod. The calculated rod reactivity worths range from 0.25 to 0.39Yo 5k. This is a much larger variation than the differences found between critical assembly measurements and calculations. The differences are due to differences in the nuclear data and are primarily cause d by differences in the calculated values of the ratios of the flux at the core centre to the core averaged flux. The ratio of the flux in the outer core to that in the inner core is particularly sensitive to nuclear data differences in this large core. These calculations show that for large fast reactors the uncertainties in the calculations of fission rate distributions associated with nuclear data uncertainties result in large uncertainties in the calculated reactivity worths of individual control rods, or rings of rods. The uncertainties are smaller for the calculated reactivity worths of uniform arrays of rods.

3.11. Control rod interaction models and power distribution influence Jimctions It is possible to use approximate methods to allow for control rod interaction effects when interpreting rod reactivity worth measurements on operating reactors. Bannerman (1985) formulates two types of interaction model, a flux interaction model and a reactivity interaction model. The case of a twodimensional representation of an array consisting of some fully inserted and some fully withdrawn control rods is considered first. Three assumptions are made (1) The flux spectrum at the position of a control rod is not changed significantly by insertion of other rods. (2) The flux perturbation caused by the insertion of several rods is the product of individual rod flux perturbation factors,/ij, at position i due to rodj. (3) The flux perturbation is proportional to the isolated rod reactivity worth w)°) of the perturbing rod j. The flux perturbation factor takes the form ./ij = 1 + % w ~ °~ .

The reactivity worth of an array of rods is expressed in the form W= ~ W - (°j i I ] . i i j = Z W- (°) , [ l (l+a,jW~°') i

j~-i

i

j~-i

309

Physics of fast reactor control rods where a 0 is a flux perturbation coefficient at position i due to a control inserted at positionj. The flux perturbation factor is defined by reference to the exact perturbation theory expressions for the reactivity worth of rod i with and without rodj present in the reactor. For arrays of partly inserted rods the formula must be modified to allow for the degree of insertion of the rods, Zj. This can be done by subdividing into axial segments, s, treating interaction effects between segments and assuming linear interpolation for dependence on the degree of insertion within a segment. The coefficients can be obtained by perturbation theory calculations for progressive insertion of each control rod into successive segments. In the case of fully inserted and full withdrawn rods a two-dimensional model is applicable. A twodimensional model can also be of acceptable accuracy for partly inserted rods (the isolated individual rod worth W~°) being replaced by the worth of the isolated partly inserted rod, l~j°)(Zj)). The interaction model expression can then be written in the form of a simple reactivity interaction model

W = Z Wl °' [I (1 +&W)°'). i

j>i

The [30 are determined from the measured or calculated values of the reactivity worths of the individual rods and of pairs of rods. A more general reactivity interaction model can be written in the form

i + ~ u~i j ,,/(01w~0) W : E (W (o) vv i vv j " i

j>i

where the coefficients a~j, b~k, etc are derived from calculations or measurements for all combinations of rod insertions. Similar methods, based on a perturbation theory approach, have been developed by Wheeler et al. (1975) and Konishi and Yamamoto (1975). Wheeler found that his method is sufficiently accurate to be used in designing PFR core arrangements which meet the operating limits on control requirements. Konishi and Yamamoto evaluated their 'disturbance parameter method' in an analysis of the ZEBRA MZC experiments. The interaction effects are in the range - 10% to + 15 4, and an accuracy of better than t.3% was obtained for the reactivity worths of arrays of rods, relative to a reference configuration. Similar approximate methods can be used to calculate the effects of control rods on reactor power distributions. An example is the Influence Function Method developed by Cowan and Derby (1974). The

power distribution at mesh point i with N control rods inserted has the form

P~ = p°Fl(fO, where the factor (f.)i is the first order power distribution influence function at point i due to the insertion of rod n. P~ is renormalized following the application of the factors (].)i. The power distribution influence functions can also be used in a perturbation theory estimation of control rod interaction effects.

4. C O N T R O L

ROD MEASUREMENTS

AND THE

VALIDATION OF CALCULATIONAL M E T H O D S Measurements of the reactivity worths of control rods have been made in both zero power critical assemblies and operating power reactors. In addition to reactivity worth measurements, fine structure reaction rate measurements have been made within and close to control rods, and whole reactor fission rate distributions have been measured for different arrays of control rods (with both fully inserted and partly inserted rods). The measurements include comparisons of different control rod materials and different rod sizes and geometries. These are supplemented by measurements for small samples having a range of sizes so that size effects (including resonance shielding effects) can be determined. A comprehensive review of measurement techniques is contained in the proceedings of the Specialists"

Meeting on Control Rod Measurement Techniques: Reactivity Worth and Power Distributions, held in Cadarache in April 1976. Reactivity worths of control rods have been measured either by critical balance (restoring the criticality of the reactor by moving calibrated control rods, adding elements to the edge of the core or adding fissile material within the core) or by subcritical reactivity measurements. Different reactivity scales are used, in particular the effective delayed neutron fraction and the calculated effects of fissile material addition to the core. Reaction rate measurements within and close to control rods have usually been made using foils, measuring the induced activity. Typical measurements are of 23sU and 23su fission in boron rods and l~lTa (n, ,,) 182Ta in tantalum rods. Whole reactor fission rate distribution measurements have been made using both foils and fission chambers distributed through the core or by moving fission chambers axially and radially along channels. The literature on measurements relating to control rods in critical facilities is extensive and the following sections describe only a selection to illustrate the types of measurement and the results of the analyses. In addition to rod reactivity worths and reaction rates,

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measurements have been made of the effects of control rods on sodium voiding reactivity effects. Gamma energy deposition measurements have also been made. Data on irradiation swelling are being obtained from power reactor irradiations. Several studies have been directed at improving the accuracy of subcritical reactivity monitoring techniques by refinements to the source multiplication method. These methods have been applied to control rod reactivity worth measurements and the effects of control rod movement on the fluxes at detector positions have been studied. The measurements made in zero power critical facilities have been used to derive bias factors, which are applied to reactor design calculations, and to estimate the uncertainties in predictions. The ratios of calculated to experimental values, C/E, of rod reactivity worths can vary with the radial position of the rod because of errors in the calculation of reactor flux distribution. Radial flux distributions are more sensitive to nuclear data errors and calculation methods approximations in large reactors than in small reactors and the flux distributions are also more sensitive to reactivity perturbations, such as the insertion of a control rod. Consequently, control rod interaction effects (the fractional differences between the reactivity worth of a group of rods and the sum of the worths of the individual rods) are larger in large reactors and there can be an uncertainty in extrapolating the results of measurements made on small critical assemblies to predictions for designs of large power reactors.

4.1. Methods of measuring the reactivity worths of

control rods In the critical balance method criticality (or reactor power) is maintained when the rod or group of rods is inserted by making one of the following changes (1) Moving calibrated or reference control rods. This method is usually only used to measure reactivity worths of single rods or small movements of a group of rods. The reference rods can be calibrated in terms of the delayed neutron fraction or the calculated effect of a compensating fissile material addition to the core. (2) Increasing the radius of the core by adding extra fuel elements at the core edge. The reactivity worth is then related to the calculated effect of edge element addition. Consistency is checked by interchanging the added edge elements and comparing measurement and calculation for the different arrangements. (3) Changing the fissile material content of a region of the core. The reactivity scale is then the calculated

effect of the fissile material addition. This is an appropriate reactivity scale because one of the functions of the regulating rods is to compensate for fuel burn-up. (4) Comparing the reactivity worths of different arrays which are calculated to have about the same reactivity worth with a balance of type (1), (2) or (3). The different reactivity scales can be interrelated in separate experiments (for example by measuring edge element worths in terms of calibrated rods, or distributed fissile material). It is necessary to recalibrate rods when the size of the core is changed or experimental rods are added. The principal method used to measure subcriticality is the source multiplication technique. The ratio of the total flux to the source is approximately inversely proportional to the subcritical reactivity. Flux calculations must be made to obtain the flux at the counter positions because of the changes in the flux shape caused by insertion of the rods and the spatial distribution of the source. The reactivity scale involves a reference subcritical condition, such as that produced by inserting a calibrated control rod. Measurements of subcriticality relative to the effective delayed neutron fraction can be made by calibrating a reference fine control rod by means of asymptotic period measurements following rod withdrawal, or by inverse kinetics analysis of the reactor power response following rod drop or rod withdrawal (fitting the response using the delayed neutron kinetics equations). There are a number of techniques for measuring subcritical reactivity relative to a calibrated reference control rod, in addition to source multiplication. These include rod drop, rod jerk, source jerk, pulsed source and reactor power noise. Account must be taken of spatial flux transients (either by calculating these or measuring them with arrays of counters) and of the spatial distribution of natural neutron sources due to spontaneous fission and (c~,n) reactions and any fixed sources introduced to increase the subcritical flux level. The different methods are reviewed and intercompared in the proceedings of the 1976 Cadarache Meeting (for example in the paper by Carpenter (1976)). Although there are uncertainties in total delayed neutron yields and in the time dependence of delayed neutron emission the accuracy of this reactivity scale is estimated to be + 5 %. The units used to denote reactivity include the following (1) Effective multiplication k-effective or k (sometimes in units of 10 Zdk or niles and 10 - S d k or milliniles, which is also denoted by mN, and by pcm, percent mille).

Physics of fast reactor control rods (2) Reactivity p = l - 1 / k , and reactivity change, 6p = - 6 ( 1 / k ) = f k / k l k 2 . (3) Reactivity in units of the effective delayed neutron fraction, fleff, also called the dollar, $; 1~o of this is called the cent. (4) The inhour. This is the period of the asymptotic exponential time dependence of reactor power (resulting from the reactivity addition when the reactor is at delayed criticial). It is related to the effective delayed neutron fraction. The equivalence is usually obtained experimentally with a reactivity addition which results in an asymptotic period of about 100 sec. 4.2. Measurements of control rod reactivity worths and comparisons with calculations 4.2.1. Measurements in Z E B R A . Measurements for both single control rods and arrays of rods in conventional two enrichment zone cores have been made in Assemblies 12, 13 and 16. The Assembly 12 series of control rod measurements (which was also called MZC) was part of the Mozart collaborative programme with Japan and the measurements were in support of the design of the Japanese fast reactor MONJU (Broomfield et al., t973). Assembly 13 was a mock-up of the U.K. Prototype Fast Reactor (Marshall et al., 1976). These two assemblies had a core diameter of about 1.5 m. Assembly BZB/2, which had a core diameter of about 2.4 m, was one of the Assembly 16 large core configurations studied as part of the BIZET collaboration programme (Rowlands et al., 1979). The BIZET programme also included heterogeneous core studies (Assembly 17 (or BZC) and Assemblies 18-21 (the BZD series)). In the MZC programme the MONJU reactor mock-up control rods consisted of 19 pin clusters of boron carbide pins and tantalum in sodium filled calandrias, both types of rod occupying the space of four normal ZEBRA fuel elements. The pin cluster diameter was about 8.5 cm. The absorber materials were natural B4C, enriched B4C (30, 80 and 90~o) and tantalum. Reactivity measurements were made using the critical balance technique, with criticality maintained by adding fuel elements at the outer corebreeder boundary. The reactivity worth of the edge elements was related to the reactivity worth of plutonium distributed over a region of the core in a separate experiment. The rod reactivity worths were thus related to the calculated reactivity worth of plutonium, the estimated accuracy of the reactivity scale being _+4'~o. The cross-sections for the control rod region were obtained using MURAL-FGL5 (Macdougall, 1977; Rowlands et al., 1973) and a two

311

region cylindrical model using flux averaging as the homogenization method. The core and breeder region plate cell averaged cross-sections were also obtained using MURAL-FGL5. The standard analysis was made using a twodimensional plan model of the reactor and diffusion theory. The reactivity worths of the control rods were overestimated by these calculations by an amount which depended on the boron content of the rods and the radial position. The overestimation for a central rod ranged from 4 ~o for a natural boron rod to 15 Tofor the rod with the highest boron content. The overestimation was lower by about 4 ~, for rods in the outer core region, relative to the values for rods at the core centre. The reactivity effects of the rod followers, relative to fuelled assemblies, were also overestimated (by this diffusion theory calculational method) by between 9 and 15Yo. For a central tantalum rod the overestimation was 10%. Rod interaction effects were well predicted, the C/E values for rod arrays being within +2jo °/ of the values for individual rods. When corrections were made for the approximations in this calculational method the C/E values became closer to unity. However, even on the basis of the standard method of analysis, it was considered that, by applying bias factors, rod reactivity worths could be predicted, for the same size of reactor and t'ypes of control rod, to an accuracy of about _+5Yo. BZB had a larger two enrichment zone core with a diameter of about 2.4 m. The control rods contained natural boron carbide pin clusters. The measurements included both critical balance and subcritical source multiplication measurements for arrays of partly and fully inserted rods (involving a central position, an inner ring of 6 and an outer ring of 12). In the BZB/2 series of critical balance measurements a series of eight different arrays having approximately the same reactivity worth (about 4Yo dkt was studied. These ranged from an array with absorbers only in the inner ring to an array with absorbers only in the outer ring. The analysis was made using X Y Z geometry diffusion theory in 16 energy groups. The relative reactivity worths of the different arrays were predicted to within about +5Yo (or +0.002 dkl. 4.2.2. Measurements in FCA. Control rod measurements have been made in the JAERI critical facility FCA in support of the design of the Experimental Fast Reactor JOYO and the Prototype Fast Breeder Reactor MONJU. Control rod experiments were carried out on Assembly V-3, the engineering mockup of JOYO, using the source multiplication method (Mizoo et al., 1976). This contained six control rod channels, the rods being in the form of clusters of seven

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B4C pins in a calandria filled with sodium. Analyses 10 to 20%, depending on the number of B4C pins, were performed using the JAER1 Fast Set Version II 1°B enrichment, B4C pin arrangement and and a two-dimensional diffusion theory code. The position. effective cross-sections for the B4C rod region were (2) The C/E value is lower by 10 to 20~o in the radial obtained using a collision probability method. The blanket than at the centre of the core. lowest limit of the subcriticality of the system was (3) The differences between the worths obtained by - 6 % dk and the C/E values ranged from 1.00 to 1.03. diffusion theory and transport theory are fairly large in the radial blanket. Monte Carlo calculations were also made using the modified correlated sampling technique. For the 4.2.3. Measurements in MASURCA. The measurecentral rod reactivity worth good agreement was obtained between the Monte Carlo and $8 transport ments made in the critical facility MASURCA at theory results, which were both smaller by a few ~o Cadarache include reactivity worths of sodium followers, steel rods, sodium/steel rods and absorber rods than the measured value. On Assembly V I I - l , the engineering mock-up for of different diameters and I°B enrichments in different M O N J U (which had a core diameter of about 1.4 m) core environments (Bouget et al., 1979). The analysis the studies concentrated on the I°B enrichment effect of the experiments has been made using 25 energy and interaction effects. Thirteen types of B4C pin group macroscopic cross-sections derived from the cluster, different in pin number (9, 16 and 32) and in C A R N A V A L I V cross-section library using the l°B enrichment (natural, 30, 55, 73, 80 and 90%) were HETAIRE cell calculation code. Comparisons of studied in the 90Z version of the assembly, which had a diffusion theory and transport theory show that zoned test region. The content of I°B in a rod ranged diffusion theory overestimates the reactivity effects of from 0.40 to 4.46 kg. The interaction effects were sodium filled channels by 3.5~o, steel rods by 6.7~o, studied with 24 different rod arrays comprising up to natural boron carbide rods by 7% and rods containing five rods with the location and worth selected to 90% enriched boron carbide by 8%. In control rod studies made in the PRE-RACINE simulate the MONJU safety rods. In the 90B sector version of the assembly up to nine B4C rods were and RACINE programmes measurements were made inserted in both the inner and the outer rings, resulting for two sizes of rod channel, 112 cm 2 and 280 cm 2, for in a subcriticality of about - 1 0 % dk. In these rod both sodium/steel diluent channels and boron carbide worth measurements, various techniques (Mizoo et al., absorber channels (Humbert, 1984). The smaller size 1976) including theoretically modified source multipli- corresponds to rods used in PHENIX and the larger cation, source jerk, rod drop and pulsed neutron size to rods for SUPER-PHENIX. Addition of edge source methods were applied. A special feature of the elements was used to maintain criticality when single 90B version was that the neutron source distribution rods were inserted at the centre of the core whereas the was strongly asymmetric. The C/E values obtained modified source multiplication method was used to with diffusion theory ranged from 1.03 to 1.09 for the measure the larger reactivity changes produced by measurement of the I°B enrichment effect, C/E arrays of rods. For diluent rods the largest rod worth discrepancy increasing with the macroscopic absorption crosssection of the rod region. The C/E values for the 24 (7%) was obtained for the smaller sized rod containing different arrays were in the range 1.04 to 1.08. The C/E 93% sodium and 7% steel. For smaller sodium values decrease with an increase in the number of rods fractions and larger rods the discrepancies are smaller, in the array, the values for the five rod arrays being being within the measurement uncertainties of + 1.5 % about 4% lower than those for the single rods. It is for sodium fractions of 66~o or less. For the smaller considered that all the above trends in the C/E values sized boron carbide rod three enrichments were would be improved by using transport theory. studied: natural boron, 47% enriched and 90% A mock-up experiment for the B4C absorber rod to enriched. Calculation overestimates rod worth by 6% be used for reactivity control in the radial blanket of for the natural boron carbide rod and by 9 and 10% for M O N J U was made on the modified version of the 47 and 90% enriched rods. This trend of increasing Assembly VII-2. Reactivity worths of B4C pin clusters C/E value with increasing boron enrichment is similar and radial reactivity worth distributions of a B4C to that found in the ZEBRA, FCA and SNEAK-11 sample and fission rates of 239pu and 238U were measurements. A comparison was made between a measured in the radial mock-up blanket. A compari- normal heterogeneous design of boron carbide rod son of measured and calculated B4C rod worth reveals (with an inner boron carbide region surrounded by a diluent region) and a homogenized rod having the the following results (Yoshida, 1979) (1) The B4C absorber rod worth is underestimated by same area (280 cm2). The calculational method used

Physics of fast reactor control rods overestimated the worths of both designs by the same ~o, the effect of homogenization of the composition being to increase rod worth by about 8~o. The PECORE programme on MASURCA had as its objective the simulation of characteristics of the Italian experimental fast reactor PEC (Artioli et al., 1979). Twelve control rods are positioned at the core-reflector boundary. The reactivity worths were measured both by extending the core to maintain criticality (and calculating the reactivity effect of the extra material) and also by subcriticality monitoring. 4.2.4. Measurements in SNEAK. Control rod measurements have been made in the Karlsruhe critical facility SNEAK (Giese and Helm, 1976)in a series of critical assemblies (Assemblies 2, 6, 9, 10 and 11) in support of the design of SNR-300, the 300 MW(e) LMFBR being built at Kalkar. One measurement procedure (used in the earlier measurements and for later measurements involving a single rod ~is to make the reactor critical for a reference configuration by adding core edge elements and then to compensate the reactivity change resulting from adding an extra rod (or changing rod insertion) by withdrawing calibrated shim-rods. Large reactivity changes are measured by making a sequence of small changes (less than about 1~o dk). As the size of the core is increased the shimrods must be recalibrated and the effective delayed neutron fraction recalculated. The reactivity worths of the experimental rods also change with core size. The experimental rods can also cause ttux tilts and consequently change the reactivity worths of the shimrods, which must be recalibrated after each rod movement. The accuracy of the technique is estimated to be about 3 5~%. The later measurements for groups of control rods used the modified source multiplication technique, the source being the neutrons from spontaneous fission and (:~, n) reactions in the plutonium fuel. The method was calibrated relative to the shim-rods and calculated corrections were applied to the detector flux measurements. The accuracy of the method was estimated to be + 5~, in the subcriticality. Source jerk measurements of subcriticality (which involve the change in flux resulting from the source change) were also made. The measurements were made for single central and off-central rods, both fully inserted and partly inserted. In SNEAK-9A there ~as an inner ring of three control rod positions, an intermediate ring of three shut-down rod positions and an outer ring of six control rod positions. The two inner rings were in the inner core region and the outer ring was in the outer core, adjacent to the inner core-outer core boundary. Measurements were made with groups of 3, 9 and 12

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rods both partly and fully inserted in this uranium oxide fuelled core. The core diameter was 1.42 m. The measurements were analyzed using the K F K - I N R 26 group cross-section set and the flux synthesis code KASY in X Y / Z geometry. Pilate et al., (1978) have used the same methods and data to analyze the control rod measurements made in both SNEAK-9A and ZPPR-4, a 1.88 m diameter core which contained 19 control rod positions (a central rod, an inner ring of 6 and an outer ring of 12). The diffusion theory calculations gave consistent values for C/E for the different control rod arrays and for both partly and fully inserted rods (to within _+4~, of the mean C/'E valuet. However, for the SNEAK measurements the diffusion theory calculations underestimated rod worth by about 11 °(i whereas the underestimate was only about 30,o for the ZPPR-4 measurements. The control rod designs were different, being a cluster of boron carbide pins in an aluminium block in the SNEAK experiments and a stack of boron carbide and sodium plates in the ZPPR measurements. Transport theory supercell calculations showed the effect of this difference to be small. It was concluded that the difference is related to the nature of the core fuel. In the SNEAK assemblies 2C and 6A/D control rod measurements were made in plutonium uranium fuelled zones and the analyses of these measurements gave C/E values close to unity, as for the ZPPR-4 measurements. SNEAK-10C was a compact core fuelled with uranium, the inner core fuel having an enrichment of 25°o and the outer core 30')/o. The core height was 0.489 m and the diameter 1.6 m. The core contained 15 four-element singularities, a central position, an inner ring of six and an outer ring of eight on the inner core-outer core boundary. Measurements were made for single rods in each ring, for complete rings and for different arrays, including the full array, for which the reactivity worth was -8.5%. The analysis has been made using different methods (Giese et al., 1984). The reference calculated results were obtained using the KA PER cell heterogeneity code to prepare the crosssections and using effective diffusion coefficients for the control rod channel, together with three-dimensional diffusion theory (the D3D code). The C/E values obtained for the central rod, the inner ring and the outer ring at 1.0. However, a C/E value of 1.07 is obtained for the full array. It is thought that this might be associated with the use of a two-dimensional ( X Y geometry) source multiplication calculation to obtain the corrections to the detector measurements of subcritical flux in the modified source multiplication measurements. A series of mock-up assemblies of the KNK 2 Fast Test Reactor was studied in the SNEAK-11 pro-

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gramme (Mollendorff and Helm, 1980). Control rod reactivity worths were measured by sub-critical methods, mainly the modified source multiplication technique using the spontaneous fission source of the fuel. The mean I°B enrichment of the rods ranged from 54 to 94% and the measurements were made for both single rods and pairs of rods. The C/E values obtained in the diffusion theory analyses ranged from 1.03 to 1.21, with a tendency for the value to increase with increasing l°B enrichment. A comparison of twodimensional diffusion theory and transport theory calculations showed that this trend would be reduced by use of transport theory. An improved treatment of the control rod follower in the diffusion theory calculations was also considered to be required. Absorber reactivity worth measurements were made during the cold critical nuclear test programme on KNK 2 (Stanculescu et al., 1979). Monte Carlo calculations were made to compare with the measurements, the C/E values being 1.05 for the primary rods and 1.02 for the secondary shut-down system.

4.2.5. Measurements in BFS. The BN4~00 mock-up experiments made in the assembly BFS 24-16 (Orlov et al., 1973) are an illustration of the measurements carried out in the BFS-I and 2 critical facilities at Obninsk. The assembly contained two rings of control rods, an inner ring of 16 and an outer ring of 12. This uranium fuelled assembly had an effective diameter of 1.915 m. Subcritical reactivity was measured using a central source and four detectors. In addition, measurements were also made using a pulsed source and by adding edge elements to compensate for reactivity changes. Extensive measurements were also made of control rod interaction effects so that the worths of different arrays could be deduced from measurements for individual rods and pairs of rods. These measurements can be made more accurately because the reactor is closer to critical. A conjugate interference coefficient was defined as the ratio of the reactivity effect of two rods to the sum of the effects of the individual rods. The reactivity effects of arrays were then derived assuming that each rod influences others independently of the configuration of additional rods. 4.2.6. Measurements in Z P P R . Control rod reactivity worth measurements have been made in the Argonne National Laboratory critical facility ZPPR for both conventional two zone cores and annular heterogeneous cores. The conventional cores have sizes corresponding to power reactors of 350, 700 and 900 MW(e). The annular heterogeneous cores correspond to 350 MW(e) reactor designs for the Clinch

River Breeder Reactor, CRBR and to 700 MW(e) sized reactors. The types of core and the numbers of control rod positions (CRPs) in the reference cores are as follows (1) Conventional two zone cores of 350 MW(e) size: ZPPR-2 (a uniform two zone core) Measurements were made for different designs of rod and for small groups of rods. Z P P R - 3 (19 CRPs) Measurements were made for up to 18 rods. (2) Conventional two zone cores having the CRBR geometry configuration: Z P P R ~ , 5 and 6 In core 4 measurements were made for up to 19 rods. In core 5 a study was made of accident configurations and the source level flux monitor subcriticality measurement technique was evaluated by comparison with other techniques. In core 6 measurements were made for a group of seven rods at different insertions. (3) Annular heterogeneous cores of CRBR size (350 MW(e)): Z P P R - 7 (up to 15 CRPs) An extensive series of investigative control rod measurements was made in ZPPR-7G. Control rod measurements were also made in 7B and 7C. ZPPR 8 The effect of replacing selected uranium breeder zones by thorium was studied. Control rod measurements were made in ZPPR-8F. (4) Conventional two zone cores of 700 MW(e) size: Z Z P R - 9 (uniform core zones) ZPPR-10A (19 CRPs) ZPPR-10B (19 CRPs with seven rods inserted in the reference core) (5) Conventional two zone cores of 900 MW(e) size: ZPPR 10C (19 CRPs) ZPPR-10D (31 CRPs) ZPPR-10D/1 (31 CRPs with one rod inserted) ZPPR-10D/2 (31 CRPs with six rods inserted) (6) CRBR heterogeneous core design studies: Z P P R - l l (19 CRPs) (7) Annular heterogeneous cores of 700 MW(e) size: ZPPR-13 (30 CRPs). In the conventional cores the 19 CRP positions are arranged in hexagonal rings, with a central rod, an inner ring of 6 and an outer ring of 12. In the case of the 31 CRP geometry cores (ZPPR-10D) a further ring of 12 is added and the radii of the inner two rings are reduced so that all the CRPs remain in the inner core or on the inner core~outer core boundary. In the conventional cores measurements were made with different arrays of up to seven control rods. In the

Physics of fast reactor control rods heterogeneous cores arrays of up to 12 rods have been studied. Measurements made in the conventional cores have been summarized by Lineberry et al. (1979) and McFarlane et al. (1980). Those made in the heterogeneous cores have been reviewed by McFarlane et al. (1984) and, for Assembly 13, by Collins et al. (1984). In most experiments the control rods have been simulated by columns of boron carbide plates or a combination of boron carbide and sodium/steel plates. In the smaller cores the CRPs occupy 2 × 2 matrix element positions. In cores 9 and 10, where there were 19 CRPs, these each comprised 3 × 3 matrix positions. In core 10D, which contained 31 CRPs, they each occupy 2 × 3 matrix positions. The cross-sectional areas of the CRPs are 122, 183 and 275 cm z for the four element, six element and nine element CRPs. In addition, measurements have been made varying the B4C density and the ~°B enrichment (ranging from natural boron to 92Vo enriched). Measurements have also been made for pin cluster geometry rods occupying four element positions. These contain 64 pin positions with different numbers of B4C pins being inserted in different experiments. For the two zone conventional cores the core volumes are 25401 (350 MWe), 4600 1 (700 MWe) and 6100 1 (900 MWe), the core diameters being approximately 1.8, 2.4 and 2.8 m. In most of the cores the modified source multiplication technique, calibrated by inverse kinetics analysis of a rod drop, was used to measure subcritical reactivity. The calculated corrections to the detector flux measurements were supplemented by the measurements made using 64 distributed z35U fission chambers. The standard design method of analysis is nine energy group, X Y geometry, diffusion theory. Core axially averaged group and zone dependent bucklings are derived separately for different core regions, control rods and CRPs using R Z geometry models. The broad group cross-sections are derived from ENDF/B-IV using the MC/-II cell code (with buckling recycle in the cases where this is considered necessary). In addition 28 group three-dimensional diffusion theory calculations have been made for reference configurations. In ZPPR-3 the measurements included a study of the effect of sodium voiding on rod reactivity worth. Rod worths were found to be smaller in the inner core and slightly larger in the outer core in the sodium voided case, the reduction for a rod in the inner core being greatest for the smallest rod (a factor of 0.88). The nine group X Y diffusion theory calculations gave fairly consistent C/E values for the normal and voided

315

cores for single rods but for multiple rods the values of C/E were about 5~o higher in the voided core. In ZPPR-4 the rods were fully inserted and in Z P P R ~ the measurements included partly inserted rods. The nine group X Y diffusion theory analysis of the ZPPR-4 results gave fairly consistent values of C/E for the worths of single rods having different boron contents (within 1~o) and at different radial positions (within 350) and also for the arrays of up to seven rods (with additional differences of up to about 3~o, making a total difference from central worths of about 5~o). The mean C/E values obtained in different analyses differ. The analysis using ENDF/B-III data gave C/E values in the range 0.97 to 1.04. A later analysis using ENDF/B-IV data gave mean C/E values for assemblies 3/IB, 4/1, 4/2, 4/3 and 4/'4 in the range 1.073 to 1.096 with RMS deviations of about 4% or less for the individual measurements in each assembly. The measurements in ZPPR-6 involved the movement of seven control rods simulating a burn-up cycle, with the following four critical configurations: rods fully inserted, 2/3 inserted, 1/3 inserted and rods fully withdrawn. Lee et al. (1978) have analyzed the measurements using X Y Z diffusion theory. The calculated values of k are systematically about 1.5 ~Jl,low, varying approximately linearly with rod insertion from an underprediction by 1.43~o for case (1) to an underprediction by 1.75o for case (4). Rod reactivity worth is thus underpredicted by about k-~o. the /O, underestimate being consistent for the three levels of insertion. An analysis was also made using an X Y model with axial bucklings. Using mid-plane bucklings the values of k and the mid-plane fission rate distributions are well predicted. The ZPPR-7 and 8 series of assemblies were heterogeneous cores with a central breeder island and three rings of breeder elements. Strong control rod interaction effects were found. For the six outer ring rods in Core 7B the interaction effect was 55~,,,. This interaction effect reduced to 30~o in Core 7C with a redistribution of plutonium (simulating burn-up), which compares with a value of about 2050 for a similar rod group in a homogeneous core. Because of the large interaction effects an extensive series of measurements was made in ZPPR-7G. The mean C/E values for control rod worths obtained using nine group diffusion theory and ENDF/B-IV data are in the range 0.98 to 1.04. This is about 8~/o lower than the values obtained for conventional cores using the same method. A study has been made of the various approximations involved in the coarse mesh diffusion theory calculations and a possible explanation for the lower C/E values in ZPPR-7 compared with Z P p R ~ ,

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is a difference in the degree of cancellation between the errors associated with diffusion theory, mesh effects, group condensation effects and the use of axial bucklings. Except for 7C the C/E values for the outer ring rods are 5-6~o higher than for the inner ring. This spatial discrepancy is connected with mispredictions of 2~,~o in radial fission rates. Control rod interactions among the inner ring rods are small, ranging from - 3 ~ to +6~o but there are large interactions (particularly for individual rods) between the inner and outer ring positions. Interaction effects range from - 20 ~ to + 65 ~o. These are well calculated by diffusion theory. Because of the strong interaction effects, the reactivity effect of removing a rod from a bank can be relatively large and this has implications in safety analysis when considering unintentional rod withdrawal. Removal of one rod from the bank of six inserted outer ring rods results in a reactivity increase of $6.6. This compares with the single rod worth of $2.0 or the mean worth for a rod in the bank of $2.8. The analyses of the control rod measurements made in the large cores of conventional design, ZPPR 9 and 10, have been reviewed by McFarlane et al. (1980). For changes in rod geometry and increases in enrichment, from natural boron to 92% enriched, the C/E values are consistent to within 2~0. Interaction effects range from - 6 ~ up to 52 ~o and are calculated to within 3 ~o. Factors affecting the interpretation of the results are the 20 to 30~o discrepancy in the prediction of the reactivity effects of the sodium/steel filled CRPs relative to core fuel and an 11 ~o overestimate of the reactivity effect of adding plutonium, when calculated using the reference design methods. After accounting for modelling approximations, the best calculated values are about 10~ high, with radial variations of 1 to 6~o about the average. Changing the 238U capture cross-section (to improve the agreement with reaction rate ratio measurements) and making corrections for the treatment of CRP leakage effects could reduce the radial variation but would increase the calculated worth values. The average worth discrepancy would then be consistent with that for a ~°B sample. Yamamoto et al. (1984) have analyzed the ZPPR 9 and 10 measurements using the Japanese J E N D L - 2 nuclear data library and an X Y geometry diffusion theory method with axial bucklings. Corrections were applied for energy condensation, mesh effects and transport theory. The rod worth C/E values increase with radial position. For example, for the central rod and the three rings of rods in ZPPR-10D the C/E values are 0.943, 0.954, 1.003 and 1.064. The radial variation in C/E for control rod worth is about twice as large as the variation in C/E for reaction rate distributions. Within ring and ring to ring interaction

effects are well predicted, to within 0.5~o of the total worth effect. The within ring interaction effect is 55.1 ~o for the second ring of six rods in ZPPR-10B. The C/E values obtained for pin geometry rods are about 2~o larger than the values for plate geometry rods, indicating a small error in the method used to calculate the homogenized cross-sections. The effect of changing rod pin array geometry from circular to rectangular is predicted to within about 1 ~o. The difference between the C/E values for natural and 92~o enriched boron carbide rods is about 1~o (0.974 and 0.959 respectively). The proposed core design for the Clinch River Breeder Reactor is an annular heterogeneous one. Lake and Doncals (1984) have analyzed the ZPPR-11 control rod measurements and considered the problem of extrapolating the results to the CRBR design. They have derived bias factors for the design calculational methods, applicable to the beginning of life core, for each of the three rings of rods (the inner ring of three primary rods, the intermediate ring of six secondary rods and the outer ring of six primary rods), the three factors being 0.912, 0.920 and 0.951. The accuracy estimated for these factors is _ 12 ~o (3 SD). A comparison of the C/E values for measurements made using the 52 pin control rod mock-up with those obtained using the standard ZPPR plate control rods is used in the derivation of these bias factors. There is a difference between the C/E values obtained for the plate and pin geometry rods, the ratios being 0.951, 0.981 and 0.989 respectively for the three rings. Control rod measurements made in the larger annular heterogeneous assembly ZPPR-13 have been reviewed by Collins et al. (1984). The three rings of CRPs contain 6, 12 and 12 rods respectively. The rods in the outer two rings are in two groups, those adjacent to gaps in the annular breeder rings and those between the rings. Subcritical measurements were made for up to 12 boron carbide rods using the modified source multiplication technique and the 64 in-core fission chambers. Comparisons were also made between the standard four element plate rods and pin-type rods. For some configurations transport theory corrected calculated values were derived. There is a trend in the C/E values with radial position, the corrected C/E values increasing from 0.986 for the inner ring to 1.063 for the outer ring (for the measurements in Z P P R - I 3B/4). The calculations for the pin-type rods gave C/E values up to 1.5~o higher than for the plate geometry rods. The control rod interaction effects were large and well predicted using diffusion theory. 4.2.7. Measurements in ZPR-9. In the FTR Engineering Mock-up Critical, EMC, programme of

Physics of fast reactor control rods measurements in the Argonne National Laboratory critical facility ZPR-9 control rod interaction effects have been studied for strongly asymmetrical configurations (Dobbin, 1976). The control rod arrangement consisted of an inner ring of three equally spaced safety rods, an intermediate ring of six control rods and an outer ring of seven peripheral shim-rods. The control rods are not equally spaced but are grouped in pairs close to the shut-off rod positions. Each rod was simulated by a group of four drawers containing a total of 9.35 kg of natural boron carbide (1.33 kg of l°B I. Twenty two different configurations were studied. The worth of a control rod is about halved when the neighbouring control rod and safety rod are inserted. The analysis was made using X Y geometry diffusion theory and a 30 group condensation of the FTR set 300. All the C/E values were within +8~o of unity. 4.2.8. Measurements on the experimentalJast reactor JOYO. Control rod measurements were made at the start-up of JOYO (Nomoto et al., 1979). There are two regulating rods and four safety rods forming a ring of six rods within the core. The measured reactivity worths are influenced by closeness to the Sb-Be neutron source which softens the neutron spectrum. The calculations slightly underestimate the measured rod reactivity worths, the underestimate being largest (about 3~o) for the two safety rods closest to the source, which was not modelled in the calculations.

4.3. Reaction rate measurements within and close to control rods As an example of such measurements we consider the activation measurements made using foils within and near to control rods in the ZEBRA Mozart Assembly MZC (Broomfield et al., 1973). The reaction rates measured in the boron carbide rods were fission in 235U and 238U and in the tantalum rods the reaction l~lTa (n, 7) 182Ta. A comparison of the measured values with calculation was made for three central rods: an 80Yo enriched boron carbine rod, 8°B, a natural boron carbide rod, BN, and a tantalum rod, Ta. The calculations were made using flux averaged cross-sections for the control rod region derived using the pin cluster collision probability code, PIJ. The fluxes obtained in the whole reactor diffusion theory or transport theory X Y geometry calculations were then combined with the PIJ flux distributions to give the reaction rates in the individual pins. The values of C/E for each reaction rate were normalized at a point about 50 cm from the centre of the rod. The diffusion theory results can be summarized as follows

317

(11 Reaction rates within the rods were overestimated, the C/E values being in the ranges 1.05 to 1.10 for 235U (n, f h 1.09 to 1.14 for z38U (n,j) and, in the Ta rod, up to 1.09 for Ta (n, ";). This is a similar pattern of C/E value to that found for the rod reactivity worths excepting that the C/E value for 23sU (n,j) within the SOB rod was 4~o higher than in the BN rod whereas the C/E value for rod worth was 10~o higher. Transport theory calculations give a marked improvement in the value of C/E for 238U (n, ft. (2) Reaction rates close to the rods are predicted to within _+20. (3t The 235U (n,.ll and 23sU (n,f) reaction rate variation from pin to pin is well predicted, the measured change in 235U fission rate from outer to central pin being - 17~,, for the 8°B rod. For a similar reactor, using the same calculational methods and applying bias factors based on these C/E values an accuracy of about _+5~o (1 SDI in the reaction rates (relative to the core fission rate) is considered to be attainable. The discussion on homogenization methods in Section 3.2 showed that using flux averaging as the homogenization method results in an overestimate of the absorption rate within the rod, the region averaged flux calculated in the homogenized whole reactor calculation being higher tlaan the flux in the fine structure calculation. Application of the homogenization scaling factors would result in improved agreement. Fission rate ratio measurements were made at the centres of diluent channels and control rods in the MASURCA PRE-RACINE programme (Bouget et al., 1979). Diffusion theory calculations overestimated the 238U/235U fission rate ratio by 10~o for sodium filled channels. The discrepancy was smaller for boron carbide control rods and depended on the size of the rod. The overestimation of the high energy flux in a diffusion theory calculation for a sodium filled channel results in an overestimation of the axial leakage in the channel and of the sodium moderation. Transport theory calculations, or adjusted diffusion theory parameters, are required for the accurate treatment of such diluent channels.

4.4. The effects of control rods on core power distributions

239pu fission rates have been measured in ZEBRA Assembly 13 (the PFR mock-up assembly) for five different critical configurations of the five boron carbide control rods C1 to C5 (Marshall et al., 1976). The rods occupy five of six hexagonal positions in the

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J.L. ROWLANDS

inner core adjacent to the inner core-outer core boundary. The fission rate measurements were made using the multi-chamber scanning system which had 150 chambers at five different heights within the core sections of 30 elements distributed through the core. The five arrays studied were: 13/1 All rods withdrawn. 13/3 C1-C5 all 34~o inserted. 13/4 C3 and C4 70~o inserted, C1, C2, C5 withdrawn. 13/5 C4 fully inserted, C1, C2, C3, C5 18~o inserted. 13/6 C1, C2, C3, C5 34~o inserted, C4 withdrawn. The rod array reactivity worths in Cores 13/3, 4, 5 and 6 were similar. Criticality was maintained by changing the number of outer core elements. The axial flux distribution was almost symmetrical in core 13/1. The average axial flux tilt in core 13/3 was 1.29 (and it was 1.49 close to rod C3). The radial flux tilt was a maximum in core 13/4 where it was 1.36. The fission rate distributions were reproduced to within about ___1~o by the six group X Y Z geometry diffusion theory calculations. The average C/E value was lower in the outer core than in the inner core by 0.5~o. A larger underprediction (about 4~o) of the fission rate in the outer core relative to the core centre was found in a similar analysis for the core MZB, which had a thinner outer core region and a higher ratio of outer core to inner core enrichment (1.5 compared with 1.2 for Core 13) (Ingram et al., 1973). C/E values in chambers close to absorber sections of rods were about 1~o high relative to those away from absorber sections. The 239pu fission rate distribution was measured for eight different control rod configurations using the multi-chamber scanning system in the larger ZEBRA Assembly BZB (Rowlands et al., 1979). The control rod positions were: the centre of the core, an inner ring of six and an outer ring of 12 near to the inner core-outer core boundary. Each control rod array studied had a combined reactivity worth of about 4~o dk. All the arrays were symmetrical apart from Array 5, which had one rod fully inserted in the outer ring, the inner ring rods 2/3 inserted and the remaining outer ring rods 1/4 inserted. The other arrays included two with rods fully inserted, Array 3 with the inner ring of six inserted and Array 8 with 9 of the outer ring of 12 inserted. In the other arrays there were different degrees of insertion in the inner and outer rings and the central position. There were changes in both the axial and radial fission rate distributions of up to about 30 ~o. The fission rate distributions were analysed using X Y Z geometry diffusion theory. The interpretation is complicated by a radial flux tilt of 4~o across the

diameter of the core which is present even for the symmetrical arrays of rods and which is not reproduced in the calculations. The core contained a sector of pin geometry fuel in the mainly plate geometry core and a sector of radial breeder having a different composition. The effect of these differences is not correctly reproduced by the calculational methods used to homogenize the cells. By relating the C/E values in each array to the values found for Array 1 the effect of this asymmetry can be reduced. For the values averaged over radial and axial regions it is found that, relative to Array 1, the mean deviations for each region are less than +0.5~o and the variations about the mean are less than ___1~o. An extensive series of reaction rate distribution measurements has been made in the ZPPR Assemblies. They include 239pu, 235U and 238U fission together with 23su capture. Detailed analyses have been made for the 600 1 assemblies 10D (rods withdrawn) and 10D/2 (rods inserted) (Carpenter et al., 1980). These cores have an approximate diameter 0f2.8 m. The analyses were made using X Y Z geometry diffusion theory with corrections for $4 transport theory. As was also found for the ZPPR-9 cores the calculations consistently overpredict the 235U and 239pu fission rates and 238U capture rates in the outer core relative to the core centre by 2 to 3~o. (Similar trends were found in the analyses made using the Japanese JENDL-2 nuclear data set (Yamamoto et al., 1984).) The overall effects of the control rods on reaction rate distributions were well predicted. However, the mispredictions of the reactivity worths of sodium filled CRPs by diffusion theory have a significant effect on the power distributions in these large cores. Exploratory calculations made using axial buckling adjustment factors to account for axial transport effects in the sodium filled channels gave reaction rate corrections of up to 2~o in ZPPR-10D. Diffusion theory calculations also gave discrepant results within breeder regions close to the corebreeder boundary. It can be concluded from these analyses that a refined treatment of control rods and control 'rod follower regions is required for the calculation of reactor core reaction rate distributions. Diffusion theory calculations can then produce reaction rate distributions for the reactor core to accuracies of about + 1~o relative to a reference configuration (for example a core with rods withdrawn). However, there can be discrepancies in the ratio of the average outer core to inner core reaction rates. These might be associated with nuclear data deficiencies or with the use of diffusion theory. Close to control rods diffusion theory overestimates 239pu and 235U fission rates by

Physics of fast reactor control rods about 1% (which is about a 10 % error in the local flux dip into the rod).

319

to larger cores than those for which the methods have been validated introduces larger uncertainties.

4.5. Concludin9 remarks on the experimental

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