The core axial power distribution is important in PWR cores because the temperature variations with power and control rod operations have a large effect on this distribution. Figure 3.50 illustrates basic behavior of the core axial power distribution in PWRs.

Fig. 3.48 Radial power distribution (BOC, hot full power, control rod fully withdrawn) (Copyright Mitsubishi Heavy Industries, Ltd., 2014 all rights reserved)

Fig. 3.49 Radial power distribution (EOC, hot full power, control rod fully withdrawn) (Copyright Mitsubishi Heavy Industries, Ltd., 2014 all rights reserved)

The constant axial offset control (CAOC) method is adopted to suppress the heat flux hot channel factor for the basic behavior of core axial power distribution. It suppresses the imbalance (referred to as axial offset, A. O., for which Axial Flux Difference ЛІ = A. O. x P is used as the index during reactor operation) in the axial power distribution to an as narrow range as possible (typical range = ±5 % ЛІ from the ЛІ at the Xe equilibrium state) and to reduce the imbalance of 135Xe and 135I distributions. The basic concept of CAOC operation is shown in Fig. 3.51. A large A. O. or ЛІ (in absolute value) leads to a large Fq. If A. O. or ЛІ is not restricted as shown in the top figure, Fq x core relative power can exceed the allowable design limit. As shown in the middle figure, if A. O. or ЛІ is restricted within the narrow range which is set as ±5 % ЛІ along the line to link two ЛІ values at core relative powers of 1.0 (at Xe equilibrium) and 0.0, it is possible to suppress Fq x core relative power and to maintain it at less than the allowable design limit as shown in the bottom figure.

Power distribution is controlled by a control rod bank (usually bank D) of the control group control rods. Figure 3.52 shows the variation in core parameters in CAOC operation. The load following operation during the change from 100 % to 50 % in core power as shown in the top figure can be performed by adjusting rod cluster control assemblies and soluble boron concentration.

(5) The power in the core upper part decreases due to the control rod insertion. (6) The Xe concentration in the upper part temporarily increases and then decreases with time. (7) Since the Xe concentration in the upper part was decreased, the power in the upper part increases when the control rod is withdrawn. (8) The Xe concentration in the upper part temporarily decreases, and then increases with time and the power in the upper part decreases.	(1) Since the moderator temperature is high in the core upper part at equilibrium state, the power in the core upper part is suppressed by the effect of the moderator temperature coefficient. The power increases by the less effect when the core power decreases. (2) Since the power in the core upper part is large. Xe is reduced due to neutron absorption. Since the power in the core lower part is small, however, the Xe reduction is mitigated in the core lower part and moreover Xe is increased by the decay of 136I. (3) The power in the high-temperature upper part is a little suppressed due to the effect of the moderator temperature coefficient by the power increase. (4) The Xe concentration in the upper part is reduced temporarily, and then the Xe distribution reverses between core top and bottom with time and the power distribution also reverses corresponding to that.
Fig. 3.50 Fundamental characteristics of axial power distribution (Copyright Mitsubishi Heavy Industries, Ltd., 2014 all rights reserved)

Bottom

Fq Design Limit

Bottom

Axial Flux Difference (АІ)

FaXP

Fq Design Limit

Control of

Axial Flux Difference (АҐ)

Fig. 3.51 Basic concept of CAOC operation [31] (Copyright Mitsubishi Heavy Industries, Ltd., 2014 all rights reserved)

Fq at normal operation is evaluated for such a load following operation. Figure 3.53 shows an example of the Fq evaluation at normal operation and a typical Fq(Z) x relative power is given as a function of core axial height Z in Fig. 3.54. Positive and Negative A. O. at BOC and EOC shows the Fq(Z) x Relative Power at the most positive and negative A. O. in the range of the limit, respectively. They are below the envelope curve in Fig. 3.54. The maximum Fq(Z) x relative power over all points of time is also below the envelope curve as well as at representative points introduced here. The envelope curve is used as the heat generation distribution at normal operation for the initial condition of LOCA analysis.

50 0.4

+10

U РЯ9Г_ — JJP- it _21 A2_r_§e t Range

Target Value

10

Lower Limit of Target Range

15

+20

Normalization

Time [hr]

Fig. 3.52 Typical variation of core parameters in load following operation [31] (Copyright Mitsubishi Heavy Industries, Ltd., 2014 all rights reserved)

Evaluations and limits for gadolinia-added fuel are also given in Fig. 3.53 here and Figs. 3.55 and 3.56 introduced later. The limits for gadolinia-added fuel are set lower than those of uranium fuel, considering the different physical properties.

Fq x Relative Power 3 +

Envelope Curve 2.32 x К(Z)

Positive A. O. at BOC Negative A. O. at BOC

Positive A. O. at EOC

Negative A. O. at EOC

Fig. 3.54 Typical Fq(Z) x relative power versus axial position [32] (17 x 17 type, 3-loop core, and normal operation) (Copyright Mitsubishi Heavy Industries, Ltd., 2014 all rights reserved)

4-0 Fq x Relative Power

(Maximum Linear Power Density of _3,5 Ufanium Fuel Rod =_ 59.1 kW/m)

(Maximum Linear Power Density of Gadolinia-Added Fuel Rod = 44.3 kW/m)

о Uranium Fuel Rod at BOC 0,5 DUranium Fuel Rod at EOC

л Gadolinia-Added Fuel Rod at BOC 0 xGadolinia^dded Fuel Rod at EOC

— 30 — 25 — 20 — 15 -10 — 5 0 5 10 15 20 25 30

Axial Flux Difference

Fig. 3.55 Typical Fq x relative power versus axial flux difference [32] (17 x 17 type, 3-loop core, and abnormal control rod withdrawal at power) (Copyright Mitsubishi Heavy Industries, Ltd., 2014 all rights reserved)

[1] Transitions in development

From the beginning, the main purposes of fast reactor development have been improvements of both safety and economy.

(1) Reactor types

For core design, making the reactivity negative or as small as possible at coolant voiding has been one of the development goals for safety. Annular type, pancake (flat) type, and modular type core concepts were proposed, mainly in the US in the early study of large scale fast reactors. Those concepts are shown in Fig. 4.11. The common aim is increasing neutron leakage to the outside. In the 1980’s, heterogeneous core concepts that had both reduced void reactivity and high breeding ratio were proposed in the US and France [14]. They are shown in Fig. 4.12. In those concepts, the blanket fuel assemblies are put into the core region or the blanket layer and upper and lower axial blankets are provided as well. The common aim is increasing neutron leakage from the core region to the blanket regions for higher breeding ratio. This kind of core concept has also been consid­ered in Japan as a large scale fast reactor core. Simulated critical experi­ments were carried out under Japan-US collaboration as part of the JUPITER plan [13]. However, the actual designs of the prototype reactor and demonstration reactor (Fig. 4.13) adopted homogeneous cores which have a moderately flat shape, except that a heterogeneous core was considered in the uncompleted Clinch River Breeder Reactor Plant (CRBRP) of the US [9].

[Advantages and disadvantages of heterogeneous cores)

Fig. 4.12 Heterogeneous core concepts [15]

(type I

140 MWt (L) 280 MWe < L) 660MWe(U

КТО MWe’ I

■мМ ИП п I ■•НМ

China tipraysuli EFR

♦ Reactor type under consideration

Fig. 4.13 FBR developments worldwide [16]

(2) Coolant

The requirements for selection of fast reactor coolant are the following.

(a) Neutron moderation is as little as possible.

(b) Absorption cross section is as small as possible.

(c) Cooling performance is high.

Requirement (c) is set because fast reactors have high power density. Various liquid metals were considered as fast reactor coolant. In the early days of fast reactor development, mercury was used in Clementine (US) and BR-2 (Soviet Union). NaK was used in the EBR-I (US), DFR (UK) and BR-5 (Soviet Union). Neither of these liquid metals has been utilized after those reactors due to the low boiling point and hence difficulty in achieving high thermal efficiency. Lead-bismuth, helium gas, etc were also considered as fast reactor coolant but their applications have been limited to design studies. Sodium does not moderate neutrons too much, and has a relatively small absorption cross section, a high boiling point and good cooling performance. Despite high chemical activity, sodium does not cause corrosion of materials. As a result, sodium was selected as the main option for fast reactor coolant in many countries including Japan [14, 17].

(3) Fuel

Besides MOX fuel, metal fuel, carbide fuel and nitride fuel have been considered as the core fuel of fast reactors.

In the early fast reactors i. e. the EBR-I and EBR-II (US) and DFR (UK), metal fuel was adopted from the viewpoint of high breeding ratio. However, the development was suspended due to the low burnup limit because of large irradiation swelling and the problem of eutectic alloy formation with clad­ding material. Later, improved alloy fuel was developed in the US in the 1980’s and the Integral Fast Reactor (IFR) concept, where utilization of metal fuel and its reprocessing are combined, was proposed. The IFR was also researched in Japan. Carbide fuel and nitride fuel provide good breeding performance due to high heavy metal density, and enable high linear heat rate due to high thermal conductivity. Most of these fuel types have only been applied to design studies so far. Although operation experience of carbide fuel was and continues to be accumulated in the Indian experimental reactor FBTR, the Indian prototype reactor PFBR adopts MOX fuel. Oxygen in MOX fuel moderates neutrons and hence decreases the breeding ratio. That also enhances the Doppler effect and hence improves safety. Although low thermal conductivity makes the MOX fuel temperature relatively high, the high melting point of MOX fuel enables its operation under such high temperature conditions. Since MOX fuel is relatively stable after being irradiated, high burnup and long lifetime are possible. Also, oxide fuel has accumulated extensive experience as LWR fuel, which has made MOX fuel the mainstream for fast reactor fuel [14, 17].

As mentioned before, FPs accumulate and decrease reactivity during reactor operation. This is known as FP poisoning. Among the FPs, 135Xe and 149Sm in particular, have significant absorption cross sections for thermal neutrons: 2.65 x 106 and 4.02 x 104 b at a neutron speed of 2,200 m/s (= neutron energy of 0.0253 eV), respectively. These absorption cross sections rapidly decline over

0. 1-1 eV. Such FP poisoning is therefore a specific characteristic of thermal reactors.

135Xe and 149Sm have a substantial impact on operation and control of thermal reactors because they give rise to a characteristic behavior in reactivity change during reactor startup, power change in operation, shutdown, and re-startup.

The reactivity change can be evaluated based on the six-factor formula [5] for understanding the FP poisoning. An infinite homogeneous thermal reactor is assumed here for convenience. Initially, the infinite multiplication factor of the unpoisoned reactor (i. e., before production of 135Xe and 149Sm) is given as Eq. (1.20).

k«,=rifpe, f= =

baF “Г ЬаМ

With poisons present (after production of 135Xe and 149Sm), becomes

*'<=0=4f’pe. f’=r, у, у

l*aF T 2-аЛ/Т Z, ap

where aF, Y, aM, and aP are the macroscopic thermal absorption cross sections of the fuel, moderator and structure, and the poisons, respectively. Thus the reactivity change due to the poison production can be written as

k’ k f’f 1

^ kook’oo ff’ ripe

£aP

vlLfpe

where v is the average number of neutrons released per fission and f is the macroscopic thermal fission cross section of the fuel.

Each atomic number density of 135Xe and 149Sm must be known to calculate the FP poisoning using Eq. (1.22). The atomic number density is determined by time — dependent equations of production and destruction as described in the following sections.

The primal data (microscopic cross sections, etc.) for the nuclear reactor calculation are stored in the evaluated nuclear data file which includes cross sections uniquely-determined in the whole range of neutron energy on the basis of fragmentally measured and/or theoretically calculated parameters. The cross sections are given for all possible nuclear reactions of more than 400 nuclides. JENDL [2], ENDF/B [3], and JEFF [4] are representative evaluated nuclear data files.

Extensive data are stored in 80-column text format in the evaluated nuclear data file as shown in Fig. 2.4 and complicated data such as resonance cross sections are also compactly-represented by mathematical formulas and their parameters.

The evaluated nuclear data, from anywhere in the world, are described based on the same format which might not make sense to biginners. The format has been changed according to advances in nuclear data and the current latest format is called ENDF-6 [5]. Various data of delayed neutron fractions, fission yields, half-lives, etc. as well as cross sections are available.

The inherent safety by which an increase in reactor power leads to a decrease in reactivity is required to work automatically against a reactor power rise. As reactivity feedback characteristics to determine self­controllability of the reactor core, there are: Doppler feedback which is concerned with an increase in neutron resonance absorption due to fuel temperature rise; void feedback which is concerned with a reduction in neutron moderation effect due to coolant boiling; and moderator tempera­ture feedback which is concerned with a decrease in moderator density due to moderator temperature rise. BWRs, in which water (serving both as moderator and coolant) boils to become steam to drive the turbines, are designed to have a negative void reactivity coefficient which indicates the reactivity change due to a change in void fraction (volume fraction of steam in coolant). In other words, the reactivity decreases and the power falls when the void fraction increases due to increase of reactor power and
decrease of core flow rate. It is the most important essential characteristic for the reactivity control of BWR. The negative void reactivity coefficient is also utilized to control the reactor power by adjusting the core flow rate.

Fuel contains nuclear materials such as uranium and plutonium. A record of change and inventory of nuclear materials is prepared and regularly reported to the government and the International Atomic Energy Agency (IAEA). Inspectors from the government and IAEA also regularly inspect it to prevent unauthorized use of nuclear materials (safeguards activity).

Spent Fuel Storage Pool

Fresh Fuel ^ Storage Rack

Fresh Fuel Storage Vault
(Dry Storage)

Fig. 3.29 Schematic view of spent fuel storage and handling facilities in nuclear reactor area

[2] Fuel storage and handling facilities [4]

Fuel storage and handling facilities are for storage and handling of fresh fuel from the carry-in to the power plant to loading, and of spent fuel from discharge from core to the carry-out from power plant. They include the fresh fuel storage, spent fuel storage pool, cask pit, refueling machine, reactor building crane, and fuel pool cooling cleanup system. Figure 3.29 shows a schematic view of the spent fuel storage and handling facilities in an ABWR and Table 3.10 presents the main specifications of spent fuel storage facilities.

Fuel storage and handling facilities are designed according to the “Review Guide for Safety Design” to secure safety in storing and handling fuel. For criticality safety, it is required that fuel storage and handling facilities must be designed to prevent criticality in any cases to be expected by geometrical safe arrangement or other suitable means. An outline and the design points of the main facilities follow below.

[1] Summary of Core Management

The flow chart of PWR core management is shown in Fig. 3.67. Since it takes a considerable time from placement of a fuel order to its shipment, the required fuel assemblies are ordered several years ahead based on the operation plan. This is referred to as the long-term fuel management planning in which core characteristics such as core reactivity, power distribution, and burnup of each fuel assembly are evaluated for multiple cycles.

Then, in the present cycle design referred to as the reload core design, the number of fresh fuel assemblies and fuel assemblies loading pattern in the core

are determined for getting a safe and economic core during the planned cycle length. Core characteristics are evaluated using nuclear design codes [40,43] in both the long-term fuel management planning and the reload core design.

Fuel assemblies are loaded in the core based on the fuel loading pattern determined in the reload core design. Core physics tests are performed to measure core characteristics such as critical boron concentration, moderator temperature coefficient, control rod worth, and power distribution, and to confirm the core safety by comparing with the design values. Operation man­agement such as core performance monitoring is carried out during the reactor operating cycle, and operation data are evaluated and reflected on the long-term fuel management or the next reload core design after the cycle.

[2] Long-Term Fuel Management Planning

The main purpose of the long-term fuel management planning is to predict the number of fresh fuel assemblies necessary for the operating cycle length in multiple cycles. The fuel loading pattern of each cycle to meet the design criteria is determined continuously over several cycles. The investigation contents of the long-term fuel management planning are similar to those of the reload core design to be discussed in the next item. Investigation of some core characteristics may be omitted.

[3] Reload Core Design

As shown in Fig. 3.68, the number of fresh fuel assemblies to be loaded and fuel assemblies loading pattern in the core are determined for getting a safe and economic core during the planned cycle length. Since control rods are almost completely withdrawn during reactor operation, core characteristics concerning safety and economy are determined mainly by fuel loading pattern in the core. The number of fresh fuel assemblies to be loaded is set to achieve the required operating cycle length and the fuel loading pattern is designed in combination of reloaded and fresh fuel assemblies. Core characteristics (nuclear design parameters) such as reactivity shutdown margin and power distribution are evaluated to meet the design criteria, considering the limiting conditions for operation (e. g., control rod insertion limit) and reactor protection system set-point (e. g., over-power AT reactor trip). The fuel loading pattern is repeat­edly changed and investigated until the core characteristics satisfy all the design criteria or the number of fresh fuel assemblies to be loaded would be changed if necessary. Various optimization methods [35] are adopted for an economic fuel loading pattern (a small number of fresh assemblies) under the design criteria. For a determined fuel loading pattern, various characteristics such as critical boron concentration for excess reactivity control and control rod worth are evaluated and the results are given to the reload core design data set necessary for operation.

[4] Core Performance Monitoring

Control rods are almost completely withdrawn during reactor operation and the variation in core characteristics as burnup progresses is mild. Regular tracking of the critical boron concentration and in-core power distribution measurements once a month are mainly carried out in the core management. The measurements

О

Fig. 3.69 Location of in-core neutron detectors (4-loop core) [32] (Copyright Mitsubishi Heavy Industries, Ltd., 2014 all rights reserved)

The control rod cell is used to obtain the average group constant of a pair of control rod inserted in graphite block. The neutron flux distribution in the cell is calculated by 2D transport codes such as TWOTRAN-II [39]. In this code, the neutron flux distribution in the strong neutron absorber of the control rod is calculated first. Using this flux distribution, the code

calculates the average group constants of a pair of control rods inserted in the graphite block. This code also calculates the average group constants of B4C pellets inserted in the control rod guide block for reserve shutdown system.

For the control rod cell calculation, the 2D X-Y model is used as shown in Fig. 4.30. The model is for half of a control rod guide block in which a control rod is inserted and the surrounding fuel blocks. The average group constants of the control rod guide block where a control rod is inserted is

r uel rod

Fig. 4.28 One dimensional cylindrical fuel cell model for calculating group constants of homogenized fuel

Center hall

Group constants of averaged fuel cell obtained by fuel cell calculation

Burnable poison rod

Unit [cm!

Burnable poison cell model

Fig. 4.29 One dimensional cylinder burnable poison cell model for calculating group constants of burnable poison

calculated by the flux-weighting method. Due to eccentricity of the control rod position from the block center, a one-half block model is used and the geometry of the neutron absorber is modeled by many micro rectangles.

(3) Core calculation

Diffusion calculation codes such as CITATION-1000VP are used for the core calculation [40, 41]. The effective multiplication factor, power distri­bution, reactivity coefficients, shutdown margin, etc are calculated using the core geometry and the average group constants obtained by the fuel cell calculation, burnup cell calculation, and control rod cell calculation. The power distribution obtained here is used for the fuel temperature caluculation.

Burnable poison

Fig. 4.30 Two dimensional control rod cell calculation model for calculating group constants of control rod

Average group constants of control rod guide column

Averaged group constants of fuel rod < fuel block)

C ontrol rod

Permanent reflector

Replaceable reflector

Fuel region

Black absorber ———-

aOl. bcm

Fig. 4.31 Horizontal cross section of three dimensional triangular mesh model for core calculation

The horizontal cross section of the 3D triangular mesh model for the core calculation is shown in Fig. 4.31. Each fuel block is horizontally divided into 6 triangular meshes and axially divided into 4 meshes. The number density of the materials such as uranium and graphite, etc. are homogenized in each block.

1.5.1 Definition of Power and Temperature Coefficients

An important cause of reactivity variation in an operating reactor is change in the temperature of the system; that is in the fuel, moderator, coolant, and structure. The “power coefficient,” defined as Eq. (1.50), describes the variation in reactivity due to a change in reactor power. It must be essentially negative since it is an important factor for the inherent stability of the reactor.

(1.50)

Figure 1.11 shows the variation in reactor power with time after a positive reactivity insertion by control rod withdrawal. If the power coefficient is negative,

the instantaneous rise of reactivity to supercritical decreases as the power increases, and then reaches criticality. If the power coefficient is positive, the reactor power will infinitely increase. If the absolute value of the negative power coefficient is too large, it is not easy to elevate the power level and a lot of control rod worth must be required as well; this will be a hard-to-operate and uneconomical reactor. The “temperature coefficient” corresponding to the temper­ature of each core component i (i = fuel, moderator, coolant, and structure), aTi, is defined as

(1.51)

where Ti is again the temperature associated with each core component. The power coefficient can be expressed with the temperature coefficients as

_ dp _ dp dTi ____ ^ dTi

ар~~дР~^~дТ~дР~ГаТ‘~дР

dT’

where is the variation in the temperature of the і core component due to a power change and it is evaluated by thermal-hydraulic analyses (the thermal — hydraulic analyses are not discussed here).

In a reactor designed to have a negative power coefficient, the reactivity of the system decreases as the power level increases. From a different viewpoint, since the reactivity increases with the decrease in power or temperature of the system, the reactivity essentially suppressed by control rods becomes a maximum at the cold shutdown for the fuel loading and discharge.

The reactivity decrease of the system due to an increase in power or temperature can be presented by using the power coefficient or temperature coefficient. Here a reactor is considered from the cold shutdown condition to the hot shutdown condition. The reactor power is zero and the temperature of the system uniformly increases. The reactivity decrease of the system can be written by integrating the temperature coefficient from the cold shutdown temperature to the hot shutdown temperature as

which is called the “temperature defect”. Since the entire reactor core is essentially characterized by an identical temperature, the temperature coefficient can be called the “isothermal temperature coefficient”.

The decrease in reactivity from the hot shutdown (zero power) condition to the normal power condition can be obtained by integrating the power coefficient in the same way, which is called the “power defect”. The power defect can be represented using Eq. (1.52) as

The temperature defect and power defect are part of the reactivity that the control rods should compensate.

As already mentioned, the diffusion equation is a balanced equation between neutron production and loss. The multi-group diffusion theory is similarly considered except that the neutron energy is discretized into multi-groups. As shown in Fig. 2.20, subsequent treatment should be made for neutrons moving
to other groups through slowing down which is not considered in the one-group diffusion theory. In particular, there are two points:

(i) Neutrons which move out of a target group to other groups by slowing down result in neutron loss in the neutron balance of the target group.

(ii) Neutrons which move in a target group from other groups by slowing down result in neutron gain in the neutron balance of the target group.

Hence, the multi-group diffusion equation introduces the cross section £g! g/ (scattering matrix) characterizing neutrons which transfer from group g to group g0 by elastic or inelastic scattering.

In the example of the three-group problem presented in Fig. 2.20, the neutron balance equation of each group can be given by

Group 1: DiV2^ — (Еа1ф1 + Е1^2ф1 + ^зФі) + ^ = 0 (2.57)

Group 2: D2V202 _ (Га. гФг + ^2^зФі) + (S2 + ^і^зФі) = 0 (2.58)

Group 3: Z)3V203 — (Та,3ф3) + (S3 + Z^3фг + Г2^3ф2) = 0 (2.59)

It is seen that neutron loss due to slowing down in a group results in neutron gains in other groups. Ъ1!2ф1 in Eq. (2.58) and Ъ1!3ф1 + £2!3ф2 in Eq. (2.59) are referred to as the “slowing-down neutron source” since they represent the neutron source due to slowing down from other groups.

Equation (2.57) for group 1 can be rewritten as the following.

— Ег, іФі + S± = 0 (2.60)

^V, l = Ea, l + ^1^2 + ^3 (2.61)

The sum of absorption cross section and group-transfer cross section, namely, the cross section which characterizes neutrons removed from the target group, is called the removal cross section. It is observed that Eq. (2.60) rewritten using the removal cross section is in the same form as the one-group diffusion equation and can be solved by the finite difference method mentioned before. Substituting the obtained ф1 for the one in the slowing-down source of Eq. (2.58) makes it possible to calculate the neutron flux of group 2, ф2, in the same way. Further, the neutron flux of group 3, ф3, can be also obtained by substituting ф1 and ф2 into Eq. (2.59).

Thus, the multi-group diffusion equations can be solved in sequence from the highest energy group. However, the solution is not obtained by this method if the thermal energy region (up to about 4 eV) is divided into multi-groups. As an example, a four-group diffusion problem using two fast groups and two thermal groups is considered.

Group 1: Э1Ч2ф1 — (ГаДфі + Т1^20і + %і^зФі + ^i-»40i) + S± = 0 (2.62)

Group 2: D2V202 _ С^а,202 + ^2^302 + ^2^402) + 0>2 + ^l-»20l) = 0 (2.63)

Group 3: D3V203 — (Ха3ф3 + Х3^40з) + (53 + Xi^30i + ^зФз + Е^зФл)

= О

(2.64)

Group 4: D4V204 — (2*^404) + (S4 + Ti^40i + ^2->402 + Т3^40з) = 0 (2.65)

The fast-group neutron fluxes ф1 and ф2 can be derived by the same procedure as in the previous three-group problem. In the thermal groups, however, an event that a neutron gains kinetic energy in a collision with a moderator nuclide in thermal vibration, that is, the upscattering of thermal neutrons should be considered. The last term of Eq. (2.64) represents that. Thus, since the neutron source terms for group 3 include the unknown neutron flux of group 4, ф4, it is impossible to directly solve the equation. Here, a guess

is taken at Eq. (2.64) and then a solution ф3^ is given.

4 3

2

Eq. (2.65) to acquire ф. Such calculations are iteratively performed until

ф3п) and ф4п) come to an agreement with each previous ф3”_1) and ф4”_1). This calculation is referred to as the “thermal iteration” calculation.

The codes developed for core design of only fast reactors may have no such upscattering and iteration calculation function or may cut back the slowing- down groups by assuming collisions with only heavy metal elements such as sodium. Since such codes cannot correctly manage transport calculation of thermal neutrons, they should not be used carelessly in a thermal reactor calculation even though the codes solve the same fundamental equation.