Technical Field

The present invention relates to an equivalent fissile method used in enrichment adjustment that is performed so that reactivity may become equivalent to the base design when isotopic compositions of uranium and plutonium are different from a design serving as a standard in manufacture of a uranium-plutonium mixed oxide fuel assembly, and more specifically, to a method for setting equivalent fissile coefficients and a method for manufacturing a fuel assembly in which adjustment of the plutonium enrichment is performed using the coefficients.

Background Art

In boiling water reactors (BWRs) and pressurized water reactors (PWRs), there is proceeding a so-called plutonium-thermal project that aims at resource saving of uranium and utilization of plutonium by loading a uranium-plutonium mixed oxide fuel (hereafter termed as an MOX fuel) that uses plutonium obtained by reprocessing of spent fuel as a nuclear fuel material in a reactor. In loading the MOX fuel, there is a method for loading it in a reactor core being intermingled with a uranium fuel as part of a reload fuel.

Incidentally, an isotopic composition of plutonium (hereafter described as Pu) contained in the MOX fuel varies mainly depending on a type, a degree of enrichment, and a burn-up of the spent fuel to be reprocessed. For example, a weight ratio of Pu239 in Pu that is obtained by reprocessing spent fuel of 9 x 9 arrangement fuel (B type) of the BWR with a burn-up of 45 GWd/t is about 51%; a weight ratio of Pu239 in Pu obtained by burning that fuel up to 50 GWd/t is about 48%.

Moreover, also in a period from a point of time of being reprocessed until being loaded in a reactor core, Pu241 suffers generation of Am241 and reduction of Pu241 by β decay of a half-life of about 14 years . Therefore, it needs to be premised that MOX powders of various compositions should be supplied in manufacture of the MOX fuel.

When the isotopic composition of Pu contained in the MOX fuel differ, even if it has the same Pu enrichment (hereafter described simply as an enrichment) as that of the design serving as a base (hereafter described simply as base design), abundance of each plutonium isotope contained in the fuel naturally differs from that of the base design.

Especially, if abundances of Pu239 and Pu241 that are fissionable nuclides are larger than those of the base design, the reactivity will become higher than that of the base design. On the other hand, if abundances of Pu240 and Am241 that have large poison effects on neutrons are larger than those of the base design, the reactivity will become lower than that of the base design. Therefore, according to the isotopic composition contained in the MOX fuel, it is necessary to adjust the enrichment so that the MOX fuel may become equivalent at least in reactivity to the base design.

Here, there is a literature (Non-patent Literature 1) that discusses the Pu enrichment of the MOX fuel for PWRs. In this Non-patent Literature 1, in the end of cycle of a PWR reactor core, it is considered so that a prescribed cycle burn-up may be attained even when the MOX fuel is loaded as part of a reload fuel by making uranium fuel loaded in the reactor core and the MOX fuel equivalent in reactivity to each other. Specifically, the consideration is to adjust the enrichment according to a Pu isotopic composition in the MOX fuel so that an infinite multiplication factor k_∞ of the uranium fuel and an infinite multiplication factor k_∞ of the MOX fuel may become an equal value in the reactor core average burn-up at the end of cycle (hereafter described as EOC). In this enrichment adjustment, an equivalent fissile method is used in the Non-patent Literature 1.

On the other hand, there is a literature (refer to Patent Literature 1) showing a method whereby enrichment distribution of inner layer and intermediate layer regions and the outer layer region of the fuel assembly is changed until it becomes a target LPF and subsequently the enrichment distributions of the respective regions that satisfy the reactivity equivalence is determined in order to freely control local power peaking while maintaining reactivity equivalence.

Moreover, as a method for calculating the equivalent fissile coefficients that is different from that of Non-patent Literature 1, Patent Literature 2 discloses a method for obtaining equivalent fissile coefficients, for example, by obtaining in advance an average enrichment of MOX fuel rods that makes reactivities equivalent for two cases where nuclide compositions are different, respectively, and solving simultaneous linear equations with two equivalent fissile coefficients that should be obtained set as unknowns.

Furthermore, as a method for calculating the equivalent fissile coefficients, Patent Literature 3 discloses a method for obtaining reactivity coefficients by obtaining correlation between a nuclide composition and a necessary enrichment at several points by direct calculation and performing fitting of them.

Incidentally, the equivalent fissile method, when the isotopic compositions of uranium and Pu raw materials are given, is to obtain the Pu enrichment that gives a proper reactivity by a mathematical formula. That is, regarding the MOX fuel in which various kinds of isotopes are intermingled, when a reactivity worth of each isotope of the raw material is converted into that of Pu239 and the equivalent Pu239 enrichment when the MOX fuel is converted into a virtual MOX fuel consisting of only Pu239 and U238 is denoted by E₂₃₉, if the isotopic compositions of uranium and Pu of a certain MOX fuel and the Pu enrichment ε (Put enrichment: {(total Pu + Am241) / (total Pu + Am241 + total U)} x 100 (wt%)) are given, the E₂₃₉ can be obtained through calculation from E₂₃₉ = ε∑_iα_iη_i + (100-ε) ∑_iβ_iη_i ... basic equation (1) of Non-patent Literature 1.

Where α_i is the Pu isotopic composition (weight ratio to total Pu + Am241), β_i is a U isotopic composition (weight ratio to total U), and η_i is Pu239 equivalent coefficients of each isotope (coefficients for converting the reactivity worth of Pu and U isotopic nuclides to that of Pu239, i.e., the equivalent fissile coefficients).

Therefore, in the case where the Pu enrichment is determined so that the MOX fuel may be able to have the proper reactivity to uranium and Pu of the given raw materials, the Pu enrichment that becomes the proper reactivity for the MOX fuel having a representative isotopic composition is obtained, the E₂₃₉ of the MOX fuel is calculated, this is fixed as a constant, and then the Pu enrichment ε is determined so that the E₂₃₉ may become a decided value in the isotopic composition of the given rawmaterials. Below, an equation (1) equivalent to a basic equation (2) of Non-patent Literature 1 that is used in this case is shown with a different notation.

[Mathematical 1] $$\epsilon =\frac{{\displaystyle \sum }_m{c}_m\cdot w_m-E_{239}}{{\displaystyle \sum }_m{c}_m\cdot w_m-{\displaystyle \sum }_n{c}_n\cdot w_n}$$

Where

ε

: Enrichment at which k_∞ in EOC reactor core average burn-up becomes a value equal to k_∞ of standard fuel

E₂₃₉

: Enrichment at which k_∞ at EOC reactor core average burn-up becomes a value equal to k_∞ of standard fuel in virtual isotopic composition consisting of only Pu239 and U238,

w

: Weight ratio that each isotope occupies in uranium or plutonium. Incidentally ${\displaystyle \sum _m}{w}_m=1,{\displaystyle \sum _n{w}_n}=1$

c

: Equivalent fissile coefficients of each isotope in uranium or plutonium.

Incidentally, they are defined as c₂₈ = 0 and c₄₉ = 1.

m

: Subscript indicating each isotope of uranium. Incidentally, they are:

• 24: U234
• 25: U235
• 26: U236
• 28: U238.

n

: Subscript indicating each isotope of plutonium (Am241 is included conveniently).

Incidentally, they are:

• 48: Pu238
• 49: Pu239
• 40: Pu240
• 41: Pu241
• 42: Pu242
• 51: Am241.

By the above-mentioned equation (1), the Pu enrichment ε can be determined so that the reactivity of the MOX fuel may become an appropriate value. As a method for obtaining equivalent fissile coefficients C necessary in doing this, page 7 of Non-patent Literature 1 gives "determining by analyzing and arranging fluctuation of the infinite multiplication factor to a change of a value of beginning life of each isotope enrichment". Although there is no description of a specific procedure for obtaining it in Non-patent Literature 1 in any way, it is considered that obtaining it is done by procedures as shown generally by a flowchart of Fig. 27. That is, it is the following procedures.

Procedure 1: Set a design (base design) serving as a standard that has been checked to satisfy a life when being loaded in a reactor core and a thermal limit value by a design calculation. If there is no applicable design, an enrichment of each fuel rod is determined so that the life when being loaded in the reactor core and the thermal limit value maybe satisfied.

Procedure 2 : Evaluate an amount of change dk₄₉ of the infinite multiplication factor k_∞ per unit change weight when changing a minute amount of an abundance weight of Pu239 with a lattice code that was used when evaluating nuclear properties of the base design. Incidentally, when changing an abundance of Pu239, an abundance of U238 is adjusted so that a gross weight of uranium and Pu in the MOX fuel at the beginning of life may be preserved. For example, when evaluating a case where the abundance of Pu239 is increased, the abundance of U238 is decreased by the same weight.

Procedure 3: Evaluate an amount of change dk_m (or dk_n) of the infinite multiplication factor k_∞ per unit change weight when changing a minute amount of an abundance weight of an objective nuclide m (or n) of the equivalent fissile coefficients that is intended to be obtained with the lattice code. Incidentally, when changing the abundance of the objective nuclide m (or n), the abundance of U238 is adjusted so that the gross weight of uranium and Pu in the MOX fuel at the beginning of life may be preserved. That is, when evaluating the case where the abundance of the objective nuclide m (or n) is increased, the abundance of U238 is reduced by the same weight.

Procedure 4: Calculate the equivalent fissile coefficients C_m (or C_n) by an equation of C_m = dk_m / dk₄₉, (or C_n = dk_n / dk₄₉).

Procedure 3 and Procedure 4 are performed as many times as the number of the objective nuclides of the equivalent fissile coefficients intended to be obtained.

Incidentally, regarding the equivalent fissile coefficients of Non-patent Literature 1, as described in page 8 of Non-patent Literature 1, it is said, "Since the isotopic compositions of plutonium used in actual MOX fuel manufacturing are extremely various, it is necessary to determine equivalent coefficients so that it maybe applicable to isotopic composition fluctuation of a wide range. However, if the set of equivalent coefficients is intended to be applied to all the isotopic compositions, lowering of the accuracy is inevitable because the basic equation 1 of the equivalent fissile method is a linear equation. In order to avoid this, it is necessary that the plutonium isotopic compositions be classified to several groups and the set of equivalent coefficients be set for each group". That is, since the equivalent fissile coefficients are not constants and may change by various factors, and Pu actually used is of the isotopic compositions of a very wide range, high accuracy cannot be obtained if the set of equivalent fissile coefficients determined in a certain specific isotopic composition is used as it is for the enrichment determination of various isotopic compositions of a wide range.

Therefore, in Non-patent Literature 1, groups 1 to 5 are set mainly for numerical value ranges of Puf fraction ((Pu239 + Pu241) / (total Pu + Am241) x 100 (wt%)), respectively, and a procedure shown in Fig. 27 is conducted for each group, as shown in a flowchart of Fig. 28. Incidentally, Group 1 stands for a Puf fraction of not less than 55% and less than 60%, Group 2 stands for a Puf fraction of not less than 60% and less than 65%, Group 3 stands for a Puf fraction of not less than 65% and less than 70%, Group 4 stands for a Puf fraction of not less than 70% and less than 75%, and Group 5 stands for a Puf fraction of not less than 75%. Thus, in the technique of Non-patent Literature 1, in order to support various isotopic compositions and determine the Pu enrichment without deteriorating the accuracy, the determination is performed by preparing the base design and the equivalent fissile coefficients individually for each group.

Although the above Non-patent Literature 1 describes a method for determining an assembly average enrichment, it does not mention a method for determining the enrichment of each fuel rod at all. As is described at page 2 of Non-patent Literature 1, this is because "since this document aims at presenting a technique of adjusting reactivity of anMOX fuel assembly, it does not discuss difference of isotopic composition and plutonium enrichment of each fuel rod but deals with only isotopic composition and plutonium enrichment of an assembly average."

However, although such handling enables the reactivity to acquire a desired value, it has a problem that there is a possibility that the LPF (local power peaking factor that is an index of the power distribution flattening in the fuel assembly) fluctuates largely from the desired value because consideration about Pu enrichment distribution in the fuel assembly is not made.

In Patent Literature 1, such a problem is solved by the following method. That is, first, the inner layer and intermediate layer regions and the outer layer region of an assembly are paid attention to, respectively, and the equivalent fissile coefficients corresponding to these regions are obtained in advance, respectively. Next, in order to adjust the LPF to the desired value, when changing the enrichment of the inner layer and intermediate layer regions, the enrichment of the outer layer region is determined so that the reactivity equivalence may be maintained. When the enrichment of this outer layer region is determined, the respective equivalent fissile coefficients of the inner layer and intermediate layer regions and the outer layer region that have been previously obtained in advance are used. By doing this, it became possible to determine the enrichment having the reactivity equivalence while satisfying the target LPF.

Prior Art Documents List

Patent Literature

• Patent Literature 1 : Japanese Patent No. 3008129
• Patent Literature 2 : Japanese Unexamined Patent Publication No. 2009-14511
• Patent Literature 3 : Japanese Unexamined Patent Publication No. 2010-261866

Non-patent Literature

Non-patent Literature 1 : "On Plutonium Enrichment of MOX Fuel for PWR, " MHI-NES-1001, published by Mitsubishi Heavy Industries, Ltd., February, Heisei 8 (1996)

Summary of Invention

Problem to be solved by the Invention

However, in an MOX fuel for BWRs that is targeted at a wide range of a Pu isotopic composition, it does not happen that a combination of technologies of Non-patent Literature 1 and Patent Literature 1 that are conventional technologies as described above makes it possible to be accurately determine enrichment distribution that has equivalences of reactivity and LPF to the base design.

First, equivalent fissile coefficients C in an equation (1) are obtained by C_m = dk_m / dk₄₉ (or C_n = dk_n / dk₄₉), where dk_m (or dk_n) is an amount of change of an infinite multiplication factor k_∞ per unit change weight when an abundance weight of an objective nuclide m (or n) of the equivalent coefficients intended to be obtained is changed by an infinitesimal quantity (the amount of change of k_∞ itself is evaluated with a lattice code).

On the other hand, as to "how much" the abundance weight of the objective nuclide m (or n) is made to change, there is arbitrariness, which has quite a few influence to accuracy of the equivalent fissile coefficients to be obtained. This is because when the amount of change of the abundance weight is a too infinitesimal quantity, a significant sensitivity as the amount of change of k_∞ cannot be obtained, although it is desirable that the amount of change of the abundance weight be set as small as possible in the viewpoint of a fact that linearity of the equivalent fissile coefficients holds. It is considered that this comes from a fact that a numerical calculation procedure of a neutron transport equation that is solved in evaluating k_∞ with the lattice code depends on a so-called iterative calculation technique. That is, this is because in the process of the iterative calculation of the neutron transport equation, the iterative calculation is aborted by the limited number of times based on the convergence determining condition over a neutron flux distribution defined in advance, and therefore, a calculation result of k_∞ always includes an error accompanying the discontinuation of fixed iterative calculation. On the contrary, if the abundance weight of the objective nuclide is changed somewhat largely so that a significant sensitivity can be obtained as the amount of change of k_∞, nonlinearity will appear in the sensitivity to k_∞. Therefore, in reality, it is necessary to determine a variation width of the abundance weight so that the whole error may become a minimum while permitting nonlinearity of sensitivity to k_∞, namely, permitting accuracy deterioration of an equivalent fissile method. However, no quantitative technique has been disclosed as the method of advanced technical judgment like this heretofore including Non-patent Literature 1.

Moreover, with the method for "setting the set of equivalent coefficients for each group" in Non-patent Literature 1, it tends to be thought that when higher accurate equivalent coefficients are required, it can be supported by setting grouping more finely. However, in practice, it is considered that no matter how finely the grouping is performed, there is a limit in obtaining high accurate equivalent fissile coefficients. This is because, although Non-patent Literature 1 uses a method whereby in addition to the grouping of each numeral value range of the Puf fraction, isotopic composition fractions of Pu239 and Pu242 are considered as coordinates in a two-dimensional orthogonal coordinate system and further region dividing is performed, plutonium isotopes other than Pu239 and Pu242 that are considered to cause significant influences on the linearity of the equivalent fissile coefficients are not considered as indices of the grouping. Especially, regarding Pu240, a poison effect on neutrons per unit number of atoms is not so large, but its abundance is relatively large; therefore, it is considered that the poison net effect on neutrons is relatively large, and has a significant influence on the linearity of the equivalent fissile coefficients.

Therefore, if the number of nuclides used for the index of the grouping is changed from two kinds of Pu239 and Pu242 to three kinds or more including Pu240, it is considered that achieving high accurate equivalent coefficients becomes possible. However, it is necessary to deal with the orthogonal coordinate system of three dimensions or more as a space dealt with as the grouping, the number of times of grouping becomes a huge number in an exponential manner, and it is practically impossible to deal with it.

Furthermore, in manufacturing of the MOX fuel, although the Pu composition varies within a certain range of tolerance in a certain manufacturing unit, and the following problems is considerable also in dealing with a case where a range of dispersion strides over a boundary of the grouping. When the dispersion of the Pu composition strides over a boundary of the grouping in a certain manufacturing unit, in order to avoid complexity on an actual operation, a method of consistently using the equivalent fissile coefficients of the grouping in which the average composition in the manufacturing unit is included is considered rational. Therefore, even when the division width of the grouping is narrower than a tolerance of Pu composition, since "the equivalent fissile coefficients of the grouping in which the average composition is included is used consistently," improvement of the accuracy of the equivalent fissile coefficients by thinning down the division width of the grouping cannot be expected. That is, the division width of the grouping comes to a limit of an order of the tolerance of Pu composition.

As described above, since increasing the accuracy of the equivalent fissile coefficients using the technology shown in Non-patent Literature 1 causes the grouping to become huge and restricts a substantial limit of accuracy to an order of the tolerance of Pu composition in the manufacturing, it can be said that it is not suitable for practical use.

Next, in the above-mentioned Patent Literature 1, as a "method for determining the enrichment distribution having a target LPF while maintaining reactivity equivalence to the base design," in the example of the fuel assembly shown in Fig. 2 quoted in the embodiment, the same raw materials are used for fuel materials of an inner layer fuel rod and an intermediate layer fuel rod, and is comprised of one kind of the Pu enrichment in the inner layer and intermediate layer regions and one kind of Pu enrichment in the outer layer region, total two kinds of Pu enrichments . Therefore, in the embodiment of this Patent Literature 1, what is necessary is just to deal with the equivalent fissile coefficients for two regions, at most. Moreover, since only two variables of the Pu enrichment of the inner layer and intermediate layer regions and the Pu enrichment of the outer layer region are dealt with, a relationship of the both enrichments that satisfy the target LPF is only a two-dimensional simple curve as shown in Fig. 3 of Patent Literature 1. Therefore, it is possible to obtain only one intersection of a curve "E" of Fig. 3 that satisfies a "target equivalent fissile quantity" where the reactivity becomes equivalent to that of the standard fuel and a curve that satisfies the target LPF.

On the other hand, a design that uses about four kinds of Pu enrichments as Pu enrichment types to be used is generally practiced for the MOX fuel for BWRs. A design example of an MOX fuel assembly for BWRs shown in Fig. 1 is a fuel rod lattice arrangement of nine lines by nine rows of fuel rods and has a large-diameter water rod W of a square tube shape occupying a region of nine fuel rods (3 × 3) in a central part, in which the MOX fuel rods P1 to P4 that exclude the top end regions and 14 uranium fuel rods U containing no MOX fuel are arranged in prescribed positions, respectively. One kind of Pu enrichment is used for each of MOX fuel rods P1 to P4, respectively. Therefore, in the example shown in this Fig. 1, four kinds of Pu enrichment types are used in the fuel assembly.

When applying the technology shown in Patent Literature 1 to the design example as shown in this Fig. 1, a region where P1 is arranged can be regarded as "inner layer and intermediate layer regions" that is called in the embodiment of Patent Literature 1, and a region where P2 to P4 are arranged can be regarded as an "outer layer region" that is called in the embodiment of Patent Literature 1. In this case, since the number of the regions to be dealt with is two regions, at most, it is considered that the technology shown in Patent Literature 1 is applicable. However, in reality, since only the average enrichment of the regions concerned is dealt with for regions where P2 to P4 are arranged, there is arbitrariness about how to determine the enrichment distribution of P2 to P4.

Especially, in the case of the fuel assembly for BWRs, there is often the case where a fuel rod where the LPF occurs is usually a fuel rod in an outer periphery region of the assembly, and P2 to P4 are arranged in this region. Therefore, the LPF is affected by enrichment distribution of P1 to P4, and the LPF cannot be necessarily set to a desired value only by distribution of the Pu enrichment of P1 of "the inner layer and intermediate layer regions" and the average Pu enrichment of P2 to P4 of the "outer layer region." In order to solve this problem, it is necessary to find out a method for determining uniquely the enrichment distribution of P2 to P4, but Patent Literature 1 does not mention such a method at all.

As a candidate of solutions against this problem that are easily conceived, a method for giving the enrichment distribution within P2 to P4 of the "outer layer region" by multiplying the enrichment distribution of P2 to P4 of the base design by a constant is conceivable. However, with such a method, when the Pu composition differs largely from the standard composition, the power peaking of any one of the fuel rods among P2 to P4 becomes excessive, and a possibility can be easily conceived that there does not exist a combination that realizes a target LPF only by adjusting the Pu enrichment of P1 and the distribution of the average enrichment of P2 to P4 . Therefore, in essence, like the technology shown in Non-patent Literature 1, a designer needs to prepare the enrichment distribution that satisfies the target LPF in advance for grouping into multiple groups. This leads to an idea that most of merits of the technology shown in Patent Literature 1 have been lost because of a point that a technology of Patent Literature 1 of adjusting the LPF to the desired value exerts only an effect of fine adjustment or so on the LPF that shows a minute change in a narrow composition range by nature.

Furthermore, as another method of application of the technology shown in Patent Literature 1 to a design example as shown in Fig. 1, a method for obtaining equivalent fissile coefficients of respective regions of P1 to P4 is conceivable. That is, it is a method whereby the number of regions to be dealt with is set to four regions. In this case, although constraint conditions are two, namely, "attaining the target LPF" and "making standard fuel and reactivity equivalent," in contrast to this, variables to be dealt with are Pu enrichments of four respective regions, i.e., four variables. Therefore, since the variables to be obtained is four for two constraint conditions, the degree of freedom is larger by two variables, and the enrichments of four regions cannot be determined uniquely. Patent Literature 1 does not mention a method for determining the enrichment at all in such a case where the degree of freedom is large.

As an easily conceivable candidate of means of solving this problem, a method of increasing the constraint conditions in number can be considered, for example, a method whereby the target LPFs at the burn-ups B1, B2, and B3 are set as the target LPF and the enrichments of P1 to P4 are determined so as to satisfy them. However, this method comes with a question as to "whether a solution that satisfies all the constraint conditions is sure to be found." This question can be put in another way as a problem as to "whether the LPFs of the burn-ups B1, B2, and B3 can be controlled freely by the enrichment distribution of P1 to P4." Generally, the LPF that comes though burn-up of the MOX fuel for BWRs is affected by the enrichment distribution of P1 to P4 and as well as an arrangement and the number of the fuel rods to which gadolinium is added as a burnable poison. Especially, in a period when gadolinium exists in a gadolinium added fuel rod, the LPF is strongly affected by consumption of gadolinium accompanying burning of the fuel. This consumption of gadolinium is seldom affected by the enrichment distribution of P1 to P4, but is affected by the Pu composition contained in the MOX fuel rod. Therefore, in a period when gadolinium exists in the gadolinium added fuel rod, it is impossible to control the LPF by the enrichment distribution of P1 to P4. Moreover, a transition accompanying burn-up of the LPF in a period after entire gadolinium in the gadolinium added fuel rod was consumed is determined mainly by the Pu composition contained in the MOX fuel rod. Therefore, it is considered physically difficult to "freely control the LPF of the burn-ups B1, B2, and B3 by the enrichment distribution of P1 to P4." That is, it is a high probability that the enrichment distribution of P1 to P4 that satisfies the target LPF in the burn-ups B1, B2, and B3 does not exist physically.

As described above, it is considered impossible to apply the technology shown in Patent Literature 1 to an MOX fuel with three kinds or more of enrichment types.

Moreover, in a method for obtaining equivalent fissile coefficients disclosed in Patent Literatures 2, the equivalent fissile coefficients are determined by simultaneous linear equations using a set of values of average enrichments of the MOX fuel rods whose number is the same at most as the number of the equivalent fissile coefficients that shouldbe obtained. This is not a sufficient number to make error components resulting from numerical computation such as the iterative calculation as described above cancel each other. Therefore, also in this Patent Literature 2, similarly with the conventional technologies including Non-patent Literature 1, it is necessary to decide the variation width of the abundance weight such that the whole error may become a minimum, namely a case of the nuclide composition while allowing the nonlinearity of the sensitivity to k_∞, that is, while allowing deterioration of the accuracy of the equivalent fissile method. However, a quantitative technique for that has not been clarified heretofore.

Moreover, Patent Literature 3 does not describe a concrete explanation about a method for obtaining required enrichment in advance at all, and there is arbitrariness when determining the enrichment distribution of multiple kinds of the MOX fuel rods in the assembly. Therefore, if a designer determines the enrichment distribution artificially to each nuclide composition, as is generally carried out, the average enrichment of the MOX fuel rods that will become equivalent in reactivity will be affected by the enrichment distribution as described above, and this influence will appear, as it is, as errors of the equivalent fissile coefficients; therefore, the equivalent fissile coefficients cannot be obtained with a high accuracy. Therefore, also by the method of this Patent Literature 3, it is impossible to minimize an error contained in the equivalent fissile method as a whole.

An object of the present invention is to provide a method capable for setting the equivalent fissile coefficients with a higher accuracy than before by minimizing errors contained in the equivalent fissile coefficients as a whole for a more efficient design and manufacture of the MOX fuel assembly. Moreover, the present invention aims at providing a method for setting equivalent fissile coefficients that make reactivity equivalence to the base design more excellent also in a wide Pu composition range. Furthermore, the present invention aims at providing a method capable for determining the enrichment distribution of each fuel rod that realizes a local power peaking factor equivalent to the base design also in the MOX fuel with three kinds or more of enrichment types.

Means for solving the Problem

In order to achieve the above-mentioned object, a method for setting equivalent fissile coefficients according to a first aspect of the present invention is a method for setting equivalent fissile coefficients for each of the predetermined nuclides that is used for determining the plutonium enrichment corresponding to a plutonium isotopic composition to be applied to the nuclear fuel material by an equivalent fissile method in targeting, as a manufacture object, an MOX fuel assembly containing uranium and plutonium as nuclear fuel materials. The method comprising the steps of: selecting a plurality of plutonium isotopic compositions that cover a range of plutonium isotopic compositions to be applied thereto up to a number larger than the whole number of the equivalent fissile coefficients; determining an enrichment distribution of each MOX fuel rod that becomes equivalent in infinite multiplication factor to a design serving as a predetermined base for each the selected plutonium isotopic composition; and obtaining equivalent fissile coefficients by calculating an equation obtained by applying a least squares method to an equivalent fissile method defined by the equation (1) using a plutonium enrichment ε based on the determined enrichment distribution and data based on a weight ratio that each isotope occupies in uranium or plutonium in each plutonium isotopic composition.

[Mathematical 2] $$\epsilon =\frac{{\displaystyle \sum }_m{c}_m\cdot w_m-E_{239}}{{\displaystyle \sum }_m{c}_m\cdot w_m-{\displaystyle \sum }_n{c}_n\cdot w_n}$$

Where

ε

: Enrichment at which k_∞ in EOC reactor core average burn-up becomes a value equal to k_∞ of standard fuel

E₂₃₉

: Enrichment at which k_∞ at EOC reactor core average burn-up becomes a value equal to k_∞ of standard fuel in virtual isotopic composition consisting of only Pu239 and U238,

w

: Weight ratio that each isotope occupies in uranium or plutonium. Incidentally ${\displaystyle \sum _m}{w}_m=1,{\displaystyle \sum _n{w}_n}=1$

c

: Equivalent fissile coefficients of each isotope in uranium or plutonium.

Incidentally, they are defined as c₂₈ = 0 and c₄₉ = 1.

m

: Subscript indicating each isotope of uranium. Incidentally, they are:

• 24: U234
• 25: U235
• 26: U236
• 28: U238.

n

: Subscript indicating each isotope of plutonium (Am241 is included conveniently).

Incidentally, they are:

• 48: Pu238
• 49: Pu239
• 40: Pu240
• 41: Pu241
• 42: Pu242
• 51: Am241.

A method for setting equivalent fissile coefficients according to a second aspect of the present invention is the method for setting equivalent fissile coefficients described in the first aspect thereof, in which the step of determining the enrichment distribution determines it by an adjustment step of iterating a step of multiplying the enrichment of each MOX fuel rod given as an initial enrichment distribution by a constant uniformly until a difference between the infinite multiplication factor for the enrichment distribution set in the step and the infinite multiplication factor of the base design becomes smaller than a predetermined convergence determining value.

A method for setting equivalent fissile coefficients according to a third aspect of the present invention is the method for setting equivalent fissile coefficients according to either the first or the second aspect thereof, in which the step of obtaining equivalent fissile coefficients obtains C through calculation by making the plutonium enrichment ε of the selected plutonium isotopic composition and the data based on the weight ratio that each isotope occupies correspond with an equation (11) as a fitting function y and the variables x of the fitting function.

[Mathematical 3] $$\mathbf{C}={\mathbf{A}}^{-\mathbf{1}}\mathbf{B}$$

Where

A

: Square matrix having a_i,j as elements,

B

: Vector having b_i as elements,

C

: Vector having nontrivial equivalent fissile coefficients and E₂₃₉ as elements,

$a_{\mathit{ij}}={\displaystyle \sum _l}\left[f_i\left(x_{1,l},x_{2,l},\dots \right)\cdot f_j\left(x_{1,l},x_{2,l},\dots \right)\right]$ $b_i={\displaystyle \sum _l}\left[y_l\cdot f_i\left(x_{1,l},x_{2,l},\dots \right)\right]$

l

: Subscript indicating a set of data to be fitted.

A method for setting equivalent fissile coefficients according to a fourth aspect of the present invention is the method for setting equivalent fissile coefficients according to the third aspect thereof, in which C is obtained through calculation by setting a function shown in an equation (12) as the fitting function y and the variables x of the fitting function that shouldbe set and making this equation (12) correspond with the equation (11).

[Mathematical 4] $$\begin{array}{ccccccc}y& :=& -\epsilon \cdot w_{49}\hfill & & & & \\ x_1& :=& w_{24}\cdot \left(1-\epsilon \right)\hfill & & & & \\ x_2& :=& w_{25}\cdot \left(1-\epsilon \right)\hfill & & & & \\ x_3& :=& w_{26}\cdot \left(1-\epsilon \right)\hfill & & & & \\ x_4& :=& w_{48}\cdot \epsilon \hfill & & & & \\ x_5& :=& w_{40}\cdot \epsilon \hfill & & & & \\ x_6& :=& w_{41}\cdot \epsilon \hfill & & & & \\ x_7& :=& w_{42}\cdot \epsilon \hfill & & & & \\ x_8& :=& w_{51}\cdot \epsilon \hfill & & & & \\ & & & & & & \\ & & & & & & \\ c_1& :=& c_{24}\hfill & ;& f_1\left(x_1,x_2,\dots x_8\right)& :=& x_1\\ c_2& :=& c_{25}\hfill & ;& f_2\left(x_1,x_2,\dots x_8\right)& :=& x_2\\ c_3& :=& c_{26}\hfill & ;& f_3\left(x_1,x_2,\dots x_8\right)& :=& x_3\\ c_4& :=& c_{48}\hfill & ;& f_4\left(x_1,x_2,\dots x_8\right)& :=& x_4\\ c_5& :=& c_{40}\hfill & ;& f_5\left(x_1,x_2,\dots x_8\right)& :=& x_5\\ c_6& :=& c_{41}\hfill & ;& f_6\left(x_1,x_2,\dots x_8\right)& :=& x_6\\ c_7& :=& c_{42}\hfill & ;& f_7\left(x_1,x_2,\dots x_8\right)& :=& x_7\\ c_8& :=& c_{51}\hfill & ;& f_8\left(x_1,x_2,\dots x_8\right)& :=& x_8\\ c_9& :=& -E_{239}\hfill & ;& f_9\left(x_1,x_2,\dots x_8\right)& :=& 1\end{array}$$

A method for setting equivalent fissile coefficients according to a fifth aspect of the present invention is the method for setting equivalent fissile coefficients according to either the second aspect or the third aspect thereof and is characterized in that the equivalent fissile coefficients that shouldbe set are made to have a dependence on a composition fraction of isotopes of one kind or more that are selected in a descending order of a macroscopic cross section in plutonium, the largest the first.

A method for setting equivalent fissile coefficients according to a sixth aspect of the present invention is the method for setting equivalent fissile coefficients according to the fifth aspect thereof and is characterized in that C is obtained through calculation by setting an equation (14a) by designating a function shown in an equation (13a) as the fitting function y and the variables x of the fitting function that should be set and making this equation (14a) correspond with the equation (11).

[Mathematical 5] $$\begin{array}{llll}{c}_{m\left(k\right)}& =0,& & k=1\\ c_{m\left(k\right)}& =c_{m\left(k\right),0}& +{\displaystyle \sum _{\mathit{kʹ}=1}^N}{c}_{m\left(k\right),n\left(\mathit{kʹ}\right)}\cdot w_{n\left(\mathit{kʹ}\right)},& k>1\\ c_{n\left(k\right)}& =1,& & k=1\\ c_{n\left(k\right)}& =c_{n\left(k\right),0}& +{\displaystyle \sum _{\mathit{kʹ}=1}^N}{c}_{n\left(k\right),n\left(\mathit{kʹ}\right)}\cdot w_{n\left(\mathit{kʹ}\right)},& k>1\end{array}$$

Where

m(k)

: Sequence (example: {28, 24, 25, 26}), arrangement of an integer symbol 28 of first term (subscript k = 1) indicating U238 that is trivial as equivalent fissile coefficient plus subsequent terms (subscript k = 2, ...) of integer terms each indicating uranium isotope nuclide that is nontrivial as equivalent fissile coefficient and should be taken into consideration,

n(k)

: Sequence (example: {49, 40, 41, 42, 51, 48}), arrangement of an integer symbol 49 of first term (subscript k = 1) indicating Pu239 that is trivial as equivalent fissile coefficient plus subsequent terms (subscript k = 2, ...) of integer terms each indicating Pu isotope nuclide that is nontrivial as equivalent fissile coefficient and should be taken into consideration (Am241 is also included conveniently) (however, for subscript k = 2 and subsequent k's, in decreasing order of macroscopic cross section),

N

: Number of Pu isotopes that should be taken into consideration wher providing dependence on composition fraction of Pu isotope to equivalent fissile coefficients,

N'

: Lesser of N and k-1 (N' : = min{N, k-1}).

[Mathematical 6] $$\begin{array}{llllllll}y& :=-\epsilon \cdot w_{49}& & & & & & \\ x_{m\left(k\right),0}& :=\left(1-\epsilon \right)\cdot w_{m\left(k\right)}& ,& x_{m\left(k\right),n\left(\mathit{kʹ}\right)}& :=& x_{m\left(k\right),0}\cdot w_{n\left(\mathit{kʹ}\right)}& ;& k>1,\mathit{kʹ}=1,\dots ,\mathit{N}\\ x_{n\left(k\right),0}& :=\epsilon \cdot w_{n\left(k\right)}& ,& x_{n\left(k\right),n\left(\mathit{kʹ}\right)}& :=& x_{n\left(k\right),0}\cdot w_{n\left(\mathit{kʹ}\right)}& ;& k>1,\mathit{kʹ}=1,\dots ,\mathit{Nʹ}\\ f_{m\left(k\right),0}& :=x_{m\left(k\right),0}& ,& f_{m\left(k\right),n\left(\mathit{kʹ}\right)}& :=& x_{m\left(k\right),n\left(\mathit{kʹ}\right)}& ;& k>1,\mathit{kʹ}=1,\dots ,\mathit{N}\\ f_{n\left(k\right),0}& :=x_{n\left(k\right),0}& ,& f_{n\left(k\right),n\left(\mathit{kʹ}\right)}& :=& x_{n\left(k\right),n\left(\mathit{kʹ}\right)}& ;& k>1,\mathit{kʹ}=1,\dots ,\mathit{Nʹ}\\ f_{49,0}& :=1& & & & & & \end{array}$$

A method for setting equivalent fissile coefficients according to a seventh aspect of the present invention is the method for setting equivalent fissile coefficients according to any one of the second to sixth aspects thereof that in targeting an MOX fuel assembly containing three kinds or more of plutonium enrichment types as the MOX fuel assembly to be manufacture object, has: a step of setting an enrichment distribution that reproduces a distribution of the physical quantity predetermined for each plutonium enrichment type regarding a single physical quantity of a target of design optimization for each of the selectedplutonium isotopic compositions based on an equation (30) using an equation (29), an equation (31), and the equation (32); an adjustment step of calculating the single physical quantity of the target of design optimization for the enrichment distribution set in the step, iterating the step until its difference from the predetermined physical quantity distribution becomes smaller than the predetermined convergence determining value, and designating the obtained enrichment distribution as the initial enrichment distribution; and an adjustment step of, when a difference between an infinite multiplication factor for the initial enrichment distribution and an infinite multiplication factor of the base design is larger than the predetermined convergence determining value, iterating a step of multiplying the initial enrichment distribution by a constant uniformly until it becomes smaller than it; and the equivalent fissile coefficients are obtained by applying the least squares method for each enrichment type based on the enrichment distribution adjusted in this adjustment step, respectively.

[Mathematical 7] $$a_{i,j}=\mathrm{\Delta }{\mathit{LDP}}_{i,j}/\mathrm{\Delta }{\epsilon }_j$$

Where

LDP_i

: Physical quantity serving as design objective of enrichment type i,

ΔLDP_i,j

: Amount of fluctuation of LDP_i at the time of fluctuation of Pu enrichment of enrichment type j

Δε_j

: Amount of fluctuation of Pu enrichment of enrichment type j,

i

: Subscript indicating enrichment type i,

j

: The subscript indicating enrichment type j.

[Mathematical 8] $${\epsilon }_i^{\left(n\right)}={\epsilon }_i^{\left(n-1\right)}+\mathrm{\Delta }{\epsilon }_i^{\left(n\right)}$$

Where

${\epsilon }_i^{\left(n-1\right)}:$  : Enrichment of enrichment type i before enrichment adjustment,

${\epsilon }_i^{\left(n\right)}$  : Enrichment of enrichment type i after enrichment adjustment.

[Mathematical 9] $$\mathbf{C}={\mathbf{A}}^{-\mathbf{1}}\mathbf{B}$$

Where

A

: Square matrix having a_ij of equation (29) as elements,

B

: Vector having ALDP_i^(n-1) as elements,

C

: Vector having Δε_i⁽ⁿ⁾ as elements.

[Mathematical 10] $$\mathrm{\Delta }{\mathit{LDP}}_i^{\left(n-1\right)}={\mathit{LDP}}_i^T-{\mathit{LDP}}_i^{\left(n-1\right)}$$

Where

${\mathit{LDP}}_i^T:$  Predetermined desired value of physical quantity serving as design objective of enrichment type i.

A method for setting equivalent fissile coefficients according to an eighth aspect of the present invention has: a first processing step of designating the equivalent fissile coefficients obtained in advance by a method described in the seventh aspect of the present invention and an E₂₃₉ enrichment as zeroth equivalent coefficients, tentatively determining a plutonium enrichment of each MOX fuel rod that becomes equivalent to the E₂₃₉ enrichment from the equation (1) using the zeroth equivalent coefficients for each of the selected plutonium isotopic compositions, and designating it as the initial enrichment distribution; a second processing step of calculating an infinite multiplication factor for the tentatively determined plutonium enrichment distribution, and performing comparison determination of its difference from the infinite multiplication factor of the base design and the predetermined convergence determining value; a third processing step of, when a difference between the infinite multiplication factors of the both is larger than the convergence determining value by the determination, adjusting the plutonium enrichment of each MOX fuel rod by multiplying the tentatively determined plutonium enrichment distribution by a constant uniformly and obtaining a new tentatively determined plutonium enrichment distribution; a step of iterating a step of returning to the second processing step for the new tentatively determined plutonium enrichment distribution obtained in the third processing step and recalculating the infinite multiplication factor and performing the determination until the difference between the infinite multiplication factors of the both becomes smaller than the convergence determining value; and a fourth processing step of determining the equivalent fissile coefficients by a method described in the seventh aspect of the present invention based on the plutonium enrichment of each MOX fuel rod at a point of time when the difference between the infinite multiplication factors becomes smaller than the convergence determining value and each plutonium composition, and designating it as the first equivalent coefficients; the setting method iterates a series of work from the first processing step to the fourth processing step, each series of work uses the equivalent fissile coefficients determined in the fourth processing step of the previous round in the first processing step of the next round, and the work of an N-th round that is the last starts the first processing step using (N-1) -th equivalent coefficients and determines N-th equivalent coefficients in the fourth processing step.

A method for setting equivalent fissile coefficients according to a ninth aspect of the present invention is the method for setting equivalent fissile coefficients according to either the seventh aspect or the eighth aspect of the present invention and is characterized in that the single physical quantity of the target of design optimization is a maximum of the fuel rod linear heat generation or a maximum of a fuel rod burn-up of each plutonium enrichment type.

A method for setting equivalent fissile coefficients according to a tenth aspect of the present invention comprises: a first processing step of in targeting an MOX fuel assembly containing three kinds or more of plutoniumenrichment types as the MOX fuel assembly to be manufacture obj ect, designating the equivalent fissile coefficients obtained in advance by a method described by any one of the second to sixth aspects of the present invention as the zeroth equivalent coefficients, and determining an E₂₃₉ enrichment of each MOX fuel rod of the base design through calculation from the equation (1) based on the zeroth equivalent coefficients and the plutonium enrichment of each MOX fuel rod of the base design; a second processing step of tentatively determining the plutonium enrichment of each MOX fuel rod that becomes equal to the E₂₃₉ enrichment of each MOX fuel rod of the base design from the equation (1) using the zeroth equivalent coefficients for each of the selected plutonium isotopic compositions and designating it as the initial enrichment distribution; the thirdprocessing step of calculating an infinite multiplication factor for the tentatively determined plutonium enrichment distribution and performing comparison determination of its difference from the infinite multiplication factor of the base design and the predetermined convergence determining value; and the fourth processing step of when the difference between the infinite multiplication factors of the both is larger than the convergence determining value, adjusting the plutonium enrichment of each MOX fuel rod by multiplying the tentatively determined plutonium enrichment distribution by a constant uniformly and obtaining a new tentatively determined plutonium enrichment distribution; a step of iterating a step of returning to the third processing step for a new tentatively determined plutonium enrichment distribution obtained in the fourth processing step and performing the determination by calculating the infinite multiplication factor again until the difference between the infinite multiplication factors of the both becomes smaller than the convergence determining value; and a fifth processing step of determining the equivalent fissile coefficients by a method described in any one of the second to sixth aspects of the present invention based on the plutonium enrichments of the respective MOX fuel rods at a point of time when the difference between the infinite multiplication factors becomes smaller than the convergence determining value and the respective plutonium isotopic compositions, and designating them as the first equivalent coefficients; and the method iterates a series of work from the first processing step to the fifth processing step, each series of work uses the equivalent fissile coefficients determined in the fifth processing step of a previous round in the first processing step of a next round, and the work of the N-th round that is the last starts the first processing step using (N-1)-th equivalent coefficients and determines N-th equivalent coefficients in the fifth processing step.

Advantageous Effects of Invention

The present invention can provide a method capable for setting equivalent fissile coefficients with the higher accuracy than the prior art by minimizing errors included in the equivalent fissile coefficients as a whole for more efficient design and manufacture of the MOX fuel assembly. Moreover, the present invention can also provide a method for setting equivalent fissile coefficients that make the reactivity equivalence to the base design more excellent also in the wide Pu composition range. Furthermore, the present invention can provide a method for determining the enrichment distribution for each fuel rod that makes a local power peaking factor equivalent to the base design also in an MOX fuel assembly with three kinds or more of enrichment types.

Brief Description of Drawings

• Fig. 1 is an outline design diagram showing an example of an MOX fuel assembly for BWRs that is targeted at in an embodiment of the present invention.
• Fig. 2 is a comparison diagram of an average enrichment of the MOX fuel rod calculated using equivalent fissile coefficients determined in a first embodiment of the present invention and an average enrichment of the MOX fuel rods calculated with a lattice code.
• Fig. 3 is a comparison diagram of an average enrichment of the MOX fuel rods calculated using equivalent fissile coefficients determined in a second embodiment of the present invention and an average enrichment of the MOX fuel rods calculated with the lattice code.
• Fig. 4 is a diagrammatic drawing showing a statistic of a relative difference between a Pu enrichment obtained using the equivalent fissile coefficients for N = 0 to 6 and a Pu enrichment by the lattice code in the second embodiment of the present invention.
• Fig. 5 is a flowchart diagram showing a processing procedure for obtaining zeroth equivalent fissile coefficients in a third embodiment of the present invention.
• Fig. 6 is a flowchart diagram showing a processing procedure for obtaining N-th equivalent fissile coefficients in the third embodiment of the present invention.
• Fig. 7 is a comparison diagram of an average enrichment of the MOX fuel rods calculated using the equivalent fissile coefficients determined in the third embodiment of the present invention and an average enrichment of the MOX fuel rods calculated with the lattice code.
• Fig. 8 is a distribution chart showing a relative difference between an LPF of the assembly and an LPF of a base design for an enrichment distribution calculated using the equivalent fissile coefficients determined in the third embodiment of the present invention.
• Fig. 9 is a flowchart diagram showing a processing procedure for obtaining zeroth equivalent fissile coefficients in a fourth embodiment of the present invention.
• Fig. 10 is a flowchart diagram showing a processing procedure for obtaining N-th equivalent fissile coefficients in the fourth embodiment of the present invention.
• Fig. 11 is a distribution chart showing a difference between an infinite multiplication factor in an EOC reactor core average burn-up calculated with the lattice code and an infinite multiplication factor of the base design for a Pu enrichment distribution determined by the equivalent fissile coefficients obtained in the fourth embodiment of the present invention.
• Fig. 12 is a distribution chart showing a relative difference between the LPF in an early stage of burn-up and the LPF of the base design for the Pu enrichment distribution determined by the equivalent fissile coefficients obtained in the fourth embodiment of the present invention.
• Fig. 13 is a first source program among FORTRAN 95 programs used for calculation to which a least squares method for obtaining equivalent fissile coefficients is applied in the first embodiment of the present invention.
• Fig. 14 is a first half part of a second source program among the FORTRAN 95 programs used for calculation to which the least squares method for obtaining equivalent fissile coefficients is applied in the first embodiment of the present invention.
• Fig. 15 is a second half part of the second source program.
• Fig. 16 is a first half part of a third source program among the FORTRAN 95 programs used for calculation to which the least squares method for obtaining equivalent fissile coefficients is applied in the first embodiment of the present invention.
• Fig. 17 is a second half part of the third source program.
• Fig. 18 is a fourth source program among the FORTRAN 95 programs used for calculation to which the least squares method for obtaining equivalent fissile coefficients is applied in the first embodiment of the present invention.
• Fig. 19 is a fifth source program among the FORTRAN 95 programs used for calculation to which the least squares method for obtaining equivalent fissile coefficients is applied in the first embodiment of the present invention.
• Fig. 20 is a first half part of a sixth source program among the FORTRAN 95 programs used for calculation to which the least squares method for obtaining equivalent fissile coefficients is applied in the first embodiment of the present invention.
• Fig. 21 is a second half part of the sixth source program.
• Fig. 22 is a seventh source program among the FORTRAN 95 programs used for calculation to which the least squares method for obtaining equivalent fissile coefficients is applied in the first embodiment of the present invention.
• Fig. 23 is an eighth source program among the FORTRAN 95 programs used for calculation to which the least squares method for obtaining equivalent fissile coefficients is applied in the first embodiment of the present invention.
• Fig. 24 is a first half part of a source program that replaces the second source program of the first embodiment among the FORTRAN 95 programs used for calculation to which the least squares method for obtaining equivalent fissile coefficients is applied in the second embodiment of the present invention.
• Fig. 25 is a second half part of the source program shown in Fig. 24.
• Fig. 26 is a source program that replaces the fifth source program of the first embodiment among the FORTRAN 95 programs used for calculation to which the least squares method for obtaining equivalent fissile coefficients is applied in the second embodiment of the present invention.
• Fig. 27 is a flowchart diagram showing a processing procedure for obtaining equivalent fissile coefficients assumed in Non-patent Literature 1 of the prior art.
• Fig. 28 is a flowchart diagram showing a processing procedure for obtaining equivalent fissile coefficients for each Pu isotopic composition group assumed in Non-patent Literature 1 of the prior art.

Description of Embodiments

The present invention is a method for setting equivalent fissile coefficients for each predetermined nuclide used for determining a Pu enrichment corresponding to a Pu isotopic composition that is applied to an MOX fuel assembly to be manufactured as a nuclear fuel material by an equivalent fissile method. First, multiple Pu isotopic compositions that cover the range of the Pu isotopic compositions to be applied thereto are selected up to a number larger than the whole number of the equivalent fissile coefficients, the enrichment distribution of each MOX fuel rod is determined by multiplying the enrichment of each MOX fuel rod of a base design by a constant uniformly until it becomes equivalent in reactivity to a design serving as a predetermined base for each of the selected Pu isotopic composition, the average Pu enrichment ε of the MOX fuel rods is obtained through calculation, and then the equivalent fissile coefficients are obtained by calculating an equation obtained by applying a least squares method to the equivalent fissile method using this average Pu enrichment ε of the MOX fuel rods and data based on a weight ratio that each isotope occupies in uranium or Pu in each Pu isotopic composition.

Since it is considered to be possible to find an original amount of change of k_∞ with error components arising from numerical calculation such as a iterative calculation cancelled mutually by obtaining the equivalent fissile coefficients of a certain nuclide based on results of a large number of calculation cases, not by using only one calculation result as a calculation case of k_∞ that obtains the equivalent fissile coefficients of a certain nuclide, this method makes it possible to obtain significant equivalent fissile coefficients with numerical calculation errors of a lattice code excluded by obtaining equivalent fissile coefficients using results of lattice codes that are larger in number than the equivalent fissile coefficients to be obtained.

The present invention minimizes errors included in the equivalent fissile coefficients as a whole systematically by applying the least squares method "that is frequently used to obtain a solution for a state of excess determination" ("New Edition Handbook of Numerical Computation," ISBN4-274-07584-2, published by Ohmsha, Ltd., September 1990) in addition to the above method.

Application of the least squares method to the equivalent fissile method can be performed as follows. First, in order to apply the least squares method to an equation (1), the equation (1) is transformed into a following equation (2).

[Mathematical 11] $$-\epsilon \cdot w_{49}\cdot w_{49}=\left(1-\epsilon \right)\cdot {\displaystyle \sum _m}\left(c_m\cdot v_m\right)+\epsilon {\displaystyle \sum _{n,n\ne 49}}\left(c_n\cdot w_n\right)-E_{239}$$

Since C₂₈ = 0 and C₄₉ = 1 hold, the above-mentioned equation (2) becomes a following equation (3).

[Mathematical 12] $$-\epsilon \cdot w_{49}={\displaystyle \sum _{m,m\ne 28}}\left[c_m\cdot v_m\cdot \left(1-\epsilon \right)\right]+{\displaystyle \sum _{n,n\ne 49}}\left[c_n\cdot w_n\cdot \epsilon \right]-E_{239}$$

In the present invention, when applying the least squares method to the equation (3), an average Pu enrichment ε of MOX fuel rods that becomes equivalent in reactivity to the base design for a certain Pu composition is obtained in advance with a lattice code. Here, as a method for determining the enrichment of each MOX fuel rod, by multiplying the enrichment of each MOX fuel rod of the base design by a constant uniformly, an average enrichment of the MOX fuel rods is adjusted, and the enrichment distribution of the MOX fuel rods that becomes equivalent in reactivity to the base design is determined.

This is because generally in the MOX fuel assembly comprised of various kinds of enrichments, even when the Pu composition and the average enrichment of the MOX fuel rods are the same, k_∞ differs slightly between designs that are different in enrichment distribution, the average enrichment of the MOX fuel rods that becomes equivalent in reactivity is affected by the enrichment distribution, and this influence appears as it is as errors of the equivalent fissile coefficients. In the present invention, in order to avoid this, the enrichment distribution of each MOX fuel rod is determined by a method that is clearly generalized with "the Pu enrichment of each MOX fuel rod of the base design is multiplied by a constant uniformly" and therefore has less arbitrariness. That is, it is important that continuous correlation holds between the Pu composition and the average enrichment of the MOX fuel rods that become equivalent in reactivity to the Pu composition.

The step of "multiplying the Pu enrichment of each MOX fuel rod of the base design by a constant uniformly" is as shown in the following a) to e).

1. a) First, enrichment distributions of the base design (enrichments ε₁ to ε₄ of respective fuel rod kinds), the average enrichment ε of the MOX fuel rods from them (standard average enrichment ε), a U/Pu composition (standard composition) used in the design concerned, and an infinite multiplication factor (standard infinite multiplication factor) k_∞, at an EOC equivalent reactor core average burn-up evaluated with the lattice code for the design concerned are given as input values.
2. b) A U/Pu composition (a target composition) different from the standard composition is given.
3. c) The infinite multiplication factor k_∞⁽⁰⁾ at the EOC equivalent reactor core average burn-up is evaluated with the lattice code by using the standard enrichment distributions (ε₁ to ε₄ and ε) as the zeroth enrichment distributions (ε₁⁽⁰⁾ to ε₄⁽⁰⁾, and ε⁽⁰⁾) and by combining them with the target composition. k_∞⁽⁰⁾ and k_∞ are compared, and if their difference is within a convergence determining value (e.g., 0.00001), the calculation will be ended it is decided that the enrichment distribution that should be obtained has been obtained. If the difference is not within the convergence determining value, the flow will shift to the next step d).
4. d) The infinite multiplication factor k_∞⁽¹⁾ is evaluated with the lattice code (U/Pu composition is the "target composition") by using an enrichment distribution calculated by a following equation D1 with a minute enrichment Δε set appropriately and ε⁽⁰⁾ as a first enrichment distribution. $${\mathrm{\epsilon }}^{\left(\mathrm{1}\right)}={\mathrm{\epsilon }}^{\left(\mathrm{0}\right)}-\mathrm{\Delta \epsilon },{{\mathrm{\epsilon }}_{\mathrm{i}}}^{\left(\mathrm{1}\right)}=\left({\mathrm{\epsilon }}^{\left(\mathrm{1}\right)}/{\mathrm{\epsilon }}^{\left(\mathrm{0}\right)}\right){\mathrm{\epsilon }}^{\left(\mathrm{0}\right)},\mathrm{i}=\mathrm{1},\mathrm{4}$$
   
   

   k_∞⁽¹⁾ and k_∞ are compared, and if their difference is a difference within the convergence determining value, the calculation will be ended it is decided that the enrichment distribution that should be obtained has been obtained. Otherwise, after calculating a parameter of a following equation D2, the flow will shift to the next step e). $${\mathrm{a}}^{\left(1\right)}=\left({{\mathrm{k}}_{\mathrm{\infty }}}^{\left(1\right)}-{{\mathrm{k}}_{\mathrm{\infty }}}^{\left(\mathrm{0}\right)}\right)/\left({\mathrm{\epsilon }}^{\left(1\right)}-{\mathrm{\epsilon }}^{\left(\mathrm{0}\right)}\right)$$

5. e) The infinite multiplication factor k_∞⁽ⁿ⁾ is evaluated with the lattice code (U/Pu composition is the "target composition") by using what was calculated by a following equation E1 as an n-th enrichment distribution. Incidentally, the evaluation value of a previous step are assumed to be an (n-1)-th evaluation value. $${\mathrm{\epsilon }}^{\left(\mathrm{n}\right)}={\mathrm{\epsilon }}^{\left(\mathrm{n}-\mathrm{1}\right)}-\left({{\mathrm{k}}_{\mathrm{\infty }}}^{\left(\mathrm{n}-\mathrm{1}\right)}-{\mathrm{k}}_{\mathrm{\infty }}\right)/{\mathrm{a}}^{\left(\mathrm{n}-\mathrm{1}\right)},{{\mathrm{\epsilon }}_{\mathrm{i}}}^{\left(\mathrm{n}\right)}=\left({\mathrm{\epsilon }}^{\left(\mathrm{n}\right)}/{\mathrm{\epsilon }}^{\left(\mathrm{n}-\mathrm{1}\right)}\right){{\mathrm{\epsilon }}_{\mathrm{i}}}^{\left(\mathrm{n}-\mathrm{1}\right)},\mathrm{i}=\mathrm{1},4$$
   
   

   k_∞⁽ⁿ⁾ and k_∞ are compared, and if their difference is a difference within the convergence determining value, the calculation will be ended it is decided that the enrichment distribution that should be obtained has been obtained. Otherwise, after calculating a parameter of a following equation E2, this step e) will be repeated again. $${\mathrm{a}}^{\left(\mathrm{n}\right)}=\left({{\mathrm{k}}_{\mathrm{\infty }}}^{\left(\mathrm{n}\right)}-{{\mathrm{k}}_{\mathrm{\infty }}}^{\left(\mathrm{n}-\mathrm{1}\right)}\right)/\left({\mathrm{\epsilon }}^{\left(\mathrm{n}\right)}-{\mathrm{\epsilon }}^{\left(\mathrm{n}-\mathrm{1}\right)}\right)$$

Moreover, as the "reactivity equivalence, " in a burn-up equivalent to an EOC reactor core average burn-up, the average Pu enrichment of the MOX fuel rods is adjusted until the infinite multiplication factor k_∞ of the base design and k_∞ of the MOX fuel assembly of a certain Pu composition come to agreement within a difference of, for example, 0.00001. Here, a reason why the " assembly average Pu enrichment" as in Non-patent Literature 1 is not used but the "average Pu enrichment of MOX fuel rods" is used is because in the MOX fuel assembly for BWRs, there exist the uranium fuel rods containing no Pu in the fuel assembly, as also shown in the example of Fig. 1. The MOX fuel for PWRs that is targeted in Non-patent Literature 1 contains no uranium fuel rod, and the assembly average Pu enrichment is equivalent to the average Pu enrichment of the MOX fuel rods.

Next, as the Pu composition that is an evaluation case, multiple Pu compositions are selected comprehensively for a specific fuel type, based on a burn-up history and a reprocessing scenario that are in conformity with reality. This is useful in a point that errors contained in the equivalent fissile coefficients can be minimized for actual MOX powder supplied from a reprocessing plant of a used nuclear fuel and in a point that a range of the errors can be statistically estimated by a statistic of the evaluated equivalent fissile coefficients. When using calculation results of multiple cases in each of which only a weight ratio of one nuclide is changed as is estimated from a description of Non-patent Literature 1, a concern that "an effect that the errors when obtaining equivalent fissile coefficients cancel each other is not necessarily expected" can be dispelled by a fact that an error accompanying the discontinuation of the iterative calculation appears in the calculation result with a tendency.

As described above, the equivalent fissile coefficients of the equation (1) are obtained by the least squares method using the average Pu enrichment ε of the MOX fuel rods that has been obtained in advance with multiple Pu compositions and the lattice code. Below, how to apply the least squares method to the equation (3) will be shown in detail. First, let a fitting formula treated in the least squares method be expressed like a following equation (4).

[Mathematical 13] $$y\left(x_1,x_2,\dots \right)={\displaystyle \sum _i}{c}_i\cdot f_i\left(x_1,x_2,\dots \right)$$

Where

x

: Variable of fitting function,

y

: Fitting function,

f

: Expansion function of fitting function,

c

: Expansion coefficient of fitting function,

i

: Subscript indicating expansion function of fitting function.

A following equation (5) is considered as data that should be fitted.

[Mathematical 14] $$\begin{array}{cccc}{y}_1& x_{1,1}& x_{2,1}& \cdots \\ y_2& x_{1,2}& x_{2,2}& \cdots \\ ⋮& ⋮& ⋮& \\ y_l& x_{1,l}& x_{2,l}& \cdots \\ ⋮& ⋮& ⋮& \end{array}$$

Where

l

: Subscript indicating a set of data to be fitted.

In the least squares method, expansion coefficients of a fitting function are determined so that a sum total of squares of residuals R between a fitting formula and fitting data may become a minimum. The residual R is expressed by a following equation (6).

[Mathematical 15] $$\begin{array}{cc}{R}_l& =y_l-y\left(x_{1,l},x_{2,l},\dots \right)\hfill \\ & =y_l-{\displaystyle \sum _i}{c}_i\cdot f_i\left(x_{1,l},x_{2,l},\dots \right)\hfill \end{array}$$

The sum total of the squares of the residuals R is expressed by a following equation (7).

[Mathematical 16] $${\displaystyle \sum _l}{R_l}^2={\displaystyle \sum _l}\left[y_l-{\displaystyle \sum _l}{c}_i\cdot f_i\left(x_{1,l},x_{2,l},\dots \right)\right]$$

Regarding expansion coefficients C_i that minimize the sum total of the squares of the residuals R, what is necessary is just to obtain variables C_i that minimize "the sum total of the squares of the residuals R" when assuming "the sum total of the squares of the residuals R" to be a function of the expansion coefficients C_i. That is, what is necessary is to obtain variables C_i each of which zeros out a differentiation of "the sum total of the squares of the residuals R" in terms of to the variable C_i. Mathematically, this can be represented by a following equation (8).

[Mathematical 17] $$\begin{array}{l}0=\frac{\partial \left\{{\displaystyle \sum _l}{R_l}^2\right\}}{\partial c_i}\\ =\frac{\partial \left\{{\displaystyle \sum _l}{\left[y_l-y\left(x_{1,l},x_{2,l},\dots \right)\right]}^2\right\}}{\partial c_i}\\ =\frac{\partial \left\{{\displaystyle \sum _l}{\left[y_l-{\displaystyle \sum _j}{c}_j\cdot f_j\left(x_{1,l},x_{2,l},\dots \right)\right]}^2\right\}}{\partial c_i}\\ ={\displaystyle \sum _l}\frac{\partial \left\{{\left[y_l-{\displaystyle \sum _j}{c}_j\cdot f_j\left(x_{1,l},x_{2,l},\dots \right)\right]}^2\right\}}{\partial c_i}\\ ={\displaystyle \sum _l}\left\{2\cdot \left[y_l-{\displaystyle \sum _j}{c}_j\cdot f_j\left(x_{1,l},x_{2,l},\dots \right)\right]\cdot \left[-f_i\left(x_{1,l},x_{2,l},\dots \right)\right]\right\}\\ =-2\cdot {\displaystyle \sum _l}\left[y_l\cdot f_i\left(x_{1,l},x_{2,l},\dots \right)-{\displaystyle \sum _j}{c}_j\cdot f_j\left(x_{1,l},x_{2,l},\dots \right)\cdot f_i\left(x_{1,l},x_{2,l},\dots \right)\right]\\ =-2\cdot \left\{{\displaystyle \sum _l}\left[y_l\cdot f_i\left(x_{1,l},x_{2,l},\dots \right)\right]-{\displaystyle \sum _l}{\displaystyle \sum _j}{c}_j\cdot f_j\left(x_{1,l},x_{2,l},\dots \right)\cdot f_i\left(x_{1,l},x_{2,l},\dots \right)\right\}\\ =-2\cdot \left\{{\displaystyle \sum _l}\left[y_l\cdot f_i\left(x_{1,l},x_{2,l},\dots \right)\right]-{\displaystyle \sum _j}\left[c_j{\displaystyle \sum _l}\cdot \left(f_j\left(x_{1,l},x_{2,l},\dots \right)\cdot f_i\left(x_{1,l},x_{2,l},\dots \right)\right)\right]\right\}\\ =-2\cdot \left\{b_i-{\displaystyle \sum _j}\left[a_{\mathit{ij}}\cdot c_j\right]\right\}\end{array}$$

Where

• $a_{\mathit{ij}}={\displaystyle \sum _l}\left[f_i\left(x_{1,l},x_{2,l},\dots \right)\cdot f_j\left(x_{1,l},x_{2,l},\dots \right)\right]$
• $b_i={\displaystyle \sum _l^l}\left[y_l\cdot f_i\left(x_{1,l},x_{2,l},\dots \right)\right]$

Therefore, what is necessary is just to obtain C_i that satisfies a relationship of a following equation (9) for each of the expansion coefficients C_i.

[Mathematical 18] $$\begin{array}{ccc}{\displaystyle \sum _j}\left[a_{1,j}\cdot c_j\right]& =& b_1\\ {\displaystyle \sum _j}\left[a_{2,j}\cdot c_j\right]& =& b_2\\ ⋮& ⋮& ⋮\\ {\displaystyle \sum _j}\left[a_{\mathit{ij}}\cdot c_j\right]& =& b_i\\ ⋮& ⋮& ⋮& \end{array}$$

The above-mentioned equation (9) is expressed like a following equation (10) using a matrix.

[Mathematical 19] $$\mathbf{AC}=\mathbf{B}$$

Where

A

: Square matrix having a_ij as elements,

B

: Vector having b_i as elements,

C

: Vector having expansion coefficients c_i of fitting function as elements.

Therefore, the expansion coefficients C_i are calculated by a following equation (11).

[Mathematical 20] $$\mathbf{C}={\mathbf{A}}^{-\mathbf{1}}\mathbf{B}$$

In order to apply the above least squares method to the equation (3), the expansion coefficients are made to correspond with the equation (4) of the least squares method as shown in the following equations (12), respectively.

[Mathematical 21] $$\begin{array}{ccccccc}y& :=& -\epsilon \cdot w_{49}\hfill & & & & \\ x_1& :=& w_{24}\cdot \left(1-\epsilon \right)\hfill & & & & \\ x_2& :=& w_{25}\cdot \left(1-\epsilon \right)\hfill & & & & \\ x_3& :=& w_{26}\cdot \left(1-\epsilon \right)\hfill & & & & \\ x_4& :=& w_{48}\cdot \epsilon \hfill & & & & \\ x_5& :=& w_{40}\cdot \epsilon \hfill & & & & \\ x_6& :=& w_{41}\cdot \epsilon \hfill & & & & \\ x_7& :=& w_{42}\cdot \epsilon \hfill & & & & \\ x_8& :=& w_{51}\cdot \epsilon \hfill & & & & \\ & & & & & & \\ & & & & & & \\ c_1& :=& c_{24}\hfill & ;& f_1\left(x_1,x_2,\dots x_8\right)& :=& x_1\\ c_2& :=& c_{25}\hfill & ;& f_2\left(x_1,x_2,\dots x_8\right)& :=& x_2\\ c_3& :=& c_{26}\hfill & ;& f_3\left(x_1,x_2,\dots x_8\right)& :=& x_3\\ c_4& :=& c_{48}\hfill & ;& f_4\left(x_1,x_2,\dots x_8\right)& :=& x_4\\ c_5& :=& c_{40}\hfill & ;& f_5\left(x_1,x_2,\dots x_8\right)& :=& x_5\\ c_6& :=& c_{41}\hfill & ;& f_6\left(x_1,x_2,\dots x_8\right)& :=& x_6\\ c_7& :=& c_{42}\hfill & ;& f_7\left(x_1,x_2,\dots x_8\right)& :=& x_7\\ c_8& :=& c_{51}\hfill & ;& f_8\left(x_1,x_2,\dots x_8\right)& :=& x_8\\ c_9& :=& -E_{239}\hfill & ;& f_9\left(x_1,x_2,\dots x_8\right)& :=& 1\end{array}$$

Therefore, the enrichment distribution of each MOX fuel rod for the each Pu composition is determined by the following steps: various kinds of Pu compositions are prepared considering a BWR fuel as a target; and the enrichment of the each MOX fuel rod of the base design is multiplied by a constant uniformly with the above-mentioned lattice code until the enrichment becomes equivalent in reactivity to the base design. Together with this, the average Pu enrichment ε of the MOX fuel rods has been obtained in advance. Subsequently, the equivalent fissile coefficients (C₂₄, C₂₅, C₂₆, C₄₈, C₄₀, C₄₁, C₄₂, and C₅₁) are obtained by the following steps: y and x₁ to x₈ of an equation (12) are set from that enrichment ε and a weight ratio w of each nuclide based on each Pu composition; and the equation (11) is calculated by bringing these pieces of data into correspondence.

According to the above method, as shown in a below-mentioned first embodiment, it is possible to determine high accurate equivalent fissile coefficients that cannot be achieved by the conventional technologies including Non-patent Literature 1.

Moreover, when setting the equivalent fissile coefficients by applying the least squares method as described above, by making each of the equivalent fissile coefficients have a dependence on the composition fraction of the isotopes of one kind or more having a largest macroscopic cross section in Pu, it is possible to determine only one the set of the equivalent fissile coefficients that have an extremely high accuracy to an extremely wide Pu composition region as follows.

The conventional technologies including Non-patent Literature 1 could not but take a symptomatic action such that region dividing of the Pu composition is made finer and finer in increasing the accuracy because of linearity of the basic equation (1) of the equivalent fissile method as described above.

First of all, it is considered that a reason why the linearity of the basic equation (1) does not hold for a wide composition range of some extent is that a neutron spectrum in the fuel assembly changes with a change of the Pu composition, which affects a reactivity worth of each nuclide, i.e., the equivalent fissile coefficients. This will be explained as follows. First, the reactivity worth of the nuclide depends on an absorption reaction rate and a nuclear fission reaction rate of the nuclide concerned, and these reaction rates depend on multiplication of an absorption cross section and a fission cross section by a neutron flux. On the other hand, since these absorption cross section and fission cross section are different depending on each nuclide, and in addition, have dependencies on energy of neutrons and the neutron spectrum is defined as a distribution to neutron energy of the neutron flux, the reactivity worth of each nuclide will change due to a change of the neutron spectrum.

The neutron spectrum itself depends on absorption and scattering cross sections of the fuel material in the fuel rod and of a structural material and a coolant in its periphery. The change of the Pu composition causes a change of the abundance of Pu isotope in the MOX fuel, i.e., a net absorption cross section of Pu isotope. The net absorption cross section of Pu isotope is obtained by multiplying the absorption cross section per atom of each isotope (termed a microscopic cross section) by the abundance in the fuel rod (atomic number density), and this physical quantity is termed the macroscopic cross section. Although the microscopic cross section differs according to each Pu isotope, when Pu239 is set to be a reference, those of Pu238 and Pu242 are about one fourth, that of Pu240 is about one half, and those of Pu241 and Am241 are about the same. Since Pu239 has a large microscopic cross section and its abundance itself is the largest as about 50 wt% to 80 wt%, Pu239 is the most dominant isotope in the macroscopic cross section in Pu. Although Pu240 has a not so much microscopic cross section, its abundance is as much as 13 wt% to 35 wt%, being relatively larger than those of other isotopes (about 10 wt% at most) except for Pu239. Therefore, it can be said that Pu240 is a comparatively dominant isotope in the macroscopic cross section in Pu, next to Pu239.

As described above, although the neutron spectrum is affected by the change of the Pu composition, since especially abundances of Pu239 and Pu240 are considered to be dominant, it is conceivable that while indices of subgrouping in Non-patent Literature 1 are nuclides of Pu239 and Pu242, grouping is performed by substituting them with nuclides of Pu239 and Pu240. However, with only this, some accuracy improvement can only be expected.

Therefore,in the presentinvention,equivalentfissile coefficients themselves are made to have neutron spectrum dependence, that is, a dependence on an isotope that is selected in a decreasing order of the macroscopic cross section. The number N of the selected isotope nuclide is unity or more, and an actual number is appropriately set depending on an application composition range and required accuracy. First, the equivalent fissile coefficients are expressed as functions shown in a following equation (13a), departing from the conventional constant values, and correspondence with the function when the least squares method is applied is as shown in an equation (14a).

[Mathematical 22] $$\begin{array}{llll}{c}_{m\left(k\right)}& =0,& & k=1\\ c_{m\left(k\right)}& =c_{m\left(k\right),0}& +{\displaystyle \sum _{\mathit{kʹ}=1}^N}{c}_{m\left(k\right),n\left(\mathit{kʹ}\right)}\cdot w_{n\left(\mathit{kʹ}\right)},& k>1\\ c_{n\left(k\right)}& =1,& & k=1\\ c_{n\left(k\right)}& =c_{n\left(k\right),0}& +{\displaystyle \sum _{\mathit{kʹ}=1}^N}{c}_{n\left(k\right),n\left(\mathit{kʹ}\right)}\cdot w_{n\left(\mathit{kʹ}\right)}& k>1\end{array}$$

Where

m(k)

: Sequence (example: {28, 24, 25, 26}), arrangement of an integer symbol 28 of first term (subscript k = 1) indicating U238 that is trivial as equivalent fissile coefficient plus subsequent terms (subscript k = 2, ...) of integer terms each indicating uranium isotope nuclide that is nontrivial as equivalent fissile coefficient and should be taken into consideration,

n(k)

: Sequence (example: {49, 40, 41, 42, 51, 48}), arrangement of an integer symbol 49 of first term (subscript k = 1) indicating Pu239 that is trivial as equivalent fissile coefficient plus subsequent terms (subscript k = 2, ...) of integer terms each indicating Pu isotope nuclide that is nontrivial as equivalent fissile coefficient and should be taken into consideration (Am241 is also included conveniently) (however, for subscript k = 2 and subsequent k's, in decreasing order of macroscopic cross section),

N

: Number of Pu isotopes that should be taken into consideration wher providing dependence on composition fraction of Pu isotope to equivalent fissile coefficients,

N'

: Lesser of N and k-1 (N' : = min{N, k-1}).

[Mathematical 23] $$\begin{array}{llll}y& :=-\epsilon \cdot w_{49}& & \\ x_{m\left(k\right),0}& :=\left(1-\epsilon \right)\cdot w_{m\left(k\right)}& ,x_{m\left(k\right),n\left(\mathit{kʹ}\right)}:=x_{m\left(k\right),0}\cdot w_{n\left(\mathit{kʹ}\right)}& ;k>1,\mathit{kʹ}=1,\mathrm{..},N\\ x_{n\left(k\right),0}& :=\epsilon \cdot w_{m\left(k\right)}& ,x_{m\left(k\right),n\left(\mathit{kʹ}\right)}:=x_{m\left(k\right),0}\cdot w_{n\left(\mathit{kʹ}\right)}& ;k>1,\mathit{kʹ}=1,\mathrm{..},\mathit{Nʹ}\\ f_{m\left(k\right),0}& :=x_{m\left(k\right),0}& ,f_{m\left(k\right),n\left(\mathit{kʹ}\right)}:=x_{m\left(k\right),n\left(\mathit{kʹ}\right)}& ;k>1,\mathit{kʹ}=1,\mathrm{..},N\\ f_{n\left(k\right),0}& :=x_{n\left(k\right),0}& ,f_{n\left(k\right),n\left(\mathit{kʹ}\right)}:=x_{n\left(k\right),n\left(\mathit{kʹ}\right)}& ;k>1,\mathit{kʹ}=1,\mathrm{..},\mathit{Nʹ}\\ f_{49,0}& :=1& & \end{array}$$

For example, in the case where isotopes Pu239, Pu240 of N = 2 are selected in the decreasing order of the macroscopic cross section, the largest the first, and these two nuclides are made to have the dependence, the equivalent fissile coefficients are expressed as functions to the composition fractions of Pu239 and Pu240 shown in a following equation (13b) departing from the conventional constant value, and subsequently the least squares method is applied thereto.

[Mathematical 24] $$\begin{array}{llll}{c}_{24}\left(w_{49},w_{40}\right)& =c_{24,0}& +c_{24,49}\cdot w_{49}& +c_{24,40}\cdot w_{40}\\ c_{25}\left(w_{49},w_{40}\right)& =c_{25,0}& +c_{25,49}\cdot w_{49}& +c_{25,40}\cdot w_{40}\\ c_{26}\left(w_{49},w_{40}\right)& =c_{26,0}& +c_{26,49}\cdot w_{49}& +c_{26,40}\cdot w_{40}\\ c_{28}& =0& & \\ c_{48}\left(w_{49},w_{40}\right)& =c_{48,0}& +c_{48,49}\cdot w_{49}& +c_{48,40}\cdot w_{40}\\ c_{49}& =1& & \\ c_{40}\left(w_{49}\right)& =c_{40,0}& +c_{40,49}\cdot w_{49}& \\ c_{41}\left(w_{49},w_{40}\right)& =c_{41,0}& +c_{41,49}\cdot w_{49}& +c_{41,40}\cdot w_{40}\\ c_{42}\left(w_{49},w_{40}\right)& =c_{42,0}& +c_{42,49}\cdot w_{49}& +c_{42,40}\cdot w_{40}\\ c_{51}\left(w_{49},w_{40}\right)& =c_{51,0}& +c_{51,49}\cdot w_{49}& +c_{51,40}\cdot w_{40}\end{array}$$

Here, the coefficients C of the right-hand side of the above-mentioned equation (13b) are all constant values. In order to deal with this equation (13b) by the least squares method, a following equation (14b) is used.

[Mathematical 25] $$\begin{array}{l}\begin{array}{cccc}y& :=-\epsilon \cdot w_{49}& & \\ x_1& :=w_{24}\cdot \left(1-\epsilon \right)& ;x_2:=w_{49}\cdot x_1& ;x_3:=w_{40}\cdot x_1\\ x_4& :=w_{25}\cdot \left(1-\epsilon \right)& ;x_5:=w_{49}\cdot x_4& ;x_6:=w_{40}\cdot x_4\\ x_7& :=w_{26}\cdot \left(1-\epsilon \right)& ;x_8:=w_{49}\cdot x_7& ;x_9:=w_{40}\cdot x_7\\ x_{10}& :=w_{48}\cdot \epsilon & ;x_{11}:=w_{49}\cdot x_{10}& ;x_{12}:=w_{40}\cdot x_{10}\\ x_{13}& :=w_{40}\cdot \epsilon & ;x_{14}:=w_{49}\cdot x_{13}& \\ x_{15}& :=w_{41}\cdot \epsilon & ;x_{16}:=w_{49}\cdot x_{15}& ;x_{17}:=w_{40}\cdot x_{15}\\ x_{18}& :=w_{42}\cdot \epsilon & ;x_{19}:=w_{49}\cdot x_{18}& ;x_{20}:=w_{40}\cdot x_{18}\\ x_{21}& :=w_{51}\cdot \epsilon & ;x_{22}:=w_{49}\cdot x_{21}& ;x_{23}:=w_{40}\cdot x_{21}\\ & & & \\ c_1& :=c_{24,0}& ;c_2:=c_{24,49}& ;c_3:=c_{24,40}\\ c_4& :=c_{25,0}& ;c_5:=c_{25,49}& ;c_6:=c_{25,40}\\ c_7& :=c_{26,0}& ;c_8:=c_{26,49}& ;c_9:=c_{26,40}\\ c_{10}& :=c_{48,0}& ;c_{11}:=c_{48,49}& ;c_{12}:=c_{48,40}\\ c_{13}& :=c_{40,0}& ;c_{14}:=c_{40,49}& \\ c_{15}& :=c_{41,0}& ;c_{16}:=c_{41,49}& ;c_{17}:=c_{41,40}\\ c_{18}& :=c_{42,0}& ;c_{19}:=c_{42,49}& ;c_{20}:=c_{42,40}\\ c_{21}& :=c_{51,0}& ;c_{22}:=c_{51,49}& ;c_{23}:=c_{51,40}\\ c_{24}& :=-E_{239}& & \\ & & & \end{array}\\ \begin{array}{c}{f}_n\left(x_1,x_2,\dots ,x_{23}\right):=x_n,n<24\\ f_n\left(x_1,x_2,\dots ,x_{23}\right):=1,n=24\end{array}\end{array}$$