Research Article Improving Nuclear Safety of Fast Reactors by … · 2019. 5. 12. · =1 1+/ + 6 =1 1+ , +1 prt = 1 +1 prt + +1 +1, ( ) where is the number of annular layers in neutron

Research ArticleImproving Nuclear Safety of Fast Reactors by Slowing DownFission Chain Reaction

G G Kulikov A N Shmelev and V A Apse

National Research Nuclear University MEPhI (Moscow Engineering Physics Institute) Kashirskoe shosse 31 Moscow 115409 Russia

Correspondence should be addressed to G G Kulikov ggkulikovmephiru

Received 23 May 2014 Revised 19 September 2014 Accepted 20 September 2014 Published 16 October 2014

Academic Editor Massimo Zucchetti

Copyright copy 2014 G G Kulikov et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Light materials with small atomic mass (light or heavy water graphite and so on) are usually used as a neutron reflector andmoderator The present paper proposes using a new heavy element as neutron moderator and reflector namely ldquoradiogenic leadrdquowith dominant content of isotope 208Pb Radiogenic lead is a stable natural lead This isotope is characterized by extremely lowmicro cross-section of radiative neutron capture (sim023mb) for thermal neutrons which is smaller than graphite and deuteriumcross-sectionsThe reflector-converter for a fast reactor core is the structure capable of transforming some part of prompt neutronsleaked from the core into the reflected neutrons with properties similar to those of delayed neutrons that is sufficiently largecontribution to reactivity at the level of effective fraction of delayed neutrons and relatively long lifetime comparable with lifetimesof radionuclides-emitters of delayed neutrons It is evaluated that the use of radiogenic leadmakes it possible to slow down the chainfission reaction on prompt neutrons in the fast reactor This can improve the fast reactor safety and reduce some requirements tothe technologies used to fabricate fuel for the fast reactor

1 Introduction

Importance of such physical characteristic as prompt neutronlifetime for nuclear reactor safety is well known for along time and was reflected in numerous publications forexample in one of such fundamental works as [1]The longerprompt neutron lifetime could produce the most favorableeffects on nuclear reactor safety under conditions of thereactivity-induced accidents

The positive role to be played by the radiogenic leadthat is lead with a dominant content of isotope 208Pb asa coolant for fast reactor safety was first noted in works[2 3] where a possibility for significant improvement of thecoolant temperature reactivity coefficient was shown Latersome possibilities for improving other neutron-physical andthermal-hydraulic parameters of power fast reactors wereconsidered at usage of the radiogenic lead as a coolant [4]This direction of researches has been developed in work [5]where in addition to the aforementioned possibilities theprospects of the radiogenic lead applications for developing

high-flux accelerator-driven systems capable to transmuteradioactive wastes and for upgrading proliferation resistanceof advanced Pu-based fuel compositions were investigatedtoo

Extension of prompt neutron lifetime in fast reactors withthe radiogenic lead as a neutron reflectorwas first proposed inworks [6 7]This paper presents the results of further studiescarried out in this direction

2 Prompt Neutron Lifetime ofDifferent Reactors

The longer prompt neutron lifetimes can substantiallyimprove kinetic response of the fast reactor to a jump-likeinsertion of relatively large (sim1 $ or even more) positivereactivity For simplicity let effective fraction 120573 of delayedneutrons be the same for all the fast reactor models underconsideration here 120573(235U) = 00065

The curves presented in Figure 1 demonstrate depen-dencies of the reactivity jump required to provide the

Hindawi Publishing CorporationInternational Journal of Nuclear EnergyVolume 2014 Article ID 373726 18 pageshttpdxdoiorg1011552014373726

2 International Journal of Nuclear Energy

power excursion with asymptotic time period 119879 (withoutany feedback effects) in the fourth-generation lead-cooledfast BREST-type [8] reactor (neutron reflectormdash50 cm thicknatural lead) in thermal VVER-type and CANDU-typereactors as well as in two hypothetical reactors whose promptneutron lifetimes are equal to 001 s and 01 s respectively

These curves demonstrate that the shorter lifetime ofprompt neutrons results in the faster power excursion atthe same reactivity jump It seems helpful to reformulatethe statement as follows If prompt neutron lifetime becamelonger then the power excursion with a certain asymptotictime period 119879 would require the larger reactivity jumpOtherwise the power excursionwill be slower Consequentlythe reactor safety can be enhanced by making lifetime ofprompt neutrons longer

For example based on one-point model of neutronkinetics it may be concluded that application of thick 208Pbreflector in the fast reactor BREST allowed us to reachprompt neutron lifetime of sim1ms If 1-$ reactivity jump ()occurs in the fast reactor then its power increases with theexcursion period nearly 1 s instead of 14ms in the fast reactorreflected by natural lead If prompt neutron lifetime in thefast reactor BREST reflected by thick 208Pb layer is prolongedup to 10ms then the power excursion period will be longerthan 1 s even at the reactivity jumps up to 2$ So long thepower excursion periods give sufficient time for coolant toremove the heat from fuel rodsThis means that feedbacks oncoolant temperature and coolant density could be apparentlyactuated

3 One-Point Model

The dependencies shown in Figure 1 were calculated withthe application of one-point kinetic model with six groupsof delayed neutrons No feedback effects were taken intoconsiderationThe one-pointmodel is based on the followingset of equations

119889119899 (119905)

119889119905

=

120588 minus 120573

Λprtsdot 119899 (119905) +

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

=

120573

119894

Λprtsdot 119899 (119905) minus 120582

119894sdot 119862

119894(119905) (119894 = 1 2 6)

(1)

where 119899 is neutron population in the reactor core 119862

119894is

quantity of nuclei-emitters for 119894th group of delayed neutronsin the reactor core 120588 = (119896ef minus 1)119896ef reactivity where 119896efis the effective neutron multiplication factor in the reactorincluding neutron reflector 120573 is effective fraction of delayedneutrons 120573

119894is effective fraction of delayed neutrons in 119894th

group Λprt mean prompt neutron lifetime (according to theterminology applied by Bell and Glasstone [1]) in the reactorincluding neutron reflector 120582

119894is decay constant of nuclei-

emitters in 119894th group of delayed neutrons Prompt neutronlifetime Λprt and generation time 120591 are linked by the rela-tionship 120591 = Λprt119896ef Since only critical andnearly critical

100

1610

25

1

1Asymptotic period (s)

1001

01001

Reactorneutron life time

Requ

ired

reac

tivity

($) CANDU0001 s

VVER10minus4 s

BREST

01 s

001 s

6 s05 middot 10minus6

Figure 1 Dependencies of the reactivity jump required to providethe power excursion with asymptotic time period 119879 (one-pointmodel no feedback effects)

(within some dollars from criticality) states are analyzedhere lifetime and generation time differ insignificantlyTheseterms are regarded below as being equivalent

Within the frames of one-point kinetic model (1) theinverse-hour equation that links the asymptotic period of thepower excursion119879with the inserted positive reactivity 120588maybe written in the following form

120588 =

Λprt

119879

+

6

sum

119894=1

120573

119894

1 + 120582

119894sdot 119879

(2)

4 Two-Point Model

As has been noted in [9] one-point model is not completelyapplicable for analysis of neutron kinetics in the reactor coressurrounded by neutron reflector The associated problemswere analyzed in [9ndash15]with application ofmultipointmodelsor with introduction of some additional (fictive) groups ofdelayed neutrons The latter case takes into account thefollowing new parameters time of neutron staying in thereactor core before leakage time for neutron transport fromthe reactor core to neutron reflector time of neutron stayingin the reflector time for neutron transport from the reflectorinto the reactor core and time from neutron arrival into thereactor core to initiation of fission reaction by this neutron Inthe case under consideration here neutron reflector is placedin the immediate vicinity to the reactor core and the reflectoris thick from physical point of view This means that timeof neutron staying in the reflector plays a dominant role incompetition with all other times If one additional group ofdelayed neutrons namely slow neutrons coming back fromthe reflector into the reactor core is introduced to study

International Journal of Nuclear Energy 3

the reflector-induced effects on neutron kinetics then thefollowing set of equations may be written

119889119899 (119905)

119889119905

=

120588 minus 120573 minus 120588

119877

Λ

119862

sdot 119899 (119905) +

6

sum

119894=1

120582

119894sdot 119862

119894(119905) + 120582

119877sdot 119862

119877(119905)

119889119862

119894(119905)

119889119905

=

120573

119894

Λ

119862

sdot 119899 (119905) minus 120582

119894sdot 119862

119894(119905) (119894 = 1 2 6)

119889119862

119877(119905)

119889119905

=

120588

119877

Λ

119862

sdot 119899 (119905) minus 120582

119877sdot 119862

119877(119905) 120582

119877=

1

Λ

119877

(3)

where in addition to the aforementioned designations 119862119877

is quantity of fictive nuclei-emitters for additional group ofdelayed neutrons that is for neutrons coming back fromthe reflector into the reactor core where they can initiatefission reactions (for brevity 119862

119877may be called a quantity of

fictive nuclei-emitters in the reflector) 120588119877

= (119896ef minus 119896

119862

ef)119896efis reactivity gain caused by the reflector that is increase ofreactivity caused by neutrons coming back from the reflectorinto the reactor core and subsequent initiation of additionalfission reactions (for brevity 120588

119877may be called as a reactivity

gain caused by the reflector) 119896ef is effective neutronmultipli-cation factor in the reactor including neutron reflector (likein one-point model) 119896119862ef is effective neutron multiplicationfactor in the reactor without neutron reflector Λ

119862is prompt

neutron lifetime in the reactor without neutron reflector Λ119877

is prompt neutron lifetime caused by the reflector-inducedeffects that is Λ

119877is a sum of neutron lifetime in the reactor

core time of neutron staying in the reflector and neutronlifetime in the reactor core after coming back from thereflector (for brevity Λ

119877may be called as a prompt neutron

lifetime in the reflector)According to the set of equations (3) additional group

of delayed neutrons which simulates neutron diffusion inthe reflector can be characterized by its effective fractionthat is by contribution of the reflector into reactivity and byits decay constant that is inverse value of prompt neutronlifetime in the reflector The inverse-hour equation that linksthe asymptotic period 119879 with the inserted positive reactivity120588 can be written in the following form for the reactor coresurrounded by neutron reflector

120588 =

Λ

119862

119879

+

120588

119877

1 + 119879Λ

119877

+

6

sum

119894=1

120573

119894

1 + 120582

119894sdot 119879

(4)

The equation differs from similar inverse-hour equation(2) in one-point model (1) by the second summand thatdescribes the role of neutron reflector Besides the firstsummand of the inverse-hour in one-point model containsmean prompt neutron lifetime in the reactor as a wholeincluding neutron reflector while the first summand of theinverse-hour equation in the model with additional groupof delayed neutrons (two-point model) contains promptneutron lifetime Λ

119862in the reactor core without neutron

reflector and the second summand contains prompt neutronlifetime in the reflector Λ

119877 Prompt neutron lifetime in the

reflector Λ

119877can be calculated from the following balance

relationship (see derivation of it in Appendix C)

Λprt = (1 minus 120588

119877) sdot Λ

119862+ 120588

119877sdot Λ

119877 (5)

where Λprt (like in one-point model) is a mean promptneutron lifetime in the reactor with neutron reflector

The balance relationship defines prompt neutron lifetimein a system as a sum of prompt neutron lifetimes in allsystem components (in the reactor core plus in the reflectorfor instance) with the weighing coefficients that characterizecontributions of these components to total criticality

It may be concluded from (4) that 120588119877plays here a role of

additional fraction of delayed neutrons that is characterizedby Λ

119877 prompt neutron lifetime in the reflector This means

in its turn that application of such a thick neutron reflectorgives a new quality to the reactormdashthe larger fraction ofdelayed neutrons and as a consequence slowing down ofchain fission reaction

One else important circumstance consists in the followingfact The larger fraction of delayed neutrons depends mainlyon neutron leakage from the reactor core and thus maybe chosen as a developer wills while fraction of nuclei-emitters of delayed neutrons may be chosen only withinvery stringent constraints Evidently generation rate of theseldquodelayedrdquo neutrons substantially depends on leakage rate offast and resonance neutrons from the reactor core That iswhy application of thick neutron reflector is a reasonableoption not only for fast reactors but also for the reactors withresonance and even thermal spectra with small sizes of thereactor core that is for the reactors with significant leakageof fast and resonance neutrons from the reactor core

The inverse-hour equation (4) for the reactor coresurrounded by neutron reflector with additional group ofdelayed neutrons coincides with the inverse-hour equationin two-point model [9] For brevity hereafter the term ldquo119899-pointrdquo will be used in designation of kineticmodels changingonly the number of ldquopointsrdquo

5 Multipoint Model

This model of neutron kinetics can be used by considering aneutron reflector as a whole (as one zone in two-pointmodel)or by considering a neutron reflector as a set of annular(nonintersecting) layers (multipoint model depending on thenumber of these layers)

119889119899 (119905)

119889119905

=

120588 minus 120573 minus sum

119869

119895=1120588

119895

119877

Λ

119862

sdot 119899 (119905) +

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

+

119869

sum

119895=1

120582

119895

119877sdot 119862

119895

119877(119905)


119889119862

119894(119905)

119889119905

=

120573

119894

Λ

119862

sdot 119899 (119905) minus 120582

119894sdot 119862

119894(119905)

(119894 = 1 2 6 119895 = 1 2 119869)

119889119862

119895

119877(119905)

119889119905

=

120588

119895

119877

Λ

119862

sdot 119899 (119905) minus 120582

119895

119877sdot 119862

119895

119877(119905) 120582

119895

119877=

1

Λ

119895

119877

(6)

Within the frames of multipoint model the inverse-hourequation (4) and the balance relationship (5) for determina-tion of prompt neutron lifetime in a neutron reflector can berewritten as follows

120588 =

Λ

119862

119879

+

119869

sum

119895=1

120588

119895

119877

1 + 119879Λ

119895

119877

+

6

sum

119894=1

120573

119894

1 + 120582

119894sdot 119879

Λ

119895+1

prt = (1 minus 120588

119895+1

119877) sdot Λ

119895

prt + 120588

119895+1

119877sdot Λ

119895+1

119877

(7)

where 119869 is the number of annular layers in neutron reflector120588

119895

119877is contribution of 119895th layer to total reactivity gain Λ119895

119877is

prompt neutron lifetime in 119895th layer that is sum of promptneutrons lifetime in the reactor core before they go to thereflector time for neutron transport from the reactor coreto 119895th layer of the reflector time of neutron staying in 119895thlayer time for neutron transport from 119895th layer of the reflectorto the reactor core and lifetime of neutrons after they cameback from the reflector 119862119895

119877is quantity of nuclei-emitters

for additional group of delayed neutrons in 119895th layer of thereflector Λ119895prt is prompt neutron lifetime in the reactor withneutron reflector consisting of the first 119895 layers adjacent tothe reactor core

The further results were obtained for six-layer reflector(thickness of the first layer is 50 cm thicknesses of the nextlayers are equal to 1m each) Kinetics parameters such as Λ

119862

and spatial dependences of Λ119877and 120588

119877were evaluated on the

base of numerical analysis using diffusion neutron transportmodel with an evaluated nuclear data library (RUSFOND-2010) in frames of one-dimension spherical geometry Con-tributions of the reflector into the reactivity gain and promptneutron lifetime are presented in Table 1 for two-point andmultipointmodels of neutron kinetics in the fast BREST-typereactor with the reactor core surrounded by neutron reflectorof different thickness (from 1m to 6m) Multipoint modelfor the case of 1m-thick reflector is in essence two-pointmodel the first point for the reactor core with 05m-thickreflector the second point for annular layer (from 05m to1m) in the reflector If thickness of the reflector increases upto 2m then the third point arises for annular layer from 1m to2mThree-point model is used in this case Correspondinglyneutron kinetics in 6m thick reflector is defined by seven-point model

It can be seen that in the case of 6m reflector (seven-point model) prompt neutron lifetime in the last 1m thickreflector layer is considerably longer (about one order ofmagnitude) than that for two-point model (one point for thereflector as a whole) This means that more correct mathe-matical models must be used to provide proper accounting

for neutron transport effects in the fast reactors surroundedby physically thick and weakly absorbing neutron reflectors

6 Spherically Symmetrical Continuous Modelof Neutron Kinetics in Reactor Surroundedby Physically Thick Neutron Reflector

In order to simplify explanation of continuous neutronkinetics model it seems reasonable to consider a sphericallysymmetrical reactor with a physically thick neutron reflectorThe reactor core is described by one point in the continuousmodel When the number of the reflector layers becomesinfinite multipoint model naturally converts into the contin-uous model where summing operation should be replaced byintegration

119889119899 (119905)

119889119905

=

120588 minus 120573 minus 120588

119877(119903out)

Λ

119862

sdot 119899 (119905)

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905) + int

119903out

119903in

120582

119877(119903

1015840) sdot 119862

119877(119903

1015840 119905) sdot 119889119903

1015840

119889119862

119894(119905)

119889119905

=

120573

119894

Λ

119862

sdot 119899 (119905) minus 120582

119894sdot 119862

119894(119905) (119894 = 1 2 6)

119889119862

119877(119903 119905)

119889119905

=

119889120588

119877(119903) 119889119903

Λ

119862

sdot 119899 (119905) minus 120582

119877(119903) sdot 119862

119877(119903 119905)

120582

119877(119903) =

1

Λ

119877(119903)

(8)

where 119903in and 119903out are inner and outer radii of the reflector120588

119877(119903) is reactivity gain caused by the reflector with outer

radius 119903 (119903in le 119903 le 119903out) 119889120588119877(119903)119889119903 is differential reactivitygain that is accretion of the reactivity gain caused byadding spherical reflector layer of unitary thickness to thereflector with outer radius 119903 119862

119877(119903 119905) is quantity of fictive

nuclei-emitters for additional group of delayed neutrons inspherical reflector layer of unitary thickness in spatial pointwith radius-vector 119903 Λ

119877(119903) is prompt neutron lifetime in

spherical reflector layer of unitary thickness in spatial pointwith radius-vector 119903

The inverse-hour equation and relationship for deter-mination of prompt neutron lifetime for continuous modelof the reflector in spherically symmetrical geometry can berewritten in the following new forms

120588 =

Λ

119862

119879

+ int

119903out

119903in

119889120588

119877(119903

1015840) 119889119903

1015840

1 + 119879Λ

119877(119903

1015840)

sdot 119889119903

1015840+

6

sum

119894=1

120573

119894

1 + 120582

119894sdot 119879

(9)

Λ

119877(119903) = Λ prt (119903) +

119889Λ prt (119903) 119889119903

119889120588

119877(119903) 119889119903

(10)

Continuous model of neutron kinetics may be regarded asa model describing continuous variation of reactivity gaincaused by continuous variation of prompt neutron lifetime inthe neutron reflector depending on spatial coordinate withinthe reflector Solution of the inverse-hour equation (9) is not


Table 1 Dependence of reactivity gain and prompt neutron lifetime on thickness of 208Pb reflector in the fast BREST-type reactor for variousmodels of neutron kinetics

Two-point modelThickness of 208Pb reflector m 1 2 3 4 5 6Reactivity gain caused by the reflector $ 301 530 651 721 758 778Prompt neutron lifetime in the reflector s 290 sdot 10minus5 104 sdot 10minus3 571 sdot 10minus3 121 sdot 10minus2 182 sdot 10minus2 233 sdot 10minus2

Multipoint modelsThe number of points 2 3 4 5 6 7Annular layer in 208Pb reflector m 05 divide 1 1 divide 2 2 divide 3 3 divide 4 4 divide 5 5 divide 6Increase of the reactivity gain caused by thickerreflector $ 301 229 121 069 037 020

Prompt neutron lifetime in the reflector layer s 290 sdot 10minus5 236 sdot 10minus3 262 sdot 10minus2 725 sdot 10minus2 136 sdot 10minus1 216 sdot 10minus1

a trivial task A possible way of the solution is proposed inAppendix D

7 Physical Parameters of Neutron Reflector inContinuous Model of Neutron Kinetics

Some relevant physical parameters of neutron reflector areconsidered here within the frames of spherically symmetricalcontinuous model of neutron kinetics and presented inFigure 2 It follows from Figure 2(a) that the reactivity gainthanks to the reflector 120588

119877(119903) grows as the reflector becomes

thicker and approaches a saturation level Correspondinglyderivative of the reactivity gain decreases and vanishesDerivative of the reactivity gain 119889120588

119877(119903)119889119903 defines the growth

rate of contribution given by prompt neutrons to chain fissionreaction as the reflector becomes thicker So it is naturalthat derivative of the reactivity gain decreases as the reflectorthickness increases because contribution of neutrons comingback from more distant reflector layers into chain fissionreaction in the reactor core vanishes

It follows from the curves shown in Figure 2(b) thatmeanprompt neutron lifetime Λprt in the fast BREST-type reactorwith 208Pb reflector as a whole becomes substantially longer(from sim1 120583sec to sim1msec) as the reflector thickens from 05mto 6m Derivative of the mean prompt neutron lifetime hasa weakly expressed peak at the reflector thickness of sim3mThe latter means an important fact that elongation of meanprompt neutron lifetime becomes slower as thickness of thereflector exceeds 3m

The following two terms of (10) are considered belownamely Λ

119877(0) prompt neutron lifetime in the reflector layer

of infinitesimal thickness adjacent to the reactor core that ispractically without the reflector andΛ prt(0) prompt neutronlifetime in the reactor core only The former is larger thanthe latter on the second summand This is related with thefollowing fact Within the frames of one-point model allprompt neutrons in the reactor core are characterized by acommon generation timeΛ

119862 averaged over the core volume

and it is assumed that Λ prt(0) = Λ

119862 Lifetime of prompt

neutrons of the reflector layer of even infinitesimal thicknessadjacent to the reactor core is relatively longer because thislifetime is defined by neutrons that came up to the very edge

0 1 2 3 4 5 60

2

4

6

8

0

2

4

6

8

10

Reac

tivity

gai

nca

used

by

the r

eflec

tor (

$)

Diff

eren

tial r

eact

ivity

gai

n ca

used

by

the r

eflec

tor (

$m

)

Thickness of 208Pb reflector (m)

120588R(r)

d120588R(r)dr

(a)

0 1 2 3 4 5

Diff

eren

tial n

eutro

n lif

etim

e (s

m)

6

0 2 4 6 8 10

ΛR(r)

Λprt

dΛprt (r)dr

1E + 0

1E minus 1

1E minus 2

1E minus 3

1E minus 4

1E minus 5

1E minus 6

1E minus 7

1E + 0

1E minus 1

1E minus 2

1E minus 3

1E minus 4

1E minus 5

1E minus 6

1E minus 7

Neu

tron

lifet

ime (

s)


Radial coordinate within the 208Pb reflector-r (m)

(b)

Figure 2 Physical parameters of the fast BREST-type reactordepending on thickness of neutron reflector made of 208Pb

of the reactor core and then came back to the reactor coreTheexplanation is confirmed by the calculated results presentedin Figure 2(b) It can be seen that prompt neutron lifetime inthe reflector layer of infinitesimal thickness Λ

119877(0) is longer


than prompt neutron lifetime in the reactor without neutronreflector Λ prt(0) by above one order of magnitude

As the reflector becomes thicker difference betweenprompt neutron lifetime in the reflector Λ

119877(119903) and mean

prompt neutron lifetime in the reactor with neutron reflectorΛ prt increasesWhen the reflector thickness reaches 6m thedifference exceeds two orders of magnitude It is noteworthythat lifetime of prompt neutrons coming back from6-m-thick208Pb reflector into the reactor core is equal to a rather longvalue above 01 s (Figure 2(b))

8 Advancement of Continuous Model from(119877 119879) Phase Space to (Λ 119879) Phase Space

When physical parameters of physically thick neutron reflec-tor were analyzed by using continuous neutron kineticsmodel the following dependencies were calculated

(i) Radial distribution (within the reflector thickness) ofthe contribution given by annular layer of unitarythickness into the reactivity gain 120588

119877(119903) (Figure 2(a))

(ii) Radial distribution (within the reflector thickness) ofprompt neutron lifetime in annular layer of unitarythickness Λ

119877(119903) (Figure 2(b))

This means that each annular layer of unitary thicknesswith inner radius 119903 is characterized by its own prompt neu-tron lifetime Λ

119877(119903) Then the contribution given by prompt

neutrons with lifetime Λ119877and longer into the reactivity gain

may be considered as a certain generalized parameter of sucha thick neutron reflectorThis parameter defines capability ofthe reflector to generate prompt neutrons with different delayand can be calculated by using the following formulas

120588

119877(Λ

119877 Λmax) equiv int

119903out

119903

119889120588

119877(119903

1015840)

119889119903

1015840sdot 119889119903

1015840= int

Λmax

Λ 119877

119889120588

119877

119889Λ

1015840

119877

sdot 119889Λ

1015840

119877

(11)

The contributions given by the neutron reflectors of differentthickness into the reactivity gain as functions of neutrondelay time in the reflector as long as Λ

119877and longer are

presented in Figure 3 If the reflector is made of 208Pb thenthe contributions of neutrons with lifetimes of 1msec andlonger into the reactivity gain can reach several dollars Forcomparison analogous dependency is shown in Figure 3 forthe neutron reflector made of natural lead In this casethe same reactivity effect caused by the reflector thickeningand as a consequence by longer neutron lifetime remainsinsignificant even in the reflector with 2 m thickness

Component of the reactivity gainmdashthe second summandof the inverse-hour equation (9)mdashmay be regarded as a usefulcharacteristic parameter of the neutron reflector-converter

120588

lowast

119877equiv int

119903out

119903in

119889120588

119877(119903

1015840) 119889119903

1015840

1 + 119879Λ

119877(119903

1015840)

sdot 119889119903

1015840

(12)

One else informative function is an integrand in formula (12)whose radial distribution is shown in Figure 4

0001 001 101

1

3

5

0

2

4

6

6

4

32

5

Delay of neutrons in the reflector for a timeperiod longer than a given value (s)

Reflector materialthickness (m)

Pbnat 2

Con

trib

utio

n of

neu

trons

of t

he re

flect

orin

to th

e rea

ctiv

ity g

ain120588

R(Λ

RΛ

max

) ($)

1E minus 4

208Pb

Figure 3 Radial distribution of the contribution given by neutronsof the reflector with lifetimes as long as Λ

119877and longer into the

reactivity gain 120588

119877(Λ

119877 Λmax) The reflector is made of 208Pb or

natural lead thicknessmdashfrom 2m to 6m

0 1 2 3 4 5 6

Diff

eren

tial c

ontr

ibut

ion

into

the r

equi

ered

reac

tivity

($m

)

1E + 0

1E + 1

1E minus 1

1E minus 2

1E minus 3

1E minus 4

1E minus 5

208 Pb T120588lowastR

Pbnat T120588lowastR

001 s244$

01 s096$10 s016$

001 s0036$01 s00037 $

10 s000038 $

Radial coordinate within the reflector-r (m)

Figure 4 Differential contribution into the reactivity jump requiredto provide the power excursion with asymptotic time period 119879

(integrand of formula (12))

To analyze the contributions of neutrons with variouslifetimes in the reflector into the reactivity gain it may behelpful to transform integration over radial coordinate intointegration on neutron lifetime in the reflector Λ

119877

int

119903out

119903in

119889120588

119877(119903

1015840) 119889119903

1015840

1 + 119879Λ

119877(119903

1015840)

sdot 119889119903

1015840= int

Λmax

Λmin

119889120588

119877119889Λ

1015840

119877

1 + 119879Λ

1015840

119877

sdot 119889Λ

1015840

119877

119889120588

119877

119889119903

=

119889120588

119877

119889Λ

119877

sdot

119889Λ

119877

119889119903

(13)


0 1 2 3 4 5 6

25

20

15

10

05

00

001

01

10Con

trib

utio

n of

neu

trons

of t

he re

flect

or in

to

the r

equi

red

reac

tivity

($)

Asymptotic period of the power excursions (s)

Radial coordinate within the reflector (m)

(a)

Con

trib

utio

n of

neu

trons

of t

he re

flect

or in

to th

e req

uier

ed re

activ

ity ($

)

Lifetime of neutrons in the reflector (s)

25

20

15

10

05

00

001

01

1010010001

10


1E minus 4

(b)

Figure 5 Component of the reactivity gain required to provide the power excursion with asymptotic time period 119879 which is defined byneutrons of the reflector layers adjacent to the reactor core (from 05m to 119903 or from Λmin to Λ

119877)

In this case the inverse-hour equation may be re-written asfollows

120588 =

Λ

119862

119879

+ int

Λmax

Λmin

Λ

1015840

119877sdot (119889120588

119877119889Λ

1015840

119877)

Λ

1015840

119877+ 119879

sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120573

119894

1 + 120582

119894sdot 119879

(14)

It is interesting to note that the second and third summandsin the inverse-hour equation have similar structures Takinginto account the fact that lifetime Λ

119877for major fraction of

neutrons of the reflector is well below 01 s we can removethis value from denominator of the second summand in theinverse-hour equation (14) for the asymptotic time periodslonger than 01 s As a result the second summand becomessubstantially simpler

int

Λmax

Λmin

Λ

1015840

119877sdot (119889120588

119877119889Λ

1015840

119877)

Λ

1015840

119877+ 119879

sdot 119889Λ

1015840

119877=

1

119879

int

Λmax

Λmin

Λ

1015840

119877sdot 119889120588

119877

=

1

119879

int

119903out

119903in

Λ

1015840

119877sdot 119889120588

119877(119903

1015840)

(15)

Prompt neutron lifetime in the reflector is used here as aweighing function for the integration Small values of theweighing function naturally decrease the calculated integralsSince the decay constants 120582

119894of delayed neutrons cover the

range from sim001 to sim3 inverse seconds the third summandin the inverse-hour equation (14) could not be simplified likethe second summand

Component of the reactivity gain required to provide thepower excursion with asymptotic time period 119879 which is

defined by neutrons of the reflector layers (from short-livedneutrons of the reflector layers adjacent to the reactor coreand to relatively longer-lived neutrons in distant depth 119903 iefrom Λmin to current Λ 119877) may be calculated as follows

120588

lowast

119877(119903) = int

119903

119903in

119889120588

119877(119903

1015840) 119889119903

1015840

1 + 119879Λ

119877(119903

1015840)

sdot 119889119903

1015840= int

Λ

Λmin

119889120588

119877119889Λ

1015840

119877

1 + 119879Λ

1015840

119877

sdot 119889Λ

1015840

119877

(16)

Radial and lifetime distributions of this component arepresented in Figure 5

As it follows from Figures 4 and 5 neutrons in distantlayers of the reflector that is neutrons with relatively longlifetime Λ

119877 give a dominant contribution into the reactivity

gainIf the asymptotic period of the power excursion is

comparable or longer than time constant of fuel rods thenthermal energy can receive a long enough time period forits partial removal from fuel rods by coolant Therefore theprocess of fuel heating up slows down and an opportunityarises for feedbacks on coolant density and temperatureto actuate Otherwise if asymptotic period of the powerexcursion is substantially shorter than time constant of fuelrods then fuel can be overheated and melted down As theseverest consequence reactor can lose its ability of workingAs for typical time constants of fuel rods in experimental fastbreeder reactor EBR-II (USA) fuel rods had time constantsabout 011 s [16] If 1-$ positive reactivity is inserted into thefast BREST-type reactor with 05m thick neutron reflectormade of natural lead then asymptotic period of the powerexcursion is considerably shorter (0014 s) than time constantof its fuel rods At the same time if even 2-$ positive reactivityis inserted into the fast BREST-type reactor with 6 m thickneutron reflector made of 208Pb then asymptotic period of


the power excursion becomes considerably longer (01 s)That is why it is so important to develop fuel rods withquick heat transport to coolant Then reactor will be morestable and less vulnerable to the power excursions with shortasymptotic periods caused by large reactivity jumps

9 Conclusions

It is demonstrated that the use of radiogenic lead as a neutronreflector-converter makes it possible to slow down the chainfission reaction on prompt neutrons in the fast reactor and itcan improve the nuclear safety of fast reactor

(1) Multipoint model of neutron kinetics in nuclear reac-tors was applied to analyze time-dependent evolutionof neutron population in fast reactor with physi-cally thick neutron reflector made of weak neutronabsorber (radiogenic lead with dominant content of208Pb)

(2) Multipoint model of neutron kinetics was appliedto investigate a possibility for substantial elongationof prompt neutron lifetime with correct accountingfor time of neutron staying in physically thick weakneutron-absorbing reflector

(3) The paper demonstrates how multipoint discretemodel of neutron kinetics can be transformed intocontinuous model of neutron kinetics in the reactorcore surrounded by the neutron reflector

(4) Numerical analyses of neutron kinetics were carriedout with application of multipoint and continuousmodels and demonstrated that physically thick andweakly neutron-absorbing reflector is able to prolongprompt neutron lifetime in the fast BREST-type reac-tor by several orders of magnitude (roughly from05 120583sec to 1msec)This effect can be interpreted as anappearance of one else group of delayed neutrons thatincreases their effective fraction and thus improvesthe reactor safety

(5) Advanced design of fuel rods with quick heat trans-port from fuel to coolant can remarkably enhance thereactor resistance against the power excursions withshort asymptotic periods caused by large reactivityjumps

(6) If the asymptotic period of the power excursioncaused by positive reactivity jump is comparablewith thermal constant of fuel rods then heat wouldhave a time to flow down from fuel to coolant itsdensity would decrease and it would be able to impactremarkably on the power excursion process (seeAppendix D)This allows us to consider the questionwhat positive reactivity jumps are permissible for thecoolant-induced reactivity feedback could actuate

Appendices

A Neutronic-Physical Propertiesand Advantages of Radiogenic Lead

The term radiogenic lead is used here for designation oflead that is produced in radioactive decay chains of thoriumand uranium isotopes After a series of alpha and betadecays 232Th transforms into stable lead isotope 208Pb 238Uinto stable lead isotope 206Pb and 235U into stable leadisotope 207Pb Therefore uranium ores contain radiogeniclead consisting mainly of 206Pb while thorium and mixedthorium-uranium ores contain radiogenic lead consistingmainly of 208Pb Sometimes the presence of natural lead inuraniumand thoriumores can change isotope composition ofradiogenic lead Anyway isotope composition of radiogeniclead depends on the elemental composition of the ores fromwhich this lead is extracted

Radiogenic lead consisting mainly of stable lead isotope208Pb can offer unique advantages which follow from uniquenuclear physics properties of 208Pb This lead isotope is adouble-magic nuclide with completely closed neutron andproton shells The excitation levels of 208Pb nuclei (Figure 6)are characterized by a high value of energy (the energyof the first level being 261MeV) while the first excitationlevels of other lead isotopes have lower energy range (057to 090MeV) This results in the fact that the threshold inenergy dependence of 208Pb inelastic scattering cross-sectionis atmuch higher energy than those of other lead isotopes [17](Figure 7)

The unique nuclear properties of 208Pb can be used toimprove parameters of chain fission reaction in a nuclearreactor First since energy threshold of inelastic neutronscattering by 208Pb (sim261MeV) is substantially higher thanthat by natural lead (sim08MeV) 208Pb can soften neutronspectrum in the high-energy range to a remarkably lowerdegree Second neutron radiative capture cross section of208Pb in thermal point (sim023mb) is smaller by two ordersof magnitude than that of natural lead (sim174mb) and evensmaller than that of reactor-grade graphite (sim39mb) Thesedifferences remain large within sufficiently wide energy range(from thermal energy to some tens of kilo-electron-volts)Energy dependence of neutron absorption cross sections ispresented in Figure 8 for natural lead stable lead isotopesgraphite and deuterium [17]

If radiogenic lead consists mainly of 208Pb (sim90 208Pbplus 9 206Pb and sim1 204Pb + 206Pb) then such a leadabsorbs neutrons is as weak as graphite within the energyrange from 001 eV to 1 keV Such isotope compositions ofradiogenic lead can be found in thorium and in mixedthorium-uranium ores

Thus on one hand 208Pb being heavy nuclide is arelatively weak neutron moderator both in elastic scatteringreactions within full neutron energy range of nuclear reactorsbecause of heavy atomic mass and in inelastic scatteringreactions with fast neutrons because of high energy thresholdof these reactions On the other hand 208Pb is an extremelyweak neutron absorber within wide enough energy range


396 6

371 5348 4

320 5

261 3

00 0

234 72

163 32

089 32

057 52

00 12

182 3

156 4

127 4

090 2

238 6

080 2

00 0 00 0

120 0

168 4

200 4

134 3

Ei Ii Ei IiEi Ii

Ei Ii

204 Pb 206 Pb207 Pb 208 Pb

M϶B M϶B M϶B

M϶B

Figure 6 The excitation levels of lead nuclei

0 2 4

Inel

astic

scat

terin

g cr

oss-

sect

ion

(b)

6 8 10

Neutron energy (MeV)

0

1

2

3

204 Pb

206 Pb207 Pb

208 Pb

Figure 7 Inelastic scattering cross-section of lead isotopes as afunction of neutron energy

D

01

Capt

ure c

ross

-sec

tion

(b)

10 1000

01

10

1E minus 3

1E minus 5

1E + 5 1E + 7

Neutron energy (eV)

204Pb

12C

206Pb

207Pb

208Pb

Pbnat

Figure 8 Capture cross sections of various nuclides as a function ofneutron energy (JENDL-40)

Some nuclear characteristics of light neutron moderators(hydrogen deuterium beryllium graphite and oxygen) andheavy materials (natural lead and lead isotope 208Pb) arepresented in Table 2 [17]

One can see that elastic cross sections of natural lead and208Pb do not differ significantly from the others nuclidesbeing between the corresponding values for hydrogen andother light nuclides Neutron slowing-down from 01MeVto 05 eV requires from 12 to 102 elastic collisions with lightnuclides while the same neutron slowing-down requires sim

1270 elastic collisions with natural lead or 208Pb The reasonis the high atomic mass of lead in comparison with the otherlight nuclides From this point of view neither natural leadnor 208Pb are effective neutron moderators

Since 208Pb is a double magic nuclide with closed protonand neutron shells radiative capture cross-section at thermalenergy and resonance integral of 208Pb is much smaller thanthe corresponding values of lighter nuclidesTherefore it canbe expected that even with multiple scattering of neutrons on208Pb during the process of their slowing-down they will beslowed down with a high probability and will create high fluxof slowed down neutrons

So thanks to very small neutron capture cross sectionthe moderating ratio (see Table 3) [17ndash19] that is the averagelogarithmic energy loss times scattering cross section dividedby thermal absorption cross section of 208Pb is much higherthan that for light moderators This means that 208Pb couldbe a more effective moderator than such well-known lightmoderators as light water beryllium oxide and graphite

It is noteworthy that mean lifetime of thermal neutronsin 208Pb is very large (sim06 s) This effect could be usedto essentially improve the safety of the fast reactor byslowing downprogression of chain fission reaction onpromptneutrons [6] This assumes the greater significance underaccidental conditions with destruction of the reactor core


Table 2 Neutron-physical characteristics of some materials

Nuclide 120590

10 eVel (b) Number of collisions

(01MeVrarr 05 eV) 120590

th(119899120574)

(mb) 119877119868

119899120574+ 1119881 (mb)

1H 301 12 332 1492D 42 17 055 0259Be 65 59 85 3812C 49 77 39 1816O 40 102 019 016Pbnat 113 1269 174 95208Pb 115 1274 023 078

Table 3 Properties of neutron moderators at 20∘C

Moderator Average logarithmicenergy loss 120585

Moderatingratio

120585ΣΣ

119886

th

Neutron age 120591(01MeVrarr 05 eV)

(cm2)

Diffusion length119871 (cm)

Mean lifetime 119879th ofthermal neutrons

(ms)H2O 095 70 6 3 02D2O 057 4590 58 147 130BeO 017 247 66 37 812C 016 242 160 56 13Pbnat 000962 06 3033 13 08208Pb 000958 477 2979 341 598

when the energy yield is defined by lifetime of promptneutrons because other kinetic parameters either out of anycontrol (the inserted positive reactivity) or cannot go out ofthe very limited range (Doppler coefficient effective fractionof delayed neutrons the reactor power)

It should be noted that 208Pb is not the only of thekind nuclide in Mendeleevrsquos periodic system whose neutron-physical properties are very specifics For example nuclide88Sr has a very small neutron capture as well inelastic crosssection even smaller than that for 208Pb nuclide and values ofnuclear moderating capabilities 120585 lowast 120590

119904are close for 88Sr and

208Pb [17]Table 4 summarizes relevant neutron-physical character-

istics of some nuclear materials including radiogenic lead Itcan be seen that natural lead is able to slow down only 304of fast neutrons (01MeV) into epi-cadmium range (05 eV)while the remaining fraction (696) is absorbed by naturallead in the slowing-down process On the contrary almostall fast neutrons (993) can be slowed down by 208Pb intoepithermal range

It is worthy to note that in both cases that is duringthe slowing-down process of fast neutrons in natural leadand in 208Pb mean distance of neutron transport and meanslowing-down time are approximately the same (sim134 cm and056ms) As for thermal neutrons mean distances of neutrondiffusion until absorption are quite different for the two leadtypes (30 cm in natural lead and 835 cm in 208Pb)

This means that first very small fraction of neutrons thatwere slowed down in natural lead reflector can come backinto the reactor core On the contrary 208Pb reflector givesthem such a possibility Second mean lifetime of thermal

neutrons in the infinite 208Pb environment (06 s) is longer bythree orders of magnitude than that in natural lead (08ms)So main process in 208Pb reflector is a neutron slowingdown not neutron absorption and these slow neutrons aftera diffusion (and time delay) have a probability to comeback into the reactor core and sustain the chain fissionreaction Mean lifetimes of slow neutrons and especiallythermal neutrons are substantially longer than those for fastand slowing-down neutrons This constitutes a potential forsignificant extension of mean prompt neutron lifetime in thechain fission reaction

It is noteworthy that neutron-physical parameters ofradiogenic lead extracted from thorium and thorium-uranium ore deposits in Brazil Australia the United Statesand the Ukraine are inferior to those of 208Pb but they aresubstantially better than those of natural lead (radiogeniclead compositions fromdifferent deposits are presented in theappendix) For comparison Table 4 presents relevant data forheavy water and graphite Some neutron-physical parametersof these materials are superior to those of 208Pb Howeverthe use of heavy water or graphite as a neutron reflector ina fast reactor is a doubtful option at least because such aneutron reflector can substantially soften neutron spectrumin the reactor core with all negative consequences

B Natural Resources of Radiogenic Lead

In nature there are two types of elemental lead with substan-tially different contents of four stable lead isotopes (204Pb206Pb 207Pb and 208Pb) The first type is a natural orcommon lead with a constant isotopic composition (14


Table 4 Neutron-physical characteristics of different materials

MaterialSlowing-downprobability

(01MeVrarr 05 eV)radic119903

2

119864=

radic6120591 (cm) radic

119903

2

th =radic6119871 (cm)

Neutron lifetime (ms)Slowing down Thermal

Pbnat 0304 134 30 056 08208Pb 0993 134 835 056 598D2O 0999 19 360 001 130Graphite 0998 31 138 003 13

Radiogenic Pb from different depositsBrazil 0930 134 186 056 29Australia 0914 134 142 056 17The USA 0885 134 129 056 14Ukraine 0915 134 145 056 18

204Pb 241 206Pb 221 207Pb and 524 208Pb) Thesecond type is a so-called radiogenic lead with very variableisotopic composition Radiogenic lead is a final product ofradioactive decay chains in uranium and thorium ores Thatis why isotopic compositions of radiogenic lead are definedby the ore age and by elemental compositions of mixedthorium-uranium ores sometimes with admixture of natural(common) lead as an impurityThe isotopes 208Pb 206Pb and207Pb are the final products of the radioactive decay chainsstarting from 232Th 238U and 235U respectively232Th 997888rarr 6120572 + 4120573 (146 sdot 10

9 years) 997888rarr

208Pb238U 997888rarr 8120572 + 6120573 (46 sdot 10

9 years) 997888rarr

206Pb235U 997888rarr 7120572 + 4120573 (07 sdot 10

9 years) 997888rarr

207Pb

(B1)

Therefore radiogenic lead with large abundance of 208Pbcould be extracted from natural thorium and thorium-Uranium ores [20ndash24] without any isotope separation pro-cedures

It should be noted that neutron capture cross-sections of206Pb although larger than those of 208Pb are significantlysmaller than those of 207Pb and 204Pb Thus at first glanceit appears that the ores containing sim93 208Pb and 6206Pb (Table 5) could provide the necessary composition ofradiogenic lead However the first estimations showed thatthe content of only 1 204Pb and 207Pb (these isotopes havehigh values of capture cross-sections) in radiogenic leadcould significantly weaken the advantages of radiogenic leadin thermal nuclear reactors

So radiogenic lead can be taken as a byproduct fromthe process of uranium and thorium ores mining Until nowextraction of uranium or thorium from minerals had beenfollowed by throwing radiogenic lead into tail repositories Iffurther studies will reveal the perspective for application ofradiogenic lead in nuclear power industry then a necessityarises to arrange byextraction of radiogenic lead from tho-rium and uranium deposits or tails Evidently the scope ofthe ores mining and processing is defined by the demands foruranium and thorium

However the demands of nuclear power industry forthorium are quite small now and will remain so in the

near future Nevertheless there is one important factor thatcan produce a substantial effect on the scope of thoriumand mixed thorium-uranium ore mining In the majority ofcases uranium and thorium ores belong to the complex-ore category that is they contain minor amounts of manyvaluable metals (rare-earth elements gold and so on)

The paper by Sinev [25] has demonstrated that thepresence of useful accompanying elements (some elementsof cerium group in particular) in uranium and thorium oresmight be a factor of high significance for making cheaperthe process of natural uranium and thorium productionByextraction of some valuable elements from uranium orescan drop the smaller limit (industrial minimum) of uraniumcontent in ores to 001 to 003 under application of theexisting technologies for natural uranium extraction Radio-genic lead can be recovered from the available tail repositoriesor as a byproduct of the processes applied for extraction of theaccompanying valuable metals from uranium and thoriumores [26]

C Derivation of Balance Relationship

The balance relationship can be used to evaluate meanprompt neutron lifetime in a multizone fast reactor with thereactor core surrounded by neutron reflector-converter toslow down the chain fission reaction under accidental powerexcursions As is expected the balance relationship allowsus to determine spatial (zone-wise) contributions into meanprompt neutron lifetime

To make the derivation as clear as possible the simplestspherical two-zone model (Figure 9) of nuclear reactor isconsidered below The model includes the reactor core sur-rounded by the neutron reflector It is assumed the neutronreflector is made of heavy material with weak neutronabsorption (lead isotope 208Pb eg)

As is well-known [1] mean prompt neutron lifetime is abilinear fractional functional of space-energy distributions ofneutron flux 119899(119903 119864) and its adjoint function 119899

+(119903 119864)

Λprt equiv⟨119899 119899

+⟩

⟨]119891Σ

119891sdot V sdot 119899 sdot 119899

+⟩

(C1)


Table 5 Main deposits of uranium thorium and mixed uranium-thorium ores Elemental compositions of minerals and isotopecompositions of radiogenic lead

Deposit UThPb (wt) 204Pb206Pb207Pb208Pb (at) Age (times109 yr)Monazite Guarapari Brazil 1359315 0005603046935 052 to 055Monazite Manitoba Canada 0315615 001102186879 183 to 318Monazite Mt Isa Mine Australia 0057303 0038544097936 100 to 119Monazite Las Vegas Nevada 0193904 0025907113898 077 to 173Uraninite Singar Mine India 6438189 mdash894644418 0885Monazite South Bug Ukraine 0287209 001604094930 18 to 20

С R

r

Core Reflector

Figure 9 Two-zone model of nuclear reactor

where the brackets ⟨sdot⟩ designate integration on space andenergy intervals

Space-energy distributions of neutron flux and its adjointfunction are defined by the following operator equations

119871119899 =

1

119870efsdot

119876119899

119871

+119899

+=

1

119870efsdot

119876

+119899

+

(C2)

where operator

119871 describes neutron transport absorptionand scattering and operator

119876 describes neutron multiplica-tion by fission reactions

As applied to the two-zone model under considerationhere (the reactor core surrounded by the neutron reflector)the following designations will be used below the bracket⟨sdot⟩

119862means integration on full volume of the reactor core the

bracket ⟨sdot⟩119877means integration on full volume of the neutron

reflector and the bracket ⟨sdot⟩

119862+119877means integration on full

volume of the reactor (core plus reflector)Two problems must be solved to evaluate prompt neu-

tron lifetimes in the reactor zones The first problem con-sists in solving (C3) and (C4) to determine space-energy

distributions of neutron flux 119899

119862+119877(119903 119864) and its adjoint func-

tion 119899

+

119862+119877(119903 119864) in two-zone reactor

119871

119862+119877119899

119862+119877=

1

119870efsdot

119876119899

119862+119877 (C3)

119871

+

119862+119877119899

+

119862+119877=

1

119870efsdot

119876

+119899

+

119862+119877 (C4)

The second problem is to determine space-energy distribu-tion of neutron flux 119899

119862(119903 119864) in the reactor core only that is

without the neutron reflector by solving (C5)

119871

119862119899

119862=

1

119870efsdot

119876119899

119862+119877 (C5)

Solution of (C5) for the bare core 119899

119862(119903 119864) characterizes

those neutrons which being generated in the reactor coredo not escape the core Solution of (C3) for the reflectedcore 119899

119862+119877(119903 119864) characterizes those neutrons which being

generated in the reactor core can escape or stay in thecore thus populating both reactor zones Their differenceΔ119899

119862+119877= (119899

119862+119877minus 119899

119862) characterizes those neutrons which

being generated in the reactor core escaped the core wentinto the neutron reflector and then in the migration processpopulated both reactor zones (Figure 10)

Some fraction of those neutrons which being generatedin the reactor core escaped the core went into the neutronreflector and then came back into the reactor core couldcontribute to time-dependent evolution of the chain fissionreaction Just these neutrons can be characterized by arelatively longer lifetime due to the lengthy processes oftheir diffusion scattering and slowing down in the weaklyabsorbing neutron reflector due to the processes of theircoming back to the reactor core and contributing into thechain fission reaction

Equation (C1) that determines mean prompt neutronlifetime can be rewritten for the two-zone model as follows

Λprt equiv⟨119899

119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C6)

The space-energy distribution of neutron flux 119899

119862+119877can be

replaced by 119899

119862+119877= 119899

119862+ Δ119899

119862+119877 After some simple algebraic


C R

r

n C+R

n C

Δn C+R

Figure 10 Model spatial distributions of neutron flux in thereflected core 119899

119862+119877(119903) in the bare core 119899

119862(119903) and their difference

Δ119899

119862+119877(119903)

operations the following expression for mean prompt neu-tron lifetime can be obtained

Λprt =⟨(119899

119862+ Δ119899

119862+119877) 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

=

⟨119899

119862 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

+

⟨Δ119899

119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C7)

If the first summand of formula (C7) is divided by andmulti-plied on ⟨]

119891Σ

119891sdot V sdot 119899

119888sdot 119899

+

119862+119877⟩

119862 the summand transforms into

the following form

⟨119899

119862 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

=

⟨119899

119862 119899

+

119862+119877⟩

119862

⟨]119891Σ

119891sdot V sdot 119899

119862sdot 119899

+

119862+119877⟩

119862

sdot

⟨]119891Σ

119891sdot V sdot 119899

119862sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C8)

Note the integration range of the first fraction is shortenedto the reactor core while the integration range of the secondfraction is widened up to the full reactor volume This maybe done because in both cases the integration is performedwith weight of neutron flux in the bare core 119899

119862(119903 119864) that is

equal to zero in the neutron reflector (Figure 10) As it canbe seen the first fraction is a mean prompt neutron lifetimein the reactor core that is mean lifetime of those neutronswhich did not escape the reactor core

Λ

119862equiv

⟨119899

119862 119899

+

119862+119877⟩

119862

⟨]119891Σ

119891sdot V sdot 119899

119862sdot 119899

+

119862+119877⟩

119862

(C9)

The second fraction is a ratio between the value of fissionneutrons produced by those neutrons which did not escape

the reactor core and the value of fission neutrons producedby all neutrons including those neutrons which escaped thereactor core went into the neutron reflector then came backinto the reactor core and contributed into the chain fissionreaction As 119899

119862= (119899

119862+119877minus Δ119899

119862+119877) then after some simple

algebraic operations the following formula can be obtained

⟨]119891Σ

119891sdot V sdot [119899

119862+119877minus Δ119899

119862+119877] sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

= 1 minus

⟨]119891Σ

119891sdot V sdot Δ119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C10)

The second term in the right part is a ratio between the valueof fission neutrons produced by those neutrons which cameback from the neutron reflector into the reactor core and thevalue of fission neutrons produced by all neutrons This isin essence a contribution into total reactivity 120588

119877from those

neutrons which came back from the neutron reflector

120588

119877=

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C11)

So the first summand of formula (C7) can be written in thefollowing form

⟨119899

119862 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

= Λ

119862sdot (1 minus 120588

119877) (C12)

If the second summand of formula (C7) is divided by andmultiplied on ⟨]

119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877 then the sum-

mand transforms into the following form

⟨Δ119899

119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

=

⟨Δ119899

119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

sdot

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C13)

The first fraction in the right part is a mean lifetime of thoseneutrons which escaped the reactor core came back andcontributed into the chain fission reaction

Λ

119877equiv

⟨Δ119899

119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C14)

The second fraction is a share of those neutrons whichescaped the reactor core and came back in total value offission neutrons In other words this is a contribution intototal reactivity caused by the presence of the neutron reflectorsurrounding the reactor core

Now we can formulate the balance relationship for two-zone reactor model The relationship can link mean prompt


neutron lifetime Λprt in the reactor as a whole with meanlifetime of those neutrons which did not escape the reactorcore (Λ

119862) with lifetime of those neutrons which escaped

the reactor core went into the neutron reflector came backinto the reactor core and contributed into the chain fissionreaction (Λ

119877) and with contribution of the neutron reflector

into total reactivity 120588

119877

Λprt = Λ

119862sdot (1 minus 120588

119877) + Λ

119877sdot 120588

119877 (C15)

The balance relationship can be used for example to deter-mine mean lifetime of those neutrons which visited theneutron reflector Λ

119877

If the neutron reflector is divided into several annularlayers then the approach presented above can be appliedto determine mean lifetime of those neutrons which visitedeach layer of the reflectorThus dependency ofmean neutronlifetime on depth of neutron penetration into the reflectorcan be calculated Then the approach can be generalizedfor continuous model of the neutron reflector divided intoinfinite number of infinitesimally thin annular layers In thiscase the balance relationship for (119895 + 1) layer can be writtenin the following recurrent form

Λ

119895+1

prt = Λ

119895

prt sdot (1 minus 120588

119895+1

119877) + Λ

119895+1

119877sdot 120588

119895+1

119877 (C16)

D Roots of the Inverse-Hour Equation forthe Fast Reactor Core Surrounded by theNeutron Reflector-Converter

The inverse-hour equation for the fast reactor core sur-rounded by the neutron reflector-converter can be written inthe following form with accounting for six groups of delayedneutrons

120588 =

Λ

119862

119879

+ int

Λmax

Λmin

Λ

1015840

119877sdot (119889120588

119877119889Λ

1015840

119877)

Λ

1015840

119877+ 119879

sdot 119889Λ

1015840

119877+

6

sum

119894=1

120573

119894

1 + 120582

119894sdot 119879

(D1)

In this equation the lower limit of integration can be takenas Λmin asymp Λ

119862 As for the upper limit of integration Λmax it

depends on thickness and structure of the neutron reflectorFor the BREST-type fast reactor core surrounded by 208Pbreflector the upper limit Λmax can be taken from the curve120588

119877(Λ

119877) shown in Figure 2

For this case the roots of the inverse-hour equation (D1)are presented in Figure 11(b) For comparison the roots of theinverse-hour equation for the ldquotraditionalrdquo (in a certain sense)fast reactor core surrounded by the fertile fuel breeding zoneare shown in Figure 11(a)

As the neutron reflector-converter adjoins closely tothe reactor core neutron lifetime in the nearest reflectorlayer differs slightly from Λ

119862(slightly longer because some

neutrons can be reflected from the nearest layer they undergoonly a few collisions before their return to the reactor core)As is seen the roots of the inverse-hour equation for the fastreactor core surrounded by the neutron reflector-convertercovered the gap between Λ

119862and the roots related with six

groups of delayed neutrons

0 Tas TTcT1 T2 T3 T4 T5 T6

(a) Fast reactor with fertile fuel breeding zone

0 TT1 T2 T3 T4 T5 T6 Tmax Tc Tas

(b) Fast reactor with neutron reflector-converter

Figure 11 Roots of the inverse-hour equation for two types ofthe reflected fast reactor core (119879asmdashroot corresponding to theasymptotic power excursion period)

D1 On Applicability of the ldquoZero Prompt Neutron Life-timerdquo Approximation to Solving the Kinetic Equations forthe Fast Reactor Core Surrounded by the Neutron Reflector-Converter Theneutron kinetics equations for the fast reactorcore surrounded by the neutron reflector-converter (withoutaccounting for any feedback effects) can be written in thefollowing form

119889119899 (119905)

119889119905

=

120588

0minus 120573 minus 120588

119877(Λmax)

Λ

119862

sdot 119899 (119905)

+ int

Λmax

Λ119862

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

= 120573

119894sdot

119899 (119905)

Λ

119862

minus 120582

119894sdot 119862

119894(119905) 119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

=

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

minus

1

Λ

119877

sdot 119862

119877(Λ

119877 119905)

119889120588

119877

119889Λ

119877

=

119889120588

119877(119903)

119889119903

sdot

119889119903

119889Λ

119877

(D2)

As is well-known prompt neutron lifetime in a fast reactor(FR) core is very short (Λ

119862asymp 10

minus7divide 10

minus6 s) while meanlifetime of delayed neutrons covers the range from 03 s to80 s that is prompt neutron lifetime is shorter than delayedneutron lifetime by 6ndash8 orders of magnitude When anaccidental power excursion occurs there are two componentsof the time-dependent process in fast reactors fast-actingcomponent and slow-acting one Numerical solution of sorigid equations is a very complicated task because correctintegration of the fast-acting component and in addition theslow-acting component requires ultra-large number of ultra-short time steps that can lead to a significant loss of accuracyUnder these conditions the ldquozero prompt neutron lifetimerdquoapproximation can be very helpful to overcome the difficulty

Unfortunately one else problem takes place in numericalanalysis of neutron kinetics in the fast reactor core sur-rounded by the neutron reflector-converter The matter isthe neutrons coming back from the reflector into the reactorcore and contributing into the power excursion process arecharacterized by lifetimes that continuously cover the rangefrom Λ

119862up to nearly the very short-lived (sixth) group


of delayed neutrons (Figure 11) In other words from puremathematical point of view it is impossible to define the fastreactor core surrounded by the neutron reflector-converter asa two-scale system with only two time-dependent fast-actingand slow-acting components

The following specific feature must be also taken intoconsideration The feature is related with time dependencyof the integrand in the second summand of the inverse-hour equation (D1) The integrand defines the contributionof the neutrons coming back from the reflector with variouslifetimes Λ

119877into the reactivity gain (Figure 5) The time

dependency allowed us to conclude that the contributions ofshort-lived neutrons coming back from the reflector into thereactivity gain needed to provide the power excursion with agiven asymptotic period 119879 (eg within the range from 001s to 10 s) would be relatively small This circumstance opensan opportunity to obtain a sufficiently correct upper evalu-ation of the power excursion by introducing the followingadditional assumption Let us assume that if the neutronscoming back from the reflector are characterized by lifetimesshorter than a certain value (Λ

0= 10

minus4 s for instance) thena new lifetime namely neutron lifetime in the reactor coreΛ

119862 is given to these neutrons Under these assumptions the

neutron kinetics equations can be written as follows

119889119899 (119905)

119889119905

=

120588

0minus 120573 minus [120588

119877(Λmax) minus 120588

119877(Λ

0)]

Λ

119862

sdot 119899 (119905)

+ int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

= 120573

119894sdot

119899 (119905)

Λ

119862

minus 120582

119894sdot 119862

119894(119905) 119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

=

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

minus

1

Λ

119877

sdot 119862

119877(Λ

119877 119905)

(D3)

The inverse-hour equation that links the reactivity gainwith the required asymptotic power excursion period 119879

transforms into the following form

120588 =

Λ

119862

119879

+ int

Λmax

Λ 0

Λ

1015840

119877sdot (119889120588

119877119889Λ

1015840

119877)

Λ

1015840

119877+ 119879

sdot 119889Λ

1015840

119877+

6

sum

119894=1

120573

119894

1 + 120582

119894sdot 119879

(D4)

The roots of the equation can be presented in Figure 12By using the standard mathematical operations (de-

scribed in [1] eg) the ldquozero prompt neutron lifetimerdquoapproximation can be obtained for system of equations (D3)

119899 (119905)

Λ

119862

=

(minus1)

120588

0minus 120573 minus [120588

119877(Λmax) minus 120588

119877(Λ

0)]

sdot [int

Λmax

Λ 0

1

Λ

1015840

119877

119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894119862

119894(119905)]

0 Tas TTcT1 T2 T3 T4 T5 T6 Tmax

(a) FR with the neutron reflector-converter

0 Tas TTcT1 T2 T3 T4 T5 T6 Tmax T0

(b) FR with the neutron reflector-converter (the neutronscoming back from the reflector with lifetimes Λ119877 lt Λ0 areunited with the neutrons in the reactor core)

0 Tas TT0T1 T2 T3 T4 T5 T6 Tmax

(c) Item ldquobrdquo (plus the ldquozero prompt neutron lifetimerdquo in thereactor core)

Figure 12 Roots of the inverse-hour equation for FR with the neu-tron reflector-converter (119879asmdashroot corresponding to the asymptoticpower excursion period)

119889119862

119894(119905)

119889119905

+ 120582

119894sdot 119862

119894(119905) = 120573

119894sdot

119899 (119905)

Λ

119862

119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

+

1

Λ

119877

sdot 119862

119877(Λ

119877 119905) =

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

(D5)

The transformed system (D5) can be solvedmuch easier thaninitial system (D3) because of the following reasons

(i) system (D5) does not contain differential equation forneutron flux 119899(119905)

(ii) time scale (lifetimes of neutrons and emitters ofdelayed neutrons) now covers a relatively longer range(from 10minus4 s to 10 s)

Solution of system (D5) can be found by using an iterativeprocess which is simplified by the fact that the second andthird equations can be solved analytically

119889119910

119889119905

+ 120582 sdot 119910 = 119860 (119905)

119910 (119905) = 119910 (0) sdot 119890

minus120582sdot119905+ int

119905

0

119889119905

1015840sdot 119890

minus120582(119905minus1199051015840)sdot 119860 (119905

1015840)

(D6)

The following feature in time dependency of neutron flux 119899(119905)must be noted At 119905 = 0 neutron flux undergoes an ordinaryjump that is 119899(119905 = +0) = 119899(119905 = minus0) + Δ119899(119905 = 0) where119905 = plusmn0 means 119905 = plusmn120576 with infinitesimal positive 120576 rarr 0At the same time concentrations of delayed neutron emittersremain continuous functions that is 119862

119894(119905 = minus0) = 119862

119894(119905 =

+0)As to the function 119862

119877(Λ 119905) defining pseudo neutron

emitters in the reflector taking into account relatively rapidmigration of prompt fission neutrons from the reactor coreinto the reflector (the migration time is usually shorter than10minus4 s [27]) initial condition for the function 119862

119877(Λ 119905) must


be formed quite similarly to initial condition for neutron fluxwith an ordinary jump at 119905 = 0

119862

119877(Λ

119877 119905 = +0)

119862

119877(Λ

119877 119905 = minus0)

=

119899 (119905 = +0)

119899 (119905 = minus0)

(D7)

D2 Feedbacks Accountability Depending on the powerexcursion rate some feedbacks can obtain a long enough timeinterval to change evolution of the process Doppler-effectfor example can change resonance neutron absorption at fuelwarming up instantly On the contrary some time intervalmust elapse before the coolant density effect caused by itswarming up actuates (heat must have a time to flow downfrom fuel to coolant) Thermal constant of fuel rods 120591therm isusually applied as a time parameter defining the process ofheat flowing down [16 28]

Thermal constant 120591therm of cylindrical fuel rodswith oxidefuel meat depends on their diameter and can vary withinthe range of 1 s divide 3 s [16] In the case of metal uranium fuelthermal constant 120591therm shortens by one order of magnitude(120591therm asymp 01 s) Thus heat from metal fuel rods flows downsubstantially quicker thanks to the better heat conductivity ofmetal fuel The shortest thermal constants can be achieved inmicro coated-fuel particles dispersed in inert matrix (120591therm =

001 s divide 003 s) [29] The concept of micro fuel particlesis under development now for advanced light-water powerreactors

So it can be a priori stated that if the asymptotic period119879 of the power excursion caused by positive reactivity jumpbelongs to the time range of 001 s divide 10 s then the coolant-induced reactivity feedback has a long enough time intervalto influence remarkably on the power excursion processThis in its turn allows us to consider the question whatpositive reactivity jumps are acceptable for the feedbackscould actuate

D3 Fast Power Excursion with Accounting for Doppler-Effectin Fuel (No Heat Flows Down to Coolant) Neutron kineticsequations in the case of fast power excursion with accountingfor Doppler-effect caused by fuel warming up can be writtenin the following form for the FR core surrounded by theneutron reflector-converter

119889119899 (119905)

119889119905

=

120588

0minus 120573 minus [120588

119877(Λmax) minus 120588

119877(Λ

0)] minus 120572

119879int

119905

0119899 (119905

1015840) sdot 119889119905

1015840

Λ

119862

sdot 119899 (119905)

+ int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

+ 120582

119894sdot 119862

119894(119905) = 120573

119894sdot

119899 (119905)

Λ

119862

119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

+

1

Λ

119877

sdot 119862

119877(Λ

119877 119905) =

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

(D8)

where 1205880is initial reactivity jump int119905

0119899(119905

1015840)119889119905

1015840 is heat accumu-lated in fuel by time moment 119905 120572

119879is reactivity coefficient of

Doppler-effectAccording to the well-known Nordheim-Fuchs-Hansen

(NFH) model [1] thermal energy yield from the promptpower excursion with accounting for the heat accumulationin fuel and actuation of the reactivity feedback causedby Doppler-effect can be evaluated by using the followingequation

119889119899 (119905)

119889119905

=

120588

0minus 120573 minus 120572

119879int

119905

0119899 (119905

1015840) sdot 119889119905

1015840

Λ

119862

sdot 119899 (119905) (D9)

Themodel presented here differs from theNFH-model by thefollowing features

(i) Effective fraction of delayed neutrons 120573 is replaced bythe larger value

120573 = 120573 + [120588

119877(Λmax) minus 120588

119877(Λ

0)] thanks

to the neutrons coming back from the reflector intothe reactor core

(ii) The right part of the first equation contains newsummands which define the neutrons coming backfrom the reflector and delayed neutrons from decaysof their emitters These summands must be addedbecause the time range for the procrastinated returnof neutrons from the reflector stretches from neutronlifetime in the reactor core up to lifetime of the mostshort-lived group of delayed neutrons

(iii) System of neutron kinetics equations must containequations for the neutrons coming back from thereflector and for delayed neutrons

The approach described in [1] for solving the NFH-equations can be also used to transform the first equation ofsystem (D8) for time evolution of neutron flux 119899(119905) At firstthe following designation is introduced and differentiatedwith respect to time

119910 (119905) equiv [120588

0minus

120573 minus 120572

119879int

119905

0

119899 (119905

1015840) sdot 119889119905

1015840]

119889119910 (119905)

119889119905

= minus120572

119879sdot 119899 (119905)

119910 (119905) sdot

119889119910 (119905)

119889119905

= minus (120572

119879sdot Λ

119862) sdot [

119910 (119905)

Λ

119862

sdot 119899 (119905)]

(D10)

Then application of the ldquozero prompt neutron lifetimerdquoapproximation to the first equation of system (D8) results inthe following integral equation for (119910(119905)Λ

119862) sdot 119899(119905)

[

119910 (119905)

Λ

119862

sdot 119899 (119905)]

= minusint

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

(D11)


Minus in the right part means that function 119910(119905) is negativethat is

120588

0lt

120573 = 120573 + [120588

119877(Λmax) minus 120588

119877(Λ

0)] (D12)

In other words initial reactivity jump must not exceedfraction of the neutrons which can contribute to the chainfission reaction after a certain time delay only These neu-trons include the following two components delayed (in atraditional sense) neutrons and prompt (by their origination)neutrons coming back from the reflector with a certain timedelay If expression (D11) is substituted into (D10) then

119910 (119905) sdot

119889119910 (119905)

119889119905

= + (120572

119879sdot Λ

119862)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

(D13)

Some simple mathematical transformations of the left partallowed us to find solution of (D13) in quadratures

1

2

119889119910

2

119889119905

= + (120572

119879sdot Λ

119862)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119910

2(119905) = 2 sdot (120572

119879sdot Λ

119862)

times int

119905

0

119889119905

1015840int

Λmax

Λ 0

1

Λ

1015840

119877

119862

119877(Λ

1015840

119877 119905

1015840) sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905

1015840) + const

(D14)

Initial condition 119910

2(0) = (120588

0minus

120573)

2

can be used to obtain thefollowing relationship const = (120588

0minus

120573)

2

Then

119910 (119905) = minus2 sdot (120572

119879sdot Λ

119862)

times int

119905

0

119889119905

1015840[int

Λmax

Λ 0

1

Λ

1015840

119877

119862

119877(Λ

1015840

119877 119905

1015840) 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894119862

119894(119905

1015840)] + (120588

0minus

120573)

2

12

(D15)

Now we obtained the following equations for time evolutionof neutron flux emitters of delayed neutrons and pseu-doemitters of the neutrons coming back from the reflector

[

119899 (119905)

Λ

119862

] =

(minus1)

119910 (119905)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

+ 120582

119894sdot 119862

119894(119905) = 120573

119894sdot [

119899 (119905)

Λ

119862

] 119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

+

1

Λ

119877

sdot 119862

119877(Λ

119877 119905) =

119889120588

119877

119889Λ

119877

sdot [

119899 (119905)

Λ

119862

]

(D16)

This system of integral and differential equations can betransformed into system of integral equations by usinganalytical expression (D6)

[

119899 (119905)

Λ

119862

] =

(minus1)

119910 (119905)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119862

119894(119905) = 119862

119894(119905 = +0) sdot 119890

minus120582119894119905

+ 120573

119894sdot int

119905

0

119889119905

1015840sdot 119890

minus120582119894(119905minus1199051015840)sdot [

119899 (119905

1015840)

Λ

119862

]

119862

119877(Λ

119877 119905) = 119862

119877(Λ

119877 119905 = +0)

sdot 119890

minus119905Λ 119877+

119889120588

119877

119889Λ

119877

sdot int

119905

0

119889119905

1015840sdot 119890

minus(1Λ 119877)(119905minus1199051015840)

sdot [

119899 (119905

1015840)

Λ

119862

]

(D17)

This nonlinear system of integral equations can be solved byan iterative method

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] G J Bell and S Glasstone Nuclear Reactor Theory VanNostrand Reinhold New York NY USA 1970

[2] A N Shmelev G G Kulikov V B Glebov D F Tsurikov andA G Morozov ldquoSafety in a fast burning reactor for long-livedactinides extracted from radioactive wastesrdquo Atomic Energyvol 73 no 6 pp 963ndash966 1992

[3] A N Shmelev G G Kulikov V A Apse V B Glebov D FTsurikov and A G Morozov ldquoRadiowaste transmutation in


nuclear reactorsrdquo in Proceedings of the Specialists Meeting Useof Fast Reactors for Actinide Transmutation IAEA-TECHDOC-693 pp 77ndash86 IAEA Obninsk Russia September 1993

[4] G G Kulikov V A Apse E G Kulikov A N Shmelev and VA Yufereva ldquoOn nuclear-physical potential of radiogenic leadrdquoin Proceedings of the 15th Seminar Volga Actual Problems ofNuclear Reactors Physics-Efficiency Safety Nonproliferation pp141ndash144 Moscow Russia September 2008

[5] A N Shmelev G G Kulikov V A Apse E G Kulikov and VV Artisyuk ldquoRadiogenic lead with dominant content of 208Pbnew coolant and neutron moderator for innovative nuclearfacilitiesrdquo Science and Technology of Nuclear Installations vol2011 Article ID 252903 12 pages 2011

[6] G G Kulikov A N Shmelev V A Apse and E G KulikovldquoLead containing isotope Pb-208mdashheavy neutron moderatorand reflector Its neutron-physical propertiesrdquo in Proceedings ofthe Scientific Session of NRNU MEPhI vol 1 p 49 MoscowRussia January 2011

[7] A N Shmelev E G Kulikov G G Kulikov and V A ApseldquoLead containing mainly isotope 208Pbmdasha neutron modera-tor coolant and reflector of neutrons Its neutron-physicalpropertiesrdquo in Proceedings of the 10th International Conferencetoward and over the Fukushima Daiichi Accident (GLOBAL rsquo11)Makuhari Japan December 2011

[8] V V Orlov A I Filin A V Lopatkin et al ldquoThe closed on-sitefuel cycle of the BREST reactorsrdquo Progress in Nuclear Energyvol 47 no 1ndash4 pp 171ndash177 2005

[9] V F KolesovAperiodical Pulsed Reactors RFNC-VNIIEF SarovRussia 2007

[10] G D Spriggs and R D Bush ldquoReflected reactors point kineticsand prompt criticalrdquo in Proceedings of the Topical Meeting onPhysics Safety and Applications of Pulse Reactors pp 265ndash273Washington DC USA November 1994

[11] G D Spriggs R D Bush and J G Williams ldquoTwo-regionkinetic models for reflected reactorsrdquo Annals of Nuclear Energyvol 1 no 3 1996

[12] C E Cohn ldquoReflected-reactor kineticsrdquo Nuclear Science andEngineering vol 13 no 1 1962

[13] D P Gamble ldquoThe effect of reflector-moderated neutrons onthe kinetics of the kinetic experimentwater boilerrdquoTransactionsof American Nuclear Society vol 3 no 1 p 122 1960

[14] R L Coats and R L Long ldquoReflector and decoupling exper-iments with fast burst reactorsrdquo in Proceedings of the NationalTopicalMeeting on Fast Burst Reactors Albuquerque NMUSAJanuary 1969

[15] R Avery ldquoTheory of decoupled reactorsrdquo in Proceedings of the2nd International Conference Peaceful Uses of Atomic Energyvol 12 pp 182ndash191 Geneva Switzerland 1958

[16] H H Hummel and D Okrent Reactivity Coefficients in LargeFast Power Reactors AmericanNuclear Society LaGrange ParkIll USA 1970

[17] K Shibata O Iwamoto T Nakagawa et al ldquoJENDL-40 a newlibrary for nuclear science and engineeringrdquo Journal of NuclearScience and Technology vol 48 no 1 pp 1ndash30 2011

[18] I S Grigoriev and E Z Meylikhov Physical Values ReferenceBook Energoatomizdat Moscow Russia 1991

[19] A D Galanin Introduction to a Theory of Thermal NuclearReactors Energoatomizdat Moscow Russia 1990

[20] J M GodoyM L D P Godoya and C C Aronne ldquoApplicationof inductively coupled plasma quadrupole mass spectrometryfor the determination of monazite ages by lead isotope ratiosrdquo

Journal of the Brazilian Chemical Society vol 18 no 5 pp 969ndash975 2007

[21] A O Nier R W Thompson and B F Murphey ldquoThe isotopicconstitution of lead and the measurement of geological timerdquoPhysical Review vol 60 no 2 pp 112ndash116 1941

[22] A Holmes ldquoThe pre-cambrian and associated rocks of thedistrict of Mozambiquerdquo Quarterly Journal of the GeologicalSociety vol 74 no 1ndash4 pp 31ndash98 1918

[23] T C Sarkar ldquoThe lead ratio of a crystal of monazite fromthe Gaya district Biharrdquo Proceedings of the Indian Academy ofSciences Section A vol 13 no 3 pp 245ndash248 1941

[24] Catalog of Isotope Dates for Minerals of Ukrainian ShieldNaukova Dumka Kiev Ukraine 1978

[25] N M Sinev Economics of Nuclear Power EnergoatomizdatMoscow Russia 1987

[26] J A Seneda C A L G de O Forbicini C A D S Queiroz etal ldquoStudy on radiogenic lead recovery from residues in thoriumfacilities using ion exchange and electrochemical processrdquoProgress in Nuclear Energy vol 52 no 3 pp 304ndash306 2010

[27] K H Beckurts and K Wirtz Neutron Physics Springer BerlinGermany 1964

[28] I A Kuznetsov and V M Poplavsky Safety of NPP with FastReactors IZDAT Moscow Russia 2012

[29] N N Ponomarev-Stepnoj N E Kukharkin A A Khrulev etal ldquoProspects of coated fuel particle application in the WWERreactorsrdquo Atomnaya Energiya vol 86 no 6 pp 443ndash449 1999

TribologyAdvances in

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International Journal of

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FuelsJournal of


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Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

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Journal of


Renewable Energy

Submit your manuscripts athttpwwwhindawicom


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RotatingMachinery


EnergyJournal of


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal ofPhotoenergy


Nuclear InstallationsScience and Technology of


Solar EnergyJournal of


Wind EnergyJournal of


Nuclear EnergyInternational Journal of


High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


power excursion with asymptotic time period 119879 (withoutany feedback effects) in the fourth-generation lead-cooledfast BREST-type [8] reactor (neutron reflectormdash50 cm thicknatural lead) in thermal VVER-type and CANDU-typereactors as well as in two hypothetical reactors whose promptneutron lifetimes are equal to 001 s and 01 s respectively

These curves demonstrate that the shorter lifetime ofprompt neutrons results in the faster power excursion atthe same reactivity jump It seems helpful to reformulatethe statement as follows If prompt neutron lifetime becamelonger then the power excursion with a certain asymptotictime period 119879 would require the larger reactivity jumpOtherwise the power excursionwill be slower Consequentlythe reactor safety can be enhanced by making lifetime ofprompt neutrons longer

For example based on one-point model of neutronkinetics it may be concluded that application of thick 208Pbreflector in the fast reactor BREST allowed us to reachprompt neutron lifetime of sim1ms If 1-$ reactivity jump ()occurs in the fast reactor then its power increases with theexcursion period nearly 1 s instead of 14ms in the fast reactorreflected by natural lead If prompt neutron lifetime in thefast reactor BREST reflected by thick 208Pb layer is prolongedup to 10ms then the power excursion period will be longerthan 1 s even at the reactivity jumps up to 2$ So long thepower excursion periods give sufficient time for coolant toremove the heat from fuel rodsThis means that feedbacks oncoolant temperature and coolant density could be apparentlyactuated

3 One-Point Model

The dependencies shown in Figure 1 were calculated withthe application of one-point kinetic model with six groupsof delayed neutrons No feedback effects were taken intoconsiderationThe one-pointmodel is based on the followingset of equations

119889119899 (119905)

119889119905

=

120588 minus 120573

Λprtsdot 119899 (119905) +

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

=

120573

119894

Λprtsdot 119899 (119905) minus 120582

119894sdot 119862

119894(119905) (119894 = 1 2 6)

(1)

where 119899 is neutron population in the reactor core 119862

119894is

quantity of nuclei-emitters for 119894th group of delayed neutronsin the reactor core 120588 = (119896ef minus 1)119896ef reactivity where 119896efis the effective neutron multiplication factor in the reactorincluding neutron reflector 120573 is effective fraction of delayedneutrons 120573

119894is effective fraction of delayed neutrons in 119894th

group Λprt mean prompt neutron lifetime (according to theterminology applied by Bell and Glasstone [1]) in the reactorincluding neutron reflector 120582

119894is decay constant of nuclei-

emitters in 119894th group of delayed neutrons Prompt neutronlifetime Λprt and generation time 120591 are linked by the rela-tionship 120591 = Λprt119896ef Since only critical andnearly critical

100

1610

25

1

1Asymptotic period (s)

1001

01001

Reactorneutron life time

Requ

ired

reac

tivity

($) CANDU0001 s

VVER10minus4 s

BREST

01 s

001 s

6 s05 middot 10minus6

Figure 1 Dependencies of the reactivity jump required to providethe power excursion with asymptotic time period 119879 (one-pointmodel no feedback effects)

(within some dollars from criticality) states are analyzedhere lifetime and generation time differ insignificantlyTheseterms are regarded below as being equivalent

Within the frames of one-point kinetic model (1) theinverse-hour equation that links the asymptotic period of thepower excursion119879with the inserted positive reactivity 120588maybe written in the following form

120588 =

Λprt

119879

+

6

sum

119894=1

120573

119894

1 + 120582

119894sdot 119879

(2)

4 Two-Point Model

As has been noted in [9] one-point model is not completelyapplicable for analysis of neutron kinetics in the reactor coressurrounded by neutron reflector The associated problemswere analyzed in [9ndash15]with application ofmultipointmodelsor with introduction of some additional (fictive) groups ofdelayed neutrons The latter case takes into account thefollowing new parameters time of neutron staying in thereactor core before leakage time for neutron transport fromthe reactor core to neutron reflector time of neutron stayingin the reflector time for neutron transport from the reflectorinto the reactor core and time from neutron arrival into thereactor core to initiation of fission reaction by this neutron Inthe case under consideration here neutron reflector is placedin the immediate vicinity to the reactor core and the reflectoris thick from physical point of view This means that timeof neutron staying in the reflector plays a dominant role incompetition with all other times If one additional group ofdelayed neutrons namely slow neutrons coming back fromthe reflector into the reactor core is introduced to study



119889119899 (119905)

119889119905

=

120588 minus 120573 minus 120588

119877

Λ

119862

sdot 119899 (119905) +

6

sum

119894=1

120582

119894sdot 119862

119894(119905) + 120582

119877sdot 119862

119877(119905)

119889119862

119894(119905)

119889119905

=

120573

119894

Λ

119862

sdot 119899 (119905) minus 120582

119894sdot 119862

119894(119905) (119894 = 1 2 6)

119889119862

119877(119905)

119889119905

=

120588

119877

Λ

119862

sdot 119899 (119905) minus 120582

119877sdot 119862

119877(119905) 120582

119877=

1

Λ

119877

(3)





= (119896ef minus 119896

119862




119862is prompt








120588 =

Λ

119862

119879

+

120588

119877

1 + 119879Λ

119877

+

6

sum

119894=1

120573

119894

1 + 120582

119894sdot 119879

(4)





reflector Λ




119877) sdot Λ

119862+ 120588

119877sdot Λ

119877 (5)









5 Multipoint Model


119889119899 (119905)

119889119905

=


119869

119895=1120588

119895

119877

Λ

119862

sdot 119899 (119905) +

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

+

119869

sum

119895=1

120582

119895

119877sdot 119862

119895

119877(119905)


119889119862

119894(119905)

119889119905

=

120573

119894

Λ

119862

sdot 119899 (119905) minus 120582

119894sdot 119862

119894(119905)

(119894 = 1 2 6 119895 = 1 2 119869)

119889119862

119895

119877(119905)

119889119905

=

120588

119895

119877

Λ

119862

sdot 119899 (119905) minus 120582

119895

119877sdot 119862

119895

119877(119905) 120582

119895

119877=

1

Λ

119895

119877

(6)


120588 =

Λ

119862

119879

+

119869

sum

119895=1

120588

119895

119877

1 + 119879Λ

119895

119877

+

6

sum

119894=1

120573

119894

1 + 120582

119894sdot 119879

Λ

119895+1

prt = (1 minus 120588

119895+1

119877) sdot Λ

119895

prt + 120588

119895+1

119877sdot Λ

119895+1

119877

(7)


119895


119877is





119862








119889119899 (119905)

119889119905

=

120588 minus 120573 minus 120588

119877(119903out)

Λ

119862

sdot 119899 (119905)

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905) + int

119903out

119903in

120582

119877(119903

1015840) sdot 119862

119877(119903

1015840 119905) sdot 119889119903

1015840

119889119862

119894(119905)

119889119905

=

120573

119894

Λ

119862

sdot 119899 (119905) minus 120582

119894sdot 119862

119894(119905) (119894 = 1 2 6)

119889119862

119877(119903 119905)

119889119905

=

119889120588

119877(119903) 119889119903

Λ

119862

sdot 119899 (119905) minus 120582

119877(119903) sdot 119862

119877(119903 119905)

120582

119877(119903) =

1

Λ

119877(119903)

(8)









120588 =

Λ

119862

119879

+ int

119903out

119903in

119889120588

119877(119903

1015840) 119889119903

1015840

1 + 119879Λ

119877(119903

1015840)

sdot 119889119903

1015840+

6

sum

119894=1

120573

119894

1 + 120582

119894sdot 119879

(9)

Λ

119877(119903) = Λ prt (119903) +

119889Λ prt (119903) 119889119903

119889120588

119877(119903) 119889119903

(10)






















0 1 2 3 4 5 60

2

4

6

8

0

2

4

6

8

10

Reac

tivity

gai

nca

used

by

the r

eflec

tor (

$)

Diff

eren

tial r

eact

ivity

gai

n ca

used

by

the r

eflec

tor (

$m

)


120588R(r)

d120588R(r)dr

(a)

0 1 2 3 4 5

Diff

eren

tial n

eutro

n lif

etim

e (s

m)

6

0 2 4 6 8 10

ΛR(r)

Λprt

dΛprt (r)dr

1E + 0

1E minus 1

1E minus 2

1E minus 3

1E minus 4

1E minus 5

1E minus 6

1E minus 7

1E + 0

1E minus 1

1E minus 2

1E minus 3

1E minus 4

1E minus 5

1E minus 6

1E minus 7

Neu

tron

lifet

ime (

s)



(b)



119877(0) is longer




119877(119903) and mean





119877(119903) (Figure 2(a))


119877(119903) (Figure 2(b))





120588

119877(Λ


119903out

119903

119889120588

119877(119903

1015840)

119889119903

1015840sdot 119889119903

1015840= int

Λmax

Λ 119877

119889120588

119877

119889Λ

1015840

119877

sdot 119889Λ

1015840

119877

(11)





120588

lowast

119877equiv int

119903out

119903in

119889120588

119877(119903

1015840) 119889119903

1015840

1 + 119879Λ

119877(119903

1015840)

sdot 119889119903

1015840

(12)


0001 001 101

1

3

5

0

2

4

6

6

4

32

5



Pbnat 2

Con

trib

utio

n of

neu

trons

of t

he re

flect

orin

to th

e rea

ctiv

ity g

ain120588

R(Λ

RΛ

max

) ($)

1E minus 4

208Pb




119877(Λ



0 1 2 3 4 5 6

Diff

eren

tial c

ontr

ibut

ion

into

the r

equi

ered

reac

tivity

($m

)

1E + 0

1E + 1

1E minus 1

1E minus 2

1E minus 3

1E minus 4

1E minus 5



001 s244$

01 s096$10 s016$

001 s0036$01 s00037 $

10 s000038 $





119877

int

119903out

119903in

119889120588

119877(119903

1015840) 119889119903

1015840

1 + 119879Λ

119877(119903

1015840)

sdot 119889119903

1015840= int

Λmax

Λmin

119889120588

119877119889Λ

1015840

119877

1 + 119879Λ

1015840

119877

sdot 119889Λ

1015840

119877

119889120588

119877

119889119903

=

119889120588

119877

119889Λ

119877

sdot

119889Λ

119877

119889119903

(13)


0 1 2 3 4 5 6

25

20

15

10

05

00

001

01

10Con

trib

utio

n of

neu

trons

of t

he re

flect

or in

to

the r

equi

red

reac

tivity

($)



(a)

Con

trib

utio

n of

neu

trons

of t

he re

flect

or in

to th

e req

uier

ed re

activ

ity ($

)


25

20

15

10

05

00

001

01

1010010001

10


1E minus 4

(b)


119877)


120588 =

Λ

119862

119879

+ int

Λmax

Λmin

Λ

1015840

119877sdot (119889120588

119877119889Λ

1015840

119877)

Λ

1015840

119877+ 119879

sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120573

119894

1 + 120582

119894sdot 119879

(14)




int

Λmax

Λmin

Λ

1015840

119877sdot (119889120588

119877119889Λ

1015840

119877)

Λ

1015840

119877+ 119879

sdot 119889Λ

1015840

119877=

1

119879

int

Λmax

Λmin

Λ

1015840

119877sdot 119889120588

119877

=

1

119879

int

119903out

119903in

Λ

1015840

119877sdot 119889120588

119877(119903

1015840)

(15)






120588

lowast

119877(119903) = int

119903

119903in

119889120588

119877(119903

1015840) 119889119903

1015840

1 + 119879Λ

119877(119903

1015840)

sdot 119889119903

1015840= int

Λ

Λmin

119889120588

119877119889Λ

1015840

119877

1 + 119879Λ

1015840

119877

sdot 119889Λ

1015840

119877

(16)








9 Conclusions








Appendices








396 6

371 5348 4

320 5

261 3

00 0

234 72

163 32

089 32

057 52

00 12

182 3

156 4

127 4

090 2

238 6

080 2

00 0 00 0

120 0

168 4

200 4

134 3

Ei Ii Ei IiEi Ii

Ei Ii

204 Pb 206 Pb207 Pb 208 Pb

M϶B M϶B M϶B

M϶B


0 2 4

Inel

astic

scat

terin

g cr

oss-

sect

ion

(b)

6 8 10


0

1

2

3

204 Pb

206 Pb207 Pb

208 Pb


D

01

Capt

ure c

ross

-sec

tion

(b)

10 1000

01

10

1E minus 3

1E minus 5

1E + 5 1E + 7

Neutron energy (eV)

204Pb

12C

206Pb

207Pb

208Pb

Pbnat










Nuclide 120590


(01MeVrarr 05 eV) 120590

th(119899120574)

(mb) 119877119868

119899120574+ 1119881 (mb)

1H 301 12 332 1492D 42 17 055 0259Be 65 59 85 3812C 49 77 39 1816O 40 102 019 016Pbnat 113 1269 174 95208Pb 115 1274 023 078



Moderatingratio

120585ΣΣ

119886

th


(cm2)



















2

119864=


119903

2






9 years) 997888rarr

208Pb238U 997888rarr 8120572 + 6120573 (46 sdot 10

9 years) 997888rarr

206Pb235U 997888rarr 7120572 + 4120573 (07 sdot 10

9 years) 997888rarr

207Pb

(B1)











+(119903 119864)

Λprt equiv⟨119899 119899

+⟩

⟨]119891Σ

119891sdot V sdot 119899 sdot 119899

+⟩

(C1)




С R

r

Core Reflector




119871119899 =

1

119870efsdot

119876119899

119871

+119899

+=

1

119870efsdot

119876

+119899

+

(C2)

where operator












tion 119899

+


119871

119862+119877119899

119862+119877=

1

119870efsdot

119876119899

119862+119877 (C3)

119871

+

119862+119877119899

+

119862+119877=

1

119870efsdot

119876

+119899

+

119862+119877 (C4)




119871

119862119899

119862=

1

119870efsdot

119876119899

119862+119877 (C5)


119862(119903 119864) characterizes




119862+119877= (119899

119862+119877minus 119899






119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C6)


119862+119877can be

replaced by 119899

119862+119877= 119899

119862+ Δ119899



C R

r

n C+R

n C

Δn C+R


119862+119877(119903) in the bare core 119899


Δ119899

119862+119877(119903)


Λprt =⟨(119899

119862+ Δ119899

119862+119877) 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

=

⟨119899

119862 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

+

⟨Δ119899

119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C7)


119891Σ

119891sdot V sdot 119899

119888sdot 119899

+

119862+119877⟩


the following form

⟨119899

119862 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

=

⟨119899

119862 119899

+

119862+119877⟩

119862

⟨]119891Σ

119891sdot V sdot 119899

119862sdot 119899

+

119862+119877⟩

119862

sdot

⟨]119891Σ

119891sdot V sdot 119899

119862sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C8)


119862(119903 119864) that is


Λ

119862equiv

⟨119899

119862 119899

+

119862+119877⟩

119862

⟨]119891Σ

119891sdot V sdot 119899

119862sdot 119899

+

119862+119877⟩

119862

(C9)



119862= (119899

119862+119877minus Δ119899



⟨]119891Σ

119891sdot V sdot [119899

119862+119877minus Δ119899

119862+119877] sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

= 1 minus

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C10)


119877from those


120588

119877=

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C11)


⟨119899

119862 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

= Λ

119862sdot (1 minus 120588

119877) (C12)


119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877 then the sum-


⟨Δ119899

119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

=

⟨Δ119899

119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

sdot

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C13)


Λ

119877equiv

⟨Δ119899

119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C14)









119877

Λprt = Λ

119862sdot (1 minus 120588

119877) + Λ

119877sdot 120588

119877 (C15)


119877


Λ

119895+1

prt = Λ

119895


119895+1

119877) + Λ

119895+1

119877sdot 120588

119895+1

119877 (C16)



120588 =

Λ

119862

119879

+ int

Λmax

Λmin

Λ

1015840

119877sdot (119889120588

119877119889Λ

1015840

119877)

Λ

1015840

119877+ 119879

sdot 119889Λ

1015840

119877+

6

sum

119894=1

120573

119894

1 + 120582

119894sdot 119879

(D1)




119877(Λ














119889119899 (119905)

119889119905

=

120588

0minus 120573 minus 120588

119877(Λmax)

Λ

119862

sdot 119899 (119905)

+ int

Λmax

Λ119862

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

= 120573

119894sdot

119899 (119905)

Λ

119862

minus 120582

119894sdot 119862

119894(119905) 119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

=

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

minus

1

Λ

119877

sdot 119862

119877(Λ

119877 119905)

119889120588

119877

119889Λ

119877

=

119889120588

119877(119903)

119889119903

sdot

119889119903

119889Λ

119877

(D2)


119862asymp 10

minus7divide 10









0= 10




119889119899 (119905)

119889119905

=

120588

0minus 120573 minus [120588

119877(Λmax) minus 120588

119877(Λ

0)]

Λ

119862

sdot 119899 (119905)

+ int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

= 120573

119894sdot

119899 (119905)

Λ

119862

minus 120582

119894sdot 119862

119894(119905) 119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

=

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

minus

1

Λ

119877

sdot 119862

119877(Λ

119877 119905)

(D3)



120588 =

Λ

119862

119879

+ int

Λmax

Λ 0

Λ

1015840

119877sdot (119889120588

119877119889Λ

1015840

119877)

Λ

1015840

119877+ 119879

sdot 119889Λ

1015840

119877+

6

sum

119894=1

120573

119894

1 + 120582

119894sdot 119879

(D4)



119899 (119905)

Λ

119862

=

(minus1)

120588

0minus 120573 minus [120588

119877(Λmax) minus 120588

119877(Λ

0)]

sdot [int

Λmax

Λ 0

1

Λ

1015840

119877

119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894119862

119894(119905)]








119889119862

119894(119905)

119889119905

+ 120582

119894sdot 119862

119894(119905) = 120573

119894sdot

119899 (119905)

Λ

119862

119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

+

1

Λ

119877

sdot 119862

119877(Λ

119877 119905) =

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

(D5)





119889119910

119889119905

+ 120582 sdot 119910 = 119860 (119905)

119910 (119905) = 119910 (0) sdot 119890


119905

0

119889119905

1015840sdot 119890

minus120582(119905minus1199051015840)sdot 119860 (119905

1015840)

(D6)


119894(119905 = minus0) = 119862

119894(119905 =




119877(Λ 119905) must



119862

119877(Λ

119877 119905 = +0)

119862

119877(Λ

119877 119905 = minus0)

=

119899 (119905 = +0)

119899 (119905 = minus0)

(D7)






119889119899 (119905)

119889119905

=

120588

0minus 120573 minus [120588

119877(Λmax) minus 120588

119877(Λ

0)] minus 120572

119879int

119905

0119899 (119905

1015840) sdot 119889119905

1015840

Λ

119862

sdot 119899 (119905)

+ int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

+ 120582

119894sdot 119862

119894(119905) = 120573

119894sdot

119899 (119905)

Λ

119862

119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

+

1

Λ

119877

sdot 119862

119877(Λ

119877 119905) =

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

(D8)


0119899(119905

1015840)119889119905





119889119899 (119905)

119889119905

=

120588

0minus 120573 minus 120572

119879int

119905

0119899 (119905

1015840) sdot 119889119905

1015840

Λ

119862

sdot 119899 (119905) (D9)



120573 = 120573 + [120588

119877(Λmax) minus 120588

119877(Λ

0)] thanks





119910 (119905) equiv [120588

0minus

120573 minus 120572

119879int

119905

0

119899 (119905

1015840) sdot 119889119905

1015840]

119889119910 (119905)

119889119905

= minus120572

119879sdot 119899 (119905)

119910 (119905) sdot

119889119910 (119905)

119889119905

= minus (120572

119879sdot Λ

119862) sdot [

119910 (119905)

Λ

119862

sdot 119899 (119905)]

(D10)


119862) sdot 119899(119905)

[

119910 (119905)

Λ

119862

sdot 119899 (119905)]

= minusint

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

(D11)



120588

0lt

120573 = 120573 + [120588

119877(Λmax) minus 120588

119877(Λ

0)] (D12)


119910 (119905) sdot

119889119910 (119905)

119889119905

= + (120572

119879sdot Λ

119862)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

(D13)


1

2

119889119910

2

119889119905

= + (120572

119879sdot Λ

119862)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119910

2(119905) = 2 sdot (120572

119879sdot Λ

119862)

times int

119905

0

119889119905

1015840int

Λmax

Λ 0

1

Λ

1015840

119877

119862

119877(Λ

1015840

119877 119905

1015840) sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905

1015840) + const

(D14)


2(0) = (120588

0minus

120573)

2


0minus

120573)

2

Then

119910 (119905) = minus2 sdot (120572

119879sdot Λ

119862)

times int

119905

0

119889119905

1015840[int

Λmax

Λ 0

1

Λ

1015840

119877

119862

119877(Λ

1015840

119877 119905

1015840) 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894119862

119894(119905

1015840)] + (120588

0minus

120573)

2

12

(D15)


[

119899 (119905)

Λ

119862

] =

(minus1)

119910 (119905)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

+ 120582

119894sdot 119862

119894(119905) = 120573

119894sdot [

119899 (119905)

Λ

119862

] 119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

+

1

Λ

119877

sdot 119862

119877(Λ

119877 119905) =

119889120588

119877

119889Λ

119877

sdot [

119899 (119905)

Λ

119862

]

(D16)


[

119899 (119905)

Λ

119862

] =

(minus1)

119910 (119905)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119862

119894(119905) = 119862

119894(119905 = +0) sdot 119890

minus120582119894119905

+ 120573

119894sdot int

119905

0

119889119905

1015840sdot 119890

minus120582119894(119905minus1199051015840)sdot [

119899 (119905

1015840)

Λ

119862

]

119862

119877(Λ

119877 119905) = 119862

119877(Λ

119877 119905 = +0)

sdot 119890

minus119905Λ 119877+

119889120588

119877

119889Λ

119877

sdot int

119905

0

119889119905

1015840sdot 119890

minus(1Λ 119877)(119905minus1199051015840)

sdot [

119899 (119905

1015840)

Λ

119862

]

(D17)




References





































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of



















119889119899 (119905)

119889119905

=

120588 minus 120573 minus 120588

119877

Λ

119862

sdot 119899 (119905) +

6

sum

119894=1

120582

119894sdot 119862

119894(119905) + 120582

119877sdot 119862

119877(119905)

119889119862

119894(119905)

119889119905

=

120573

119894

Λ

119862

sdot 119899 (119905) minus 120582

119894sdot 119862

119894(119905) (119894 = 1 2 6)

119889119862

119877(119905)

119889119905

=

120588

119877

Λ

119862

sdot 119899 (119905) minus 120582

119877sdot 119862

119877(119905) 120582

119877=

1

Λ

119877

(3)





= (119896ef minus 119896

119862




119862is prompt








120588 =

Λ

119862

119879

+

120588

119877

1 + 119879Λ

119877

+

6

sum

119894=1

120573

119894

1 + 120582

119894sdot 119879

(4)





reflector Λ




119877) sdot Λ

119862+ 120588

119877sdot Λ

119877 (5)









5 Multipoint Model


119889119899 (119905)

119889119905

=


119869

119895=1120588

119895

119877

Λ

119862

sdot 119899 (119905) +

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

+

119869

sum

119895=1

120582

119895

119877sdot 119862

119895

119877(119905)


119889119862

119894(119905)

119889119905

=

120573

119894

Λ

119862

sdot 119899 (119905) minus 120582

119894sdot 119862

119894(119905)

(119894 = 1 2 6 119895 = 1 2 119869)

119889119862

119895

119877(119905)

119889119905

=

120588

119895

119877

Λ

119862

sdot 119899 (119905) minus 120582

119895

119877sdot 119862

119895

119877(119905) 120582

119895

119877=

1

Λ

119895

119877

(6)


120588 =

Λ

119862

119879

+

119869

sum

119895=1

120588

119895

119877

1 + 119879Λ

119895

119877

+

6

sum

119894=1

120573

119894

1 + 120582

119894sdot 119879

Λ

119895+1

prt = (1 minus 120588

119895+1

119877) sdot Λ

119895

prt + 120588

119895+1

119877sdot Λ

119895+1

119877

(7)


119895


119877is





119862








119889119899 (119905)

119889119905

=

120588 minus 120573 minus 120588

119877(119903out)

Λ

119862

sdot 119899 (119905)

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905) + int

119903out

119903in

120582

119877(119903

1015840) sdot 119862

119877(119903

1015840 119905) sdot 119889119903

1015840

119889119862

119894(119905)

119889119905

=

120573

119894

Λ

119862

sdot 119899 (119905) minus 120582

119894sdot 119862

119894(119905) (119894 = 1 2 6)

119889119862

119877(119903 119905)

119889119905

=

119889120588

119877(119903) 119889119903

Λ

119862

sdot 119899 (119905) minus 120582

119877(119903) sdot 119862

119877(119903 119905)

120582

119877(119903) =

1

Λ

119877(119903)

(8)









120588 =

Λ

119862

119879

+ int

119903out

119903in

119889120588

119877(119903

1015840) 119889119903

1015840

1 + 119879Λ

119877(119903

1015840)

sdot 119889119903

1015840+

6

sum

119894=1

120573

119894

1 + 120582

119894sdot 119879

(9)

Λ

119877(119903) = Λ prt (119903) +

119889Λ prt (119903) 119889119903

119889120588

119877(119903) 119889119903

(10)






















0 1 2 3 4 5 60

2

4

6

8

0

2

4

6

8

10

Reac

tivity

gai

nca

used

by

the r

eflec

tor (

$)

Diff

eren

tial r

eact

ivity

gai

n ca

used

by

the r

eflec

tor (

$m

)


120588R(r)

d120588R(r)dr

(a)

0 1 2 3 4 5

Diff

eren

tial n

eutro

n lif

etim

e (s

m)

6

0 2 4 6 8 10

ΛR(r)

Λprt

dΛprt (r)dr

1E + 0

1E minus 1

1E minus 2

1E minus 3

1E minus 4

1E minus 5

1E minus 6

1E minus 7

1E + 0

1E minus 1

1E minus 2

1E minus 3

1E minus 4

1E minus 5

1E minus 6

1E minus 7

Neu

tron

lifet

ime (

s)



(b)



119877(0) is longer




119877(119903) and mean





119877(119903) (Figure 2(a))


119877(119903) (Figure 2(b))





120588

119877(Λ


119903out

119903

119889120588

119877(119903

1015840)

119889119903

1015840sdot 119889119903

1015840= int

Λmax

Λ 119877

119889120588

119877

119889Λ

1015840

119877

sdot 119889Λ

1015840

119877

(11)





120588

lowast

119877equiv int

119903out

119903in

119889120588

119877(119903

1015840) 119889119903

1015840

1 + 119879Λ

119877(119903

1015840)

sdot 119889119903

1015840

(12)


0001 001 101

1

3

5

0

2

4

6

6

4

32

5



Pbnat 2

Con

trib

utio

n of

neu

trons

of t

he re

flect

orin

to th

e rea

ctiv

ity g

ain120588

R(Λ

RΛ

max

) ($)

1E minus 4

208Pb




119877(Λ



0 1 2 3 4 5 6

Diff

eren

tial c

ontr

ibut

ion

into

the r

equi

ered

reac

tivity

($m

)

1E + 0

1E + 1

1E minus 1

1E minus 2

1E minus 3

1E minus 4

1E minus 5



001 s244$

01 s096$10 s016$

001 s0036$01 s00037 $

10 s000038 $





119877

int

119903out

119903in

119889120588

119877(119903

1015840) 119889119903

1015840

1 + 119879Λ

119877(119903

1015840)

sdot 119889119903

1015840= int

Λmax

Λmin

119889120588

119877119889Λ

1015840

119877

1 + 119879Λ

1015840

119877

sdot 119889Λ

1015840

119877

119889120588

119877

119889119903

=

119889120588

119877

119889Λ

119877

sdot

119889Λ

119877

119889119903

(13)


0 1 2 3 4 5 6

25

20

15

10

05

00

001

01

10Con

trib

utio

n of

neu

trons

of t

he re

flect

or in

to

the r

equi

red

reac

tivity

($)



(a)

Con

trib

utio

n of

neu

trons

of t

he re

flect

or in

to th

e req

uier

ed re

activ

ity ($

)


25

20

15

10

05

00

001

01

1010010001

10


1E minus 4

(b)


119877)


120588 =

Λ

119862

119879

+ int

Λmax

Λmin

Λ

1015840

119877sdot (119889120588

119877119889Λ

1015840

119877)

Λ

1015840

119877+ 119879

sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120573

119894

1 + 120582

119894sdot 119879

(14)




int

Λmax

Λmin

Λ

1015840

119877sdot (119889120588

119877119889Λ

1015840

119877)

Λ

1015840

119877+ 119879

sdot 119889Λ

1015840

119877=

1

119879

int

Λmax

Λmin

Λ

1015840

119877sdot 119889120588

119877

=

1

119879

int

119903out

119903in

Λ

1015840

119877sdot 119889120588

119877(119903

1015840)

(15)






120588

lowast

119877(119903) = int

119903

119903in

119889120588

119877(119903

1015840) 119889119903

1015840

1 + 119879Λ

119877(119903

1015840)

sdot 119889119903

1015840= int

Λ

Λmin

119889120588

119877119889Λ

1015840

119877

1 + 119879Λ

1015840

119877

sdot 119889Λ

1015840

119877

(16)








9 Conclusions








Appendices








396 6

371 5348 4

320 5

261 3

00 0

234 72

163 32

089 32

057 52

00 12

182 3

156 4

127 4

090 2

238 6

080 2

00 0 00 0

120 0

168 4

200 4

134 3

Ei Ii Ei IiEi Ii

Ei Ii

204 Pb 206 Pb207 Pb 208 Pb

M϶B M϶B M϶B

M϶B


0 2 4

Inel

astic

scat

terin

g cr

oss-

sect

ion

(b)

6 8 10


0

1

2

3

204 Pb

206 Pb207 Pb

208 Pb


D

01

Capt

ure c

ross

-sec

tion

(b)

10 1000

01

10

1E minus 3

1E minus 5

1E + 5 1E + 7

Neutron energy (eV)

204Pb

12C

206Pb

207Pb

208Pb

Pbnat










Nuclide 120590


(01MeVrarr 05 eV) 120590

th(119899120574)

(mb) 119877119868

119899120574+ 1119881 (mb)

1H 301 12 332 1492D 42 17 055 0259Be 65 59 85 3812C 49 77 39 1816O 40 102 019 016Pbnat 113 1269 174 95208Pb 115 1274 023 078



Moderatingratio

120585ΣΣ

119886

th


(cm2)



















2

119864=


119903

2






9 years) 997888rarr

208Pb238U 997888rarr 8120572 + 6120573 (46 sdot 10

9 years) 997888rarr

206Pb235U 997888rarr 7120572 + 4120573 (07 sdot 10

9 years) 997888rarr

207Pb

(B1)











+(119903 119864)

Λprt equiv⟨119899 119899

+⟩

⟨]119891Σ

119891sdot V sdot 119899 sdot 119899

+⟩

(C1)




С R

r

Core Reflector




119871119899 =

1

119870efsdot

119876119899

119871

+119899

+=

1

119870efsdot

119876

+119899

+

(C2)

where operator












tion 119899

+


119871

119862+119877119899

119862+119877=

1

119870efsdot

119876119899

119862+119877 (C3)

119871

+

119862+119877119899

+

119862+119877=

1

119870efsdot

119876

+119899

+

119862+119877 (C4)




119871

119862119899

119862=

1

119870efsdot

119876119899

119862+119877 (C5)


119862(119903 119864) characterizes




119862+119877= (119899

119862+119877minus 119899






119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C6)


119862+119877can be

replaced by 119899

119862+119877= 119899

119862+ Δ119899



C R

r

n C+R

n C

Δn C+R


119862+119877(119903) in the bare core 119899


Δ119899

119862+119877(119903)


Λprt =⟨(119899

119862+ Δ119899

119862+119877) 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

=

⟨119899

119862 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

+

⟨Δ119899

119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C7)


119891Σ

119891sdot V sdot 119899

119888sdot 119899

+

119862+119877⟩


the following form

⟨119899

119862 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

=

⟨119899

119862 119899

+

119862+119877⟩

119862

⟨]119891Σ

119891sdot V sdot 119899

119862sdot 119899

+

119862+119877⟩

119862

sdot

⟨]119891Σ

119891sdot V sdot 119899

119862sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C8)


119862(119903 119864) that is


Λ

119862equiv

⟨119899

119862 119899

+

119862+119877⟩

119862

⟨]119891Σ

119891sdot V sdot 119899

119862sdot 119899

+

119862+119877⟩

119862

(C9)



119862= (119899

119862+119877minus Δ119899



⟨]119891Σ

119891sdot V sdot [119899

119862+119877minus Δ119899

119862+119877] sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

= 1 minus

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C10)


119877from those


120588

119877=

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C11)


⟨119899

119862 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

= Λ

119862sdot (1 minus 120588

119877) (C12)


119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877 then the sum-


⟨Δ119899

119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

=

⟨Δ119899

119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

sdot

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C13)


Λ

119877equiv

⟨Δ119899

119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C14)









119877

Λprt = Λ

119862sdot (1 minus 120588

119877) + Λ

119877sdot 120588

119877 (C15)


119877


Λ

119895+1

prt = Λ

119895


119895+1

119877) + Λ

119895+1

119877sdot 120588

119895+1

119877 (C16)



120588 =

Λ

119862

119879

+ int

Λmax

Λmin

Λ

1015840

119877sdot (119889120588

119877119889Λ

1015840

119877)

Λ

1015840

119877+ 119879

sdot 119889Λ

1015840

119877+

6

sum

119894=1

120573

119894

1 + 120582

119894sdot 119879

(D1)




119877(Λ














119889119899 (119905)

119889119905

=

120588

0minus 120573 minus 120588

119877(Λmax)

Λ

119862

sdot 119899 (119905)

+ int

Λmax

Λ119862

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

= 120573

119894sdot

119899 (119905)

Λ

119862

minus 120582

119894sdot 119862

119894(119905) 119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

=

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

minus

1

Λ

119877

sdot 119862

119877(Λ

119877 119905)

119889120588

119877

119889Λ

119877

=

119889120588

119877(119903)

119889119903

sdot

119889119903

119889Λ

119877

(D2)


119862asymp 10

minus7divide 10









0= 10




119889119899 (119905)

119889119905

=

120588

0minus 120573 minus [120588

119877(Λmax) minus 120588

119877(Λ

0)]

Λ

119862

sdot 119899 (119905)

+ int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

= 120573

119894sdot

119899 (119905)

Λ

119862

minus 120582

119894sdot 119862

119894(119905) 119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

=

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

minus

1

Λ

119877

sdot 119862

119877(Λ

119877 119905)

(D3)



120588 =

Λ

119862

119879

+ int

Λmax

Λ 0

Λ

1015840

119877sdot (119889120588

119877119889Λ

1015840

119877)

Λ

1015840

119877+ 119879

sdot 119889Λ

1015840

119877+

6

sum

119894=1

120573

119894

1 + 120582

119894sdot 119879

(D4)



119899 (119905)

Λ

119862

=

(minus1)

120588

0minus 120573 minus [120588

119877(Λmax) minus 120588

119877(Λ

0)]

sdot [int

Λmax

Λ 0

1

Λ

1015840

119877

119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894119862

119894(119905)]








119889119862

119894(119905)

119889119905

+ 120582

119894sdot 119862

119894(119905) = 120573

119894sdot

119899 (119905)

Λ

119862

119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

+

1

Λ

119877

sdot 119862

119877(Λ

119877 119905) =

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

(D5)





119889119910

119889119905

+ 120582 sdot 119910 = 119860 (119905)

119910 (119905) = 119910 (0) sdot 119890


119905

0

119889119905

1015840sdot 119890

minus120582(119905minus1199051015840)sdot 119860 (119905

1015840)

(D6)


119894(119905 = minus0) = 119862

119894(119905 =




119877(Λ 119905) must



119862

119877(Λ

119877 119905 = +0)

119862

119877(Λ

119877 119905 = minus0)

=

119899 (119905 = +0)

119899 (119905 = minus0)

(D7)






119889119899 (119905)

119889119905

=

120588

0minus 120573 minus [120588

119877(Λmax) minus 120588

119877(Λ

0)] minus 120572

119879int

119905

0119899 (119905

1015840) sdot 119889119905

1015840

Λ

119862

sdot 119899 (119905)

+ int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

+ 120582

119894sdot 119862

119894(119905) = 120573

119894sdot

119899 (119905)

Λ

119862

119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

+

1

Λ

119877

sdot 119862

119877(Λ

119877 119905) =

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

(D8)


0119899(119905

1015840)119889119905





119889119899 (119905)

119889119905

=

120588

0minus 120573 minus 120572

119879int

119905

0119899 (119905

1015840) sdot 119889119905

1015840

Λ

119862

sdot 119899 (119905) (D9)



120573 = 120573 + [120588

119877(Λmax) minus 120588

119877(Λ

0)] thanks





119910 (119905) equiv [120588

0minus

120573 minus 120572

119879int

119905

0

119899 (119905

1015840) sdot 119889119905

1015840]

119889119910 (119905)

119889119905

= minus120572

119879sdot 119899 (119905)

119910 (119905) sdot

119889119910 (119905)

119889119905

= minus (120572

119879sdot Λ

119862) sdot [

119910 (119905)

Λ

119862

sdot 119899 (119905)]

(D10)


119862) sdot 119899(119905)

[

119910 (119905)

Λ

119862

sdot 119899 (119905)]

= minusint

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

(D11)



120588

0lt

120573 = 120573 + [120588

119877(Λmax) minus 120588

119877(Λ

0)] (D12)


119910 (119905) sdot

119889119910 (119905)

119889119905

= + (120572

119879sdot Λ

119862)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

(D13)


1

2

119889119910

2

119889119905

= + (120572

119879sdot Λ

119862)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119910

2(119905) = 2 sdot (120572

119879sdot Λ

119862)

times int

119905

0

119889119905

1015840int

Λmax

Λ 0

1

Λ

1015840

119877

119862

119877(Λ

1015840

119877 119905

1015840) sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905

1015840) + const

(D14)


2(0) = (120588

0minus

120573)

2


0minus

120573)

2

Then

119910 (119905) = minus2 sdot (120572

119879sdot Λ

119862)

times int

119905

0

119889119905

1015840[int

Λmax

Λ 0

1

Λ

1015840

119877

119862

119877(Λ

1015840

119877 119905

1015840) 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894119862

119894(119905

1015840)] + (120588

0minus

120573)

2

12

(D15)


[

119899 (119905)

Λ

119862

] =

(minus1)

119910 (119905)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

+ 120582

119894sdot 119862

119894(119905) = 120573

119894sdot [

119899 (119905)

Λ

119862

] 119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

+

1

Λ

119877

sdot 119862

119877(Λ

119877 119905) =

119889120588

119877

119889Λ

119877

sdot [

119899 (119905)

Λ

119862

]

(D16)


[

119899 (119905)

Λ

119862

] =

(minus1)

119910 (119905)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119862

119894(119905) = 119862

119894(119905 = +0) sdot 119890

minus120582119894119905

+ 120573

119894sdot int

119905

0

119889119905

1015840sdot 119890

minus120582119894(119905minus1199051015840)sdot [

119899 (119905

1015840)

Λ

119862

]

119862

119877(Λ

119877 119905) = 119862

119877(Λ

119877 119905 = +0)

sdot 119890

minus119905Λ 119877+

119889120588

119877

119889Λ

119877

sdot int

119905

0

119889119905

1015840sdot 119890

minus(1Λ 119877)(119905minus1199051015840)

sdot [

119899 (119905

1015840)

Λ

119862

]

(D17)




References





































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of


















119889119862

119894(119905)

119889119905

=

120573

119894

Λ

119862

sdot 119899 (119905) minus 120582

119894sdot 119862

119894(119905)

(119894 = 1 2 6 119895 = 1 2 119869)

119889119862

119895

119877(119905)

119889119905

=

120588

119895

119877

Λ

119862

sdot 119899 (119905) minus 120582

119895

119877sdot 119862

119895

119877(119905) 120582

119895

119877=

1

Λ

119895

119877

(6)


120588 =

Λ

119862

119879

+

119869

sum

119895=1

120588

119895

119877

1 + 119879Λ

119895

119877

+

6

sum

119894=1

120573

119894

1 + 120582

119894sdot 119879

Λ

119895+1

prt = (1 minus 120588

119895+1

119877) sdot Λ

119895

prt + 120588

119895+1

119877sdot Λ

119895+1

119877

(7)


119895


119877is





119862








119889119899 (119905)

119889119905

=

120588 minus 120573 minus 120588

119877(119903out)

Λ

119862

sdot 119899 (119905)

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905) + int

119903out

119903in

120582

119877(119903

1015840) sdot 119862

119877(119903

1015840 119905) sdot 119889119903

1015840

119889119862

119894(119905)

119889119905

=

120573

119894

Λ

119862

sdot 119899 (119905) minus 120582

119894sdot 119862

119894(119905) (119894 = 1 2 6)

119889119862

119877(119903 119905)

119889119905

=

119889120588

119877(119903) 119889119903

Λ

119862

sdot 119899 (119905) minus 120582

119877(119903) sdot 119862

119877(119903 119905)

120582

119877(119903) =

1

Λ

119877(119903)

(8)









120588 =

Λ

119862

119879

+ int

119903out

119903in

119889120588

119877(119903

1015840) 119889119903

1015840

1 + 119879Λ

119877(119903

1015840)

sdot 119889119903

1015840+

6

sum

119894=1

120573

119894

1 + 120582

119894sdot 119879

(9)

Λ

119877(119903) = Λ prt (119903) +

119889Λ prt (119903) 119889119903

119889120588

119877(119903) 119889119903

(10)






















0 1 2 3 4 5 60

2

4

6

8

0

2

4

6

8

10

Reac

tivity

gai

nca

used

by

the r

eflec

tor (

$)

Diff

eren

tial r

eact

ivity

gai

n ca

used

by

the r

eflec

tor (

$m

)


120588R(r)

d120588R(r)dr

(a)

0 1 2 3 4 5

Diff

eren

tial n

eutro

n lif

etim

e (s

m)

6

0 2 4 6 8 10

ΛR(r)

Λprt

dΛprt (r)dr

1E + 0

1E minus 1

1E minus 2

1E minus 3

1E minus 4

1E minus 5

1E minus 6

1E minus 7

1E + 0

1E minus 1

1E minus 2

1E minus 3

1E minus 4

1E minus 5

1E minus 6

1E minus 7

Neu

tron

lifet

ime (

s)



(b)



119877(0) is longer




119877(119903) and mean





119877(119903) (Figure 2(a))


119877(119903) (Figure 2(b))





120588

119877(Λ


119903out

119903

119889120588

119877(119903

1015840)

119889119903

1015840sdot 119889119903

1015840= int

Λmax

Λ 119877

119889120588

119877

119889Λ

1015840

119877

sdot 119889Λ

1015840

119877

(11)





120588

lowast

119877equiv int

119903out

119903in

119889120588

119877(119903

1015840) 119889119903

1015840

1 + 119879Λ

119877(119903

1015840)

sdot 119889119903

1015840

(12)


0001 001 101

1

3

5

0

2

4

6

6

4

32

5



Pbnat 2

Con

trib

utio

n of

neu

trons

of t

he re

flect

orin

to th

e rea

ctiv

ity g

ain120588

R(Λ

RΛ

max

) ($)

1E minus 4

208Pb




119877(Λ



0 1 2 3 4 5 6

Diff

eren

tial c

ontr

ibut

ion

into

the r

equi

ered

reac

tivity

($m

)

1E + 0

1E + 1

1E minus 1

1E minus 2

1E minus 3

1E minus 4

1E minus 5



001 s244$

01 s096$10 s016$

001 s0036$01 s00037 $

10 s000038 $





119877

int

119903out

119903in

119889120588

119877(119903

1015840) 119889119903

1015840

1 + 119879Λ

119877(119903

1015840)

sdot 119889119903

1015840= int

Λmax

Λmin

119889120588

119877119889Λ

1015840

119877

1 + 119879Λ

1015840

119877

sdot 119889Λ

1015840

119877

119889120588

119877

119889119903

=

119889120588

119877

119889Λ

119877

sdot

119889Λ

119877

119889119903

(13)


0 1 2 3 4 5 6

25

20

15

10

05

00

001

01

10Con

trib

utio

n of

neu

trons

of t

he re

flect

or in

to

the r

equi

red

reac

tivity

($)



(a)

Con

trib

utio

n of

neu

trons

of t

he re

flect

or in

to th

e req

uier

ed re

activ

ity ($

)


25

20

15

10

05

00

001

01

1010010001

10


1E minus 4

(b)


119877)


120588 =

Λ

119862

119879

+ int

Λmax

Λmin

Λ

1015840

119877sdot (119889120588

119877119889Λ

1015840

119877)

Λ

1015840

119877+ 119879

sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120573

119894

1 + 120582

119894sdot 119879

(14)




int

Λmax

Λmin

Λ

1015840

119877sdot (119889120588

119877119889Λ

1015840

119877)

Λ

1015840

119877+ 119879

sdot 119889Λ

1015840

119877=

1

119879

int

Λmax

Λmin

Λ

1015840

119877sdot 119889120588

119877

=

1

119879

int

119903out

119903in

Λ

1015840

119877sdot 119889120588

119877(119903

1015840)

(15)






120588

lowast

119877(119903) = int

119903

119903in

119889120588

119877(119903

1015840) 119889119903

1015840

1 + 119879Λ

119877(119903

1015840)

sdot 119889119903

1015840= int

Λ

Λmin

119889120588

119877119889Λ

1015840

119877

1 + 119879Λ

1015840

119877

sdot 119889Λ

1015840

119877

(16)








9 Conclusions








Appendices








396 6

371 5348 4

320 5

261 3

00 0

234 72

163 32

089 32

057 52

00 12

182 3

156 4

127 4

090 2

238 6

080 2

00 0 00 0

120 0

168 4

200 4

134 3

Ei Ii Ei IiEi Ii

Ei Ii

204 Pb 206 Pb207 Pb 208 Pb

M϶B M϶B M϶B

M϶B


0 2 4

Inel

astic

scat

terin

g cr

oss-

sect

ion

(b)

6 8 10


0

1

2

3

204 Pb

206 Pb207 Pb

208 Pb


D

01

Capt

ure c

ross

-sec

tion

(b)

10 1000

01

10

1E minus 3

1E minus 5

1E + 5 1E + 7

Neutron energy (eV)

204Pb

12C

206Pb

207Pb

208Pb

Pbnat










Nuclide 120590


(01MeVrarr 05 eV) 120590

th(119899120574)

(mb) 119877119868

119899120574+ 1119881 (mb)

1H 301 12 332 1492D 42 17 055 0259Be 65 59 85 3812C 49 77 39 1816O 40 102 019 016Pbnat 113 1269 174 95208Pb 115 1274 023 078



Moderatingratio

120585ΣΣ

119886

th


(cm2)



















2

119864=


119903

2






9 years) 997888rarr

208Pb238U 997888rarr 8120572 + 6120573 (46 sdot 10

9 years) 997888rarr

206Pb235U 997888rarr 7120572 + 4120573 (07 sdot 10

9 years) 997888rarr

207Pb

(B1)











+(119903 119864)

Λprt equiv⟨119899 119899

+⟩

⟨]119891Σ

119891sdot V sdot 119899 sdot 119899

+⟩

(C1)




С R

r

Core Reflector




119871119899 =

1

119870efsdot

119876119899

119871

+119899

+=

1

119870efsdot

119876

+119899

+

(C2)

where operator












tion 119899

+


119871

119862+119877119899

119862+119877=

1

119870efsdot

119876119899

119862+119877 (C3)

119871

+

119862+119877119899

+

119862+119877=

1

119870efsdot

119876

+119899

+

119862+119877 (C4)




119871

119862119899

119862=

1

119870efsdot

119876119899

119862+119877 (C5)


119862(119903 119864) characterizes




119862+119877= (119899

119862+119877minus 119899






119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C6)


119862+119877can be

replaced by 119899

119862+119877= 119899

119862+ Δ119899



C R

r

n C+R

n C

Δn C+R


119862+119877(119903) in the bare core 119899


Δ119899

119862+119877(119903)


Λprt =⟨(119899

119862+ Δ119899

119862+119877) 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

=

⟨119899

119862 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

+

⟨Δ119899

119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C7)


119891Σ

119891sdot V sdot 119899

119888sdot 119899

+

119862+119877⟩


the following form

⟨119899

119862 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

=

⟨119899

119862 119899

+

119862+119877⟩

119862

⟨]119891Σ

119891sdot V sdot 119899

119862sdot 119899

+

119862+119877⟩

119862

sdot

⟨]119891Σ

119891sdot V sdot 119899

119862sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C8)


119862(119903 119864) that is


Λ

119862equiv

⟨119899

119862 119899

+

119862+119877⟩

119862

⟨]119891Σ

119891sdot V sdot 119899

119862sdot 119899

+

119862+119877⟩

119862

(C9)



119862= (119899

119862+119877minus Δ119899



⟨]119891Σ

119891sdot V sdot [119899

119862+119877minus Δ119899

119862+119877] sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

= 1 minus

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C10)


119877from those


120588

119877=

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C11)


⟨119899

119862 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

= Λ

119862sdot (1 minus 120588

119877) (C12)


119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877 then the sum-


⟨Δ119899

119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

=

⟨Δ119899

119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

sdot

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C13)


Λ

119877equiv

⟨Δ119899

119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C14)









119877

Λprt = Λ

119862sdot (1 minus 120588

119877) + Λ

119877sdot 120588

119877 (C15)


119877


Λ

119895+1

prt = Λ

119895


119895+1

119877) + Λ

119895+1

119877sdot 120588

119895+1

119877 (C16)



120588 =

Λ

119862

119879

+ int

Λmax

Λmin

Λ

1015840

119877sdot (119889120588

119877119889Λ

1015840

119877)

Λ

1015840

119877+ 119879

sdot 119889Λ

1015840

119877+

6

sum

119894=1

120573

119894

1 + 120582

119894sdot 119879

(D1)




119877(Λ














119889119899 (119905)

119889119905

=

120588

0minus 120573 minus 120588

119877(Λmax)

Λ

119862

sdot 119899 (119905)

+ int

Λmax

Λ119862

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

= 120573

119894sdot

119899 (119905)

Λ

119862

minus 120582

119894sdot 119862

119894(119905) 119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

=

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

minus

1

Λ

119877

sdot 119862

119877(Λ

119877 119905)

119889120588

119877

119889Λ

119877

=

119889120588

119877(119903)

119889119903

sdot

119889119903

119889Λ

119877

(D2)


119862asymp 10

minus7divide 10









0= 10




119889119899 (119905)

119889119905

=

120588

0minus 120573 minus [120588

119877(Λmax) minus 120588

119877(Λ

0)]

Λ

119862

sdot 119899 (119905)

+ int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

= 120573

119894sdot

119899 (119905)

Λ

119862

minus 120582

119894sdot 119862

119894(119905) 119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

=

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

minus

1

Λ

119877

sdot 119862

119877(Λ

119877 119905)

(D3)



120588 =

Λ

119862

119879

+ int

Λmax

Λ 0

Λ

1015840

119877sdot (119889120588

119877119889Λ

1015840

119877)

Λ

1015840

119877+ 119879

sdot 119889Λ

1015840

119877+

6

sum

119894=1

120573

119894

1 + 120582

119894sdot 119879

(D4)



119899 (119905)

Λ

119862

=

(minus1)

120588

0minus 120573 minus [120588

119877(Λmax) minus 120588

119877(Λ

0)]

sdot [int

Λmax

Λ 0

1

Λ

1015840

119877

119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894119862

119894(119905)]








119889119862

119894(119905)

119889119905

+ 120582

119894sdot 119862

119894(119905) = 120573

119894sdot

119899 (119905)

Λ

119862

119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

+

1

Λ

119877

sdot 119862

119877(Λ

119877 119905) =

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

(D5)





119889119910

119889119905

+ 120582 sdot 119910 = 119860 (119905)

119910 (119905) = 119910 (0) sdot 119890


119905

0

119889119905

1015840sdot 119890

minus120582(119905minus1199051015840)sdot 119860 (119905

1015840)

(D6)


119894(119905 = minus0) = 119862

119894(119905 =




119877(Λ 119905) must



119862

119877(Λ

119877 119905 = +0)

119862

119877(Λ

119877 119905 = minus0)

=

119899 (119905 = +0)

119899 (119905 = minus0)

(D7)






119889119899 (119905)

119889119905

=

120588

0minus 120573 minus [120588

119877(Λmax) minus 120588

119877(Λ

0)] minus 120572

119879int

119905

0119899 (119905

1015840) sdot 119889119905

1015840

Λ

119862

sdot 119899 (119905)

+ int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

+ 120582

119894sdot 119862

119894(119905) = 120573

119894sdot

119899 (119905)

Λ

119862

119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

+

1

Λ

119877

sdot 119862

119877(Λ

119877 119905) =

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

(D8)


0119899(119905

1015840)119889119905





119889119899 (119905)

119889119905

=

120588

0minus 120573 minus 120572

119879int

119905

0119899 (119905

1015840) sdot 119889119905

1015840

Λ

119862

sdot 119899 (119905) (D9)



120573 = 120573 + [120588

119877(Λmax) minus 120588

119877(Λ

0)] thanks





119910 (119905) equiv [120588

0minus

120573 minus 120572

119879int

119905

0

119899 (119905

1015840) sdot 119889119905

1015840]

119889119910 (119905)

119889119905

= minus120572

119879sdot 119899 (119905)

119910 (119905) sdot

119889119910 (119905)

119889119905

= minus (120572

119879sdot Λ

119862) sdot [

119910 (119905)

Λ

119862

sdot 119899 (119905)]

(D10)


119862) sdot 119899(119905)

[

119910 (119905)

Λ

119862

sdot 119899 (119905)]

= minusint

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

(D11)



120588

0lt

120573 = 120573 + [120588

119877(Λmax) minus 120588

119877(Λ

0)] (D12)


119910 (119905) sdot

119889119910 (119905)

119889119905

= + (120572

119879sdot Λ

119862)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

(D13)


1

2

119889119910

2

119889119905

= + (120572

119879sdot Λ

119862)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119910

2(119905) = 2 sdot (120572

119879sdot Λ

119862)

times int

119905

0

119889119905

1015840int

Λmax

Λ 0

1

Λ

1015840

119877

119862

119877(Λ

1015840

119877 119905

1015840) sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905

1015840) + const

(D14)


2(0) = (120588

0minus

120573)

2


0minus

120573)

2

Then

119910 (119905) = minus2 sdot (120572

119879sdot Λ

119862)

times int

119905

0

119889119905

1015840[int

Λmax

Λ 0

1

Λ

1015840

119877

119862

119877(Λ

1015840

119877 119905

1015840) 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894119862

119894(119905

1015840)] + (120588

0minus

120573)

2

12

(D15)


[

119899 (119905)

Λ

119862

] =

(minus1)

119910 (119905)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

+ 120582

119894sdot 119862

119894(119905) = 120573

119894sdot [

119899 (119905)

Λ

119862

] 119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

+

1

Λ

119877

sdot 119862

119877(Λ

119877 119905) =

119889120588

119877

119889Λ

119877

sdot [

119899 (119905)

Λ

119862

]

(D16)


[

119899 (119905)

Λ

119862

] =

(minus1)

119910 (119905)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119862

119894(119905) = 119862

119894(119905 = +0) sdot 119890

minus120582119894119905

+ 120573

119894sdot int

119905

0

119889119905

1015840sdot 119890

minus120582119894(119905minus1199051015840)sdot [

119899 (119905

1015840)

Λ

119862

]

119862

119877(Λ

119877 119905) = 119862

119877(Λ

119877 119905 = +0)

sdot 119890

minus119905Λ 119877+

119889120588

119877

119889Λ

119877

sdot int

119905

0

119889119905

1015840sdot 119890

minus(1Λ 119877)(119905minus1199051015840)

sdot [

119899 (119905

1015840)

Λ

119862

]

(D17)




References





































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of





































0 1 2 3 4 5 60

2

4

6

8

0

2

4

6

8

10

Reac

tivity

gai

nca

used

by

the r

eflec

tor (

$)

Diff

eren

tial r

eact

ivity

gai

n ca

used

by

the r

eflec

tor (

$m

)


120588R(r)

d120588R(r)dr

(a)

0 1 2 3 4 5

Diff

eren

tial n

eutro

n lif

etim

e (s

m)

6

0 2 4 6 8 10

ΛR(r)

Λprt

dΛprt (r)dr

1E + 0

1E minus 1

1E minus 2

1E minus 3

1E minus 4

1E minus 5

1E minus 6

1E minus 7

1E + 0

1E minus 1

1E minus 2

1E minus 3

1E minus 4

1E minus 5

1E minus 6

1E minus 7

Neu

tron

lifet

ime (

s)



(b)



119877(0) is longer




119877(119903) and mean





119877(119903) (Figure 2(a))


119877(119903) (Figure 2(b))





120588

119877(Λ


119903out

119903

119889120588

119877(119903

1015840)

119889119903

1015840sdot 119889119903

1015840= int

Λmax

Λ 119877

119889120588

119877

119889Λ

1015840

119877

sdot 119889Λ

1015840

119877

(11)





120588

lowast

119877equiv int

119903out

119903in

119889120588

119877(119903

1015840) 119889119903

1015840

1 + 119879Λ

119877(119903

1015840)

sdot 119889119903

1015840

(12)


0001 001 101

1

3

5

0

2

4

6

6

4

32

5



Pbnat 2

Con

trib

utio

n of

neu

trons

of t

he re

flect

orin

to th

e rea

ctiv

ity g

ain120588

R(Λ

RΛ

max

) ($)

1E minus 4

208Pb




119877(Λ



0 1 2 3 4 5 6

Diff

eren

tial c

ontr

ibut

ion

into

the r

equi

ered

reac

tivity

($m

)

1E + 0

1E + 1

1E minus 1

1E minus 2

1E minus 3

1E minus 4

1E minus 5



001 s244$

01 s096$10 s016$

001 s0036$01 s00037 $

10 s000038 $





119877

int

119903out

119903in

119889120588

119877(119903

1015840) 119889119903

1015840

1 + 119879Λ

119877(119903

1015840)

sdot 119889119903

1015840= int

Λmax

Λmin

119889120588

119877119889Λ

1015840

119877

1 + 119879Λ

1015840

119877

sdot 119889Λ

1015840

119877

119889120588

119877

119889119903

=

119889120588

119877

119889Λ

119877

sdot

119889Λ

119877

119889119903

(13)


0 1 2 3 4 5 6

25

20

15

10

05

00

001

01

10Con

trib

utio

n of

neu

trons

of t

he re

flect

or in

to

the r

equi

red

reac

tivity

($)



(a)

Con

trib

utio

n of

neu

trons

of t

he re

flect

or in

to th

e req

uier

ed re

activ

ity ($

)


25

20

15

10

05

00

001

01

1010010001

10


1E minus 4

(b)


119877)


120588 =

Λ

119862

119879

+ int

Λmax

Λmin

Λ

1015840

119877sdot (119889120588

119877119889Λ

1015840

119877)

Λ

1015840

119877+ 119879

sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120573

119894

1 + 120582

119894sdot 119879

(14)




int

Λmax

Λmin

Λ

1015840

119877sdot (119889120588

119877119889Λ

1015840

119877)

Λ

1015840

119877+ 119879

sdot 119889Λ

1015840

119877=

1

119879

int

Λmax

Λmin

Λ

1015840

119877sdot 119889120588

119877

=

1

119879

int

119903out

119903in

Λ

1015840

119877sdot 119889120588

119877(119903

1015840)

(15)






120588

lowast

119877(119903) = int

119903

119903in

119889120588

119877(119903

1015840) 119889119903

1015840

1 + 119879Λ

119877(119903

1015840)

sdot 119889119903

1015840= int

Λ

Λmin

119889120588

119877119889Λ

1015840

119877

1 + 119879Λ

1015840

119877

sdot 119889Λ

1015840

119877

(16)








9 Conclusions








Appendices








396 6

371 5348 4

320 5

261 3

00 0

234 72

163 32

089 32

057 52

00 12

182 3

156 4

127 4

090 2

238 6

080 2

00 0 00 0

120 0

168 4

200 4

134 3

Ei Ii Ei IiEi Ii

Ei Ii

204 Pb 206 Pb207 Pb 208 Pb

M϶B M϶B M϶B

M϶B


0 2 4

Inel

astic

scat

terin

g cr

oss-

sect

ion

(b)

6 8 10


0

1

2

3

204 Pb

206 Pb207 Pb

208 Pb


D

01

Capt

ure c

ross

-sec

tion

(b)

10 1000

01

10

1E minus 3

1E minus 5

1E + 5 1E + 7

Neutron energy (eV)

204Pb

12C

206Pb

207Pb

208Pb

Pbnat










Nuclide 120590


(01MeVrarr 05 eV) 120590

th(119899120574)

(mb) 119877119868

119899120574+ 1119881 (mb)

1H 301 12 332 1492D 42 17 055 0259Be 65 59 85 3812C 49 77 39 1816O 40 102 019 016Pbnat 113 1269 174 95208Pb 115 1274 023 078



Moderatingratio

120585ΣΣ

119886

th


(cm2)



















2

119864=


119903

2






9 years) 997888rarr

208Pb238U 997888rarr 8120572 + 6120573 (46 sdot 10

9 years) 997888rarr

206Pb235U 997888rarr 7120572 + 4120573 (07 sdot 10

9 years) 997888rarr

207Pb

(B1)











+(119903 119864)

Λprt equiv⟨119899 119899

+⟩

⟨]119891Σ

119891sdot V sdot 119899 sdot 119899

+⟩

(C1)




С R

r

Core Reflector




119871119899 =

1

119870efsdot

119876119899

119871

+119899

+=

1

119870efsdot

119876

+119899

+

(C2)

where operator












tion 119899

+


119871

119862+119877119899

119862+119877=

1

119870efsdot

119876119899

119862+119877 (C3)

119871

+

119862+119877119899

+

119862+119877=

1

119870efsdot

119876

+119899

+

119862+119877 (C4)




119871

119862119899

119862=

1

119870efsdot

119876119899

119862+119877 (C5)


119862(119903 119864) characterizes




119862+119877= (119899

119862+119877minus 119899






119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C6)


119862+119877can be

replaced by 119899

119862+119877= 119899

119862+ Δ119899



C R

r

n C+R

n C

Δn C+R


119862+119877(119903) in the bare core 119899


Δ119899

119862+119877(119903)


Λprt =⟨(119899

119862+ Δ119899

119862+119877) 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

=

⟨119899

119862 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

+

⟨Δ119899

119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C7)


119891Σ

119891sdot V sdot 119899

119888sdot 119899

+

119862+119877⟩


the following form

⟨119899

119862 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

=

⟨119899

119862 119899

+

119862+119877⟩

119862

⟨]119891Σ

119891sdot V sdot 119899

119862sdot 119899

+

119862+119877⟩

119862

sdot

⟨]119891Σ

119891sdot V sdot 119899

119862sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C8)


119862(119903 119864) that is


Λ

119862equiv

⟨119899

119862 119899

+

119862+119877⟩

119862

⟨]119891Σ

119891sdot V sdot 119899

119862sdot 119899

+

119862+119877⟩

119862

(C9)



119862= (119899

119862+119877minus Δ119899



⟨]119891Σ

119891sdot V sdot [119899

119862+119877minus Δ119899

119862+119877] sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

= 1 minus

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C10)


119877from those


120588

119877=

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C11)


⟨119899

119862 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

= Λ

119862sdot (1 minus 120588

119877) (C12)


119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877 then the sum-


⟨Δ119899

119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

=

⟨Δ119899

119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

sdot

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C13)


Λ

119877equiv

⟨Δ119899

119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C14)









119877

Λprt = Λ

119862sdot (1 minus 120588

119877) + Λ

119877sdot 120588

119877 (C15)


119877


Λ

119895+1

prt = Λ

119895


119895+1

119877) + Λ

119895+1

119877sdot 120588

119895+1

119877 (C16)



120588 =

Λ

119862

119879

+ int

Λmax

Λmin

Λ

1015840

119877sdot (119889120588

119877119889Λ

1015840

119877)

Λ

1015840

119877+ 119879

sdot 119889Λ

1015840

119877+

6

sum

119894=1

120573

119894

1 + 120582

119894sdot 119879

(D1)




119877(Λ














119889119899 (119905)

119889119905

=

120588

0minus 120573 minus 120588

119877(Λmax)

Λ

119862

sdot 119899 (119905)

+ int

Λmax

Λ119862

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

= 120573

119894sdot

119899 (119905)

Λ

119862

minus 120582

119894sdot 119862

119894(119905) 119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

=

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

minus

1

Λ

119877

sdot 119862

119877(Λ

119877 119905)

119889120588

119877

119889Λ

119877

=

119889120588

119877(119903)

119889119903

sdot

119889119903

119889Λ

119877

(D2)


119862asymp 10

minus7divide 10









0= 10




119889119899 (119905)

119889119905

=

120588

0minus 120573 minus [120588

119877(Λmax) minus 120588

119877(Λ

0)]

Λ

119862

sdot 119899 (119905)

+ int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

= 120573

119894sdot

119899 (119905)

Λ

119862

minus 120582

119894sdot 119862

119894(119905) 119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

=

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

minus

1

Λ

119877

sdot 119862

119877(Λ

119877 119905)

(D3)



120588 =

Λ

119862

119879

+ int

Λmax

Λ 0

Λ

1015840

119877sdot (119889120588

119877119889Λ

1015840

119877)

Λ

1015840

119877+ 119879

sdot 119889Λ

1015840

119877+

6

sum

119894=1

120573

119894

1 + 120582

119894sdot 119879

(D4)



119899 (119905)

Λ

119862

=

(minus1)

120588

0minus 120573 minus [120588

119877(Λmax) minus 120588

119877(Λ

0)]

sdot [int

Λmax

Λ 0

1

Λ

1015840

119877

119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894119862

119894(119905)]








119889119862

119894(119905)

119889119905

+ 120582

119894sdot 119862

119894(119905) = 120573

119894sdot

119899 (119905)

Λ

119862

119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

+

1

Λ

119877

sdot 119862

119877(Λ

119877 119905) =

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

(D5)





119889119910

119889119905

+ 120582 sdot 119910 = 119860 (119905)

119910 (119905) = 119910 (0) sdot 119890


119905

0

119889119905

1015840sdot 119890

minus120582(119905minus1199051015840)sdot 119860 (119905

1015840)

(D6)


119894(119905 = minus0) = 119862

119894(119905 =




119877(Λ 119905) must



119862

119877(Λ

119877 119905 = +0)

119862

119877(Λ

119877 119905 = minus0)

=

119899 (119905 = +0)

119899 (119905 = minus0)

(D7)






119889119899 (119905)

119889119905

=

120588

0minus 120573 minus [120588

119877(Λmax) minus 120588

119877(Λ

0)] minus 120572

119879int

119905

0119899 (119905

1015840) sdot 119889119905

1015840

Λ

119862

sdot 119899 (119905)

+ int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

+ 120582

119894sdot 119862

119894(119905) = 120573

119894sdot

119899 (119905)

Λ

119862

119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

+

1

Λ

119877

sdot 119862

119877(Λ

119877 119905) =

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

(D8)


0119899(119905

1015840)119889119905





119889119899 (119905)

119889119905

=

120588

0minus 120573 minus 120572

119879int

119905

0119899 (119905

1015840) sdot 119889119905

1015840

Λ

119862

sdot 119899 (119905) (D9)



120573 = 120573 + [120588

119877(Λmax) minus 120588

119877(Λ

0)] thanks





119910 (119905) equiv [120588

0minus

120573 minus 120572

119879int

119905

0

119899 (119905

1015840) sdot 119889119905

1015840]

119889119910 (119905)

119889119905

= minus120572

119879sdot 119899 (119905)

119910 (119905) sdot

119889119910 (119905)

119889119905

= minus (120572

119879sdot Λ

119862) sdot [

119910 (119905)

Λ

119862

sdot 119899 (119905)]

(D10)


119862) sdot 119899(119905)

[

119910 (119905)

Λ

119862

sdot 119899 (119905)]

= minusint

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

(D11)



120588

0lt

120573 = 120573 + [120588

119877(Λmax) minus 120588

119877(Λ

0)] (D12)


119910 (119905) sdot

119889119910 (119905)

119889119905

= + (120572

119879sdot Λ

119862)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

(D13)


1

2

119889119910

2

119889119905

= + (120572

119879sdot Λ

119862)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119910

2(119905) = 2 sdot (120572

119879sdot Λ

119862)

times int

119905

0

119889119905

1015840int

Λmax

Λ 0

1

Λ

1015840

119877

119862

119877(Λ

1015840

119877 119905

1015840) sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905

1015840) + const

(D14)


2(0) = (120588

0minus

120573)

2


0minus

120573)

2

Then

119910 (119905) = minus2 sdot (120572

119879sdot Λ

119862)

times int

119905

0

119889119905

1015840[int

Λmax

Λ 0

1

Λ

1015840

119877

119862

119877(Λ

1015840

119877 119905

1015840) 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894119862

119894(119905

1015840)] + (120588

0minus

120573)

2

12

(D15)


[

119899 (119905)

Λ

119862

] =

(minus1)

119910 (119905)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

+ 120582

119894sdot 119862

119894(119905) = 120573

119894sdot [

119899 (119905)

Λ

119862

] 119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

+

1

Λ

119877

sdot 119862

119877(Λ

119877 119905) =

119889120588

119877

119889Λ

119877

sdot [

119899 (119905)

Λ

119862

]

(D16)


[

119899 (119905)

Λ

119862

] =

(minus1)

119910 (119905)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119862

119894(119905) = 119862

119894(119905 = +0) sdot 119890

minus120582119894119905

+ 120573

119894sdot int

119905

0

119889119905

1015840sdot 119890

minus120582119894(119905minus1199051015840)sdot [

119899 (119905

1015840)

Λ

119862

]

119862

119877(Λ

119877 119905) = 119862

119877(Λ

119877 119905 = +0)

sdot 119890

minus119905Λ 119877+

119889120588

119877

119889Λ

119877

sdot int

119905

0

119889119905

1015840sdot 119890

minus(1Λ 119877)(119905minus1199051015840)

sdot [

119899 (119905

1015840)

Λ

119862

]

(D17)




References





































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of




















119877(119903) and mean





119877(119903) (Figure 2(a))


119877(119903) (Figure 2(b))





120588

119877(Λ


119903out

119903

119889120588

119877(119903

1015840)

119889119903

1015840sdot 119889119903

1015840= int

Λmax

Λ 119877

119889120588

119877

119889Λ

1015840

119877

sdot 119889Λ

1015840

119877

(11)





120588

lowast

119877equiv int

119903out

119903in

119889120588

119877(119903

1015840) 119889119903

1015840

1 + 119879Λ

119877(119903

1015840)

sdot 119889119903

1015840

(12)


0001 001 101

1

3

5

0

2

4

6

6

4

32

5



Pbnat 2

Con

trib

utio

n of

neu

trons

of t

he re

flect

orin

to th

e rea

ctiv

ity g

ain120588

R(Λ

RΛ

max

) ($)

1E minus 4

208Pb




119877(Λ



0 1 2 3 4 5 6

Diff

eren

tial c

ontr

ibut

ion

into

the r

equi

ered

reac

tivity

($m

)

1E + 0

1E + 1

1E minus 1

1E minus 2

1E minus 3

1E minus 4

1E minus 5



001 s244$

01 s096$10 s016$

001 s0036$01 s00037 $

10 s000038 $





119877

int

119903out

119903in

119889120588

119877(119903

1015840) 119889119903

1015840

1 + 119879Λ

119877(119903

1015840)

sdot 119889119903

1015840= int

Λmax

Λmin

119889120588

119877119889Λ

1015840

119877

1 + 119879Λ

1015840

119877

sdot 119889Λ

1015840

119877

119889120588

119877

119889119903

=

119889120588

119877

119889Λ

119877

sdot

119889Λ

119877

119889119903

(13)


0 1 2 3 4 5 6

25

20

15

10

05

00

001

01

10Con

trib

utio

n of

neu

trons

of t

he re

flect

or in

to

the r

equi

red

reac

tivity

($)



(a)

Con

trib

utio

n of

neu

trons

of t

he re

flect

or in

to th

e req

uier

ed re

activ

ity ($

)


25

20

15

10

05

00

001

01

1010010001

10


1E minus 4

(b)


119877)


120588 =

Λ

119862

119879

+ int

Λmax

Λmin

Λ

1015840

119877sdot (119889120588

119877119889Λ

1015840

119877)

Λ

1015840

119877+ 119879

sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120573

119894

1 + 120582

119894sdot 119879

(14)




int

Λmax

Λmin

Λ

1015840

119877sdot (119889120588

119877119889Λ

1015840

119877)

Λ

1015840

119877+ 119879

sdot 119889Λ

1015840

119877=

1

119879

int

Λmax

Λmin

Λ

1015840

119877sdot 119889120588

119877

=

1

119879

int

119903out

119903in

Λ

1015840

119877sdot 119889120588

119877(119903

1015840)

(15)






120588

lowast

119877(119903) = int

119903

119903in

119889120588

119877(119903

1015840) 119889119903

1015840

1 + 119879Λ

119877(119903

1015840)

sdot 119889119903

1015840= int

Λ

Λmin

119889120588

119877119889Λ

1015840

119877

1 + 119879Λ

1015840

119877

sdot 119889Λ

1015840

119877

(16)








9 Conclusions








Appendices








396 6

371 5348 4

320 5

261 3

00 0

234 72

163 32

089 32

057 52

00 12

182 3

156 4

127 4

090 2

238 6

080 2

00 0 00 0

120 0

168 4

200 4

134 3

Ei Ii Ei IiEi Ii

Ei Ii

204 Pb 206 Pb207 Pb 208 Pb

M϶B M϶B M϶B

M϶B


0 2 4

Inel

astic

scat

terin

g cr

oss-

sect

ion

(b)

6 8 10


0

1

2

3

204 Pb

206 Pb207 Pb

208 Pb


D

01

Capt

ure c

ross

-sec

tion

(b)

10 1000

01

10

1E minus 3

1E minus 5

1E + 5 1E + 7

Neutron energy (eV)

204Pb

12C

206Pb

207Pb

208Pb

Pbnat










Nuclide 120590


(01MeVrarr 05 eV) 120590

th(119899120574)

(mb) 119877119868

119899120574+ 1119881 (mb)

1H 301 12 332 1492D 42 17 055 0259Be 65 59 85 3812C 49 77 39 1816O 40 102 019 016Pbnat 113 1269 174 95208Pb 115 1274 023 078



Moderatingratio

120585ΣΣ

119886

th


(cm2)



















2

119864=


119903

2






9 years) 997888rarr

208Pb238U 997888rarr 8120572 + 6120573 (46 sdot 10

9 years) 997888rarr

206Pb235U 997888rarr 7120572 + 4120573 (07 sdot 10

9 years) 997888rarr

207Pb

(B1)











+(119903 119864)

Λprt equiv⟨119899 119899

+⟩

⟨]119891Σ

119891sdot V sdot 119899 sdot 119899

+⟩

(C1)




С R

r

Core Reflector




119871119899 =

1

119870efsdot

119876119899

119871

+119899

+=

1

119870efsdot

119876

+119899

+

(C2)

where operator












tion 119899

+


119871

119862+119877119899

119862+119877=

1

119870efsdot

119876119899

119862+119877 (C3)

119871

+

119862+119877119899

+

119862+119877=

1

119870efsdot

119876

+119899

+

119862+119877 (C4)




119871

119862119899

119862=

1

119870efsdot

119876119899

119862+119877 (C5)


119862(119903 119864) characterizes




119862+119877= (119899

119862+119877minus 119899






119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C6)


119862+119877can be

replaced by 119899

119862+119877= 119899

119862+ Δ119899



C R

r

n C+R

n C

Δn C+R


119862+119877(119903) in the bare core 119899


Δ119899

119862+119877(119903)


Λprt =⟨(119899

119862+ Δ119899

119862+119877) 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

=

⟨119899

119862 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

+

⟨Δ119899

119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C7)


119891Σ

119891sdot V sdot 119899

119888sdot 119899

+

119862+119877⟩


the following form

⟨119899

119862 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

=

⟨119899

119862 119899

+

119862+119877⟩

119862

⟨]119891Σ

119891sdot V sdot 119899

119862sdot 119899

+

119862+119877⟩

119862

sdot

⟨]119891Σ

119891sdot V sdot 119899

119862sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C8)


119862(119903 119864) that is


Λ

119862equiv

⟨119899

119862 119899

+

119862+119877⟩

119862

⟨]119891Σ

119891sdot V sdot 119899

119862sdot 119899

+

119862+119877⟩

119862

(C9)



119862= (119899

119862+119877minus Δ119899



⟨]119891Σ

119891sdot V sdot [119899

119862+119877minus Δ119899

119862+119877] sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

= 1 minus

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C10)


119877from those


120588

119877=

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C11)


⟨119899

119862 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

= Λ

119862sdot (1 minus 120588

119877) (C12)


119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877 then the sum-


⟨Δ119899

119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

=

⟨Δ119899

119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

sdot

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C13)


Λ

119877equiv

⟨Δ119899

119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C14)









119877

Λprt = Λ

119862sdot (1 minus 120588

119877) + Λ

119877sdot 120588

119877 (C15)


119877


Λ

119895+1

prt = Λ

119895


119895+1

119877) + Λ

119895+1

119877sdot 120588

119895+1

119877 (C16)



120588 =

Λ

119862

119879

+ int

Λmax

Λmin

Λ

1015840

119877sdot (119889120588

119877119889Λ

1015840

119877)

Λ

1015840

119877+ 119879

sdot 119889Λ

1015840

119877+

6

sum

119894=1

120573

119894

1 + 120582

119894sdot 119879

(D1)




119877(Λ














119889119899 (119905)

119889119905

=

120588

0minus 120573 minus 120588

119877(Λmax)

Λ

119862

sdot 119899 (119905)

+ int

Λmax

Λ119862

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

= 120573

119894sdot

119899 (119905)

Λ

119862

minus 120582

119894sdot 119862

119894(119905) 119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

=

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

minus

1

Λ

119877

sdot 119862

119877(Λ

119877 119905)

119889120588

119877

119889Λ

119877

=

119889120588

119877(119903)

119889119903

sdot

119889119903

119889Λ

119877

(D2)


119862asymp 10

minus7divide 10









0= 10




119889119899 (119905)

119889119905

=

120588

0minus 120573 minus [120588

119877(Λmax) minus 120588

119877(Λ

0)]

Λ

119862

sdot 119899 (119905)

+ int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

= 120573

119894sdot

119899 (119905)

Λ

119862

minus 120582

119894sdot 119862

119894(119905) 119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

=

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

minus

1

Λ

119877

sdot 119862

119877(Λ

119877 119905)

(D3)



120588 =

Λ

119862

119879

+ int

Λmax

Λ 0

Λ

1015840

119877sdot (119889120588

119877119889Λ

1015840

119877)

Λ

1015840

119877+ 119879

sdot 119889Λ

1015840

119877+

6

sum

119894=1

120573

119894

1 + 120582

119894sdot 119879

(D4)



119899 (119905)

Λ

119862

=

(minus1)

120588

0minus 120573 minus [120588

119877(Λmax) minus 120588

119877(Λ

0)]

sdot [int

Λmax

Λ 0

1

Λ

1015840

119877

119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894119862

119894(119905)]








119889119862

119894(119905)

119889119905

+ 120582

119894sdot 119862

119894(119905) = 120573

119894sdot

119899 (119905)

Λ

119862

119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

+

1

Λ

119877

sdot 119862

119877(Λ

119877 119905) =

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

(D5)





119889119910

119889119905

+ 120582 sdot 119910 = 119860 (119905)

119910 (119905) = 119910 (0) sdot 119890


119905

0

119889119905

1015840sdot 119890

minus120582(119905minus1199051015840)sdot 119860 (119905

1015840)

(D6)


119894(119905 = minus0) = 119862

119894(119905 =




119877(Λ 119905) must



119862

119877(Λ

119877 119905 = +0)

119862

119877(Λ

119877 119905 = minus0)

=

119899 (119905 = +0)

119899 (119905 = minus0)

(D7)






119889119899 (119905)

119889119905

=

120588

0minus 120573 minus [120588

119877(Λmax) minus 120588

119877(Λ

0)] minus 120572

119879int

119905

0119899 (119905

1015840) sdot 119889119905

1015840

Λ

119862

sdot 119899 (119905)

+ int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

+ 120582

119894sdot 119862

119894(119905) = 120573

119894sdot

119899 (119905)

Λ

119862

119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

+

1

Λ

119877

sdot 119862

119877(Λ

119877 119905) =

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

(D8)


0119899(119905

1015840)119889119905





119889119899 (119905)

119889119905

=

120588

0minus 120573 minus 120572

119879int

119905

0119899 (119905

1015840) sdot 119889119905

1015840

Λ

119862

sdot 119899 (119905) (D9)



120573 = 120573 + [120588

119877(Λmax) minus 120588

119877(Λ

0)] thanks





119910 (119905) equiv [120588

0minus

120573 minus 120572

119879int

119905

0

119899 (119905

1015840) sdot 119889119905

1015840]

119889119910 (119905)

119889119905

= minus120572

119879sdot 119899 (119905)

119910 (119905) sdot

119889119910 (119905)

119889119905

= minus (120572

119879sdot Λ

119862) sdot [

119910 (119905)

Λ

119862

sdot 119899 (119905)]

(D10)


119862) sdot 119899(119905)

[

119910 (119905)

Λ

119862

sdot 119899 (119905)]

= minusint

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

(D11)



120588

0lt

120573 = 120573 + [120588

119877(Λmax) minus 120588

119877(Λ

0)] (D12)


119910 (119905) sdot

119889119910 (119905)

119889119905

= + (120572

119879sdot Λ

119862)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

(D13)


1

2

119889119910

2

119889119905

= + (120572

119879sdot Λ

119862)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119910

2(119905) = 2 sdot (120572

119879sdot Λ

119862)

times int

119905

0

119889119905

1015840int

Λmax

Λ 0

1

Λ

1015840

119877

119862

119877(Λ

1015840

119877 119905

1015840) sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905

1015840) + const

(D14)


2(0) = (120588

0minus

120573)

2


0minus

120573)

2

Then

119910 (119905) = minus2 sdot (120572

119879sdot Λ

119862)

times int

119905

0

119889119905

1015840[int

Λmax

Λ 0

1

Λ

1015840

119877

119862

119877(Λ

1015840

119877 119905

1015840) 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894119862

119894(119905

1015840)] + (120588

0minus

120573)

2

12

(D15)


[

119899 (119905)

Λ

119862

] =

(minus1)

119910 (119905)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

+ 120582

119894sdot 119862

119894(119905) = 120573

119894sdot [

119899 (119905)

Λ

119862

] 119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

+

1

Λ

119877

sdot 119862

119877(Λ

119877 119905) =

119889120588

119877

119889Λ

119877

sdot [

119899 (119905)

Λ

119862

]

(D16)


[

119899 (119905)

Λ

119862

] =

(minus1)

119910 (119905)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119862

119894(119905) = 119862

119894(119905 = +0) sdot 119890

minus120582119894119905

+ 120573

119894sdot int

119905

0

119889119905

1015840sdot 119890

minus120582119894(119905minus1199051015840)sdot [

119899 (119905

1015840)

Λ

119862

]

119862

119877(Λ

119877 119905) = 119862

119877(Λ

119877 119905 = +0)

sdot 119890

minus119905Λ 119877+

119889120588

119877

119889Λ

119877

sdot int

119905

0

119889119905

1015840sdot 119890

minus(1Λ 119877)(119905minus1199051015840)

sdot [

119899 (119905

1015840)

Λ

119862

]

(D17)




References





































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of


















0 1 2 3 4 5 6

25

20

15

10

05

00

001

01

10Con

trib

utio

n of

neu

trons

of t

he re

flect

or in

to

the r

equi

red

reac

tivity

($)



(a)

Con

trib

utio

n of

neu

trons

of t

he re

flect

or in

to th

e req

uier

ed re

activ

ity ($

)


25

20

15

10

05

00

001

01

1010010001

10


1E minus 4

(b)


119877)


120588 =

Λ

119862

119879

+ int

Λmax

Λmin

Λ

1015840

119877sdot (119889120588

119877119889Λ

1015840

119877)

Λ

1015840

119877+ 119879

sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120573

119894

1 + 120582

119894sdot 119879

(14)




int

Λmax

Λmin

Λ

1015840

119877sdot (119889120588

119877119889Λ

1015840

119877)

Λ

1015840

119877+ 119879

sdot 119889Λ

1015840

119877=

1

119879

int

Λmax

Λmin

Λ

1015840

119877sdot 119889120588

119877

=

1

119879

int

119903out

119903in

Λ

1015840

119877sdot 119889120588

119877(119903

1015840)

(15)






120588

lowast

119877(119903) = int

119903

119903in

119889120588

119877(119903

1015840) 119889119903

1015840

1 + 119879Λ

119877(119903

1015840)

sdot 119889119903

1015840= int

Λ

Λmin

119889120588

119877119889Λ

1015840

119877

1 + 119879Λ

1015840

119877

sdot 119889Λ

1015840

119877

(16)








9 Conclusions








Appendices








396 6

371 5348 4

320 5

261 3

00 0

234 72

163 32

089 32

057 52

00 12

182 3

156 4

127 4

090 2

238 6

080 2

00 0 00 0

120 0

168 4

200 4

134 3

Ei Ii Ei IiEi Ii

Ei Ii

204 Pb 206 Pb207 Pb 208 Pb

M϶B M϶B M϶B

M϶B


0 2 4

Inel

astic

scat

terin

g cr

oss-

sect

ion

(b)

6 8 10


0

1

2

3

204 Pb

206 Pb207 Pb

208 Pb


D

01

Capt

ure c

ross

-sec

tion

(b)

10 1000

01

10

1E minus 3

1E minus 5

1E + 5 1E + 7

Neutron energy (eV)

204Pb

12C

206Pb

207Pb

208Pb

Pbnat










Nuclide 120590


(01MeVrarr 05 eV) 120590

th(119899120574)

(mb) 119877119868

119899120574+ 1119881 (mb)

1H 301 12 332 1492D 42 17 055 0259Be 65 59 85 3812C 49 77 39 1816O 40 102 019 016Pbnat 113 1269 174 95208Pb 115 1274 023 078



Moderatingratio

120585ΣΣ

119886

th


(cm2)



















2

119864=


119903

2






9 years) 997888rarr

208Pb238U 997888rarr 8120572 + 6120573 (46 sdot 10

9 years) 997888rarr

206Pb235U 997888rarr 7120572 + 4120573 (07 sdot 10

9 years) 997888rarr

207Pb

(B1)











+(119903 119864)

Λprt equiv⟨119899 119899

+⟩

⟨]119891Σ

119891sdot V sdot 119899 sdot 119899

+⟩

(C1)




С R

r

Core Reflector




119871119899 =

1

119870efsdot

119876119899

119871

+119899

+=

1

119870efsdot

119876

+119899

+

(C2)

where operator












tion 119899

+


119871

119862+119877119899

119862+119877=

1

119870efsdot

119876119899

119862+119877 (C3)

119871

+

119862+119877119899

+

119862+119877=

1

119870efsdot

119876

+119899

+

119862+119877 (C4)




119871

119862119899

119862=

1

119870efsdot

119876119899

119862+119877 (C5)


119862(119903 119864) characterizes




119862+119877= (119899

119862+119877minus 119899






119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C6)


119862+119877can be

replaced by 119899

119862+119877= 119899

119862+ Δ119899



C R

r

n C+R

n C

Δn C+R


119862+119877(119903) in the bare core 119899


Δ119899

119862+119877(119903)


Λprt =⟨(119899

119862+ Δ119899

119862+119877) 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

=

⟨119899

119862 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

+

⟨Δ119899

119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C7)


119891Σ

119891sdot V sdot 119899

119888sdot 119899

+

119862+119877⟩


the following form

⟨119899

119862 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

=

⟨119899

119862 119899

+

119862+119877⟩

119862

⟨]119891Σ

119891sdot V sdot 119899

119862sdot 119899

+

119862+119877⟩

119862

sdot

⟨]119891Σ

119891sdot V sdot 119899

119862sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C8)


119862(119903 119864) that is


Λ

119862equiv

⟨119899

119862 119899

+

119862+119877⟩

119862

⟨]119891Σ

119891sdot V sdot 119899

119862sdot 119899

+

119862+119877⟩

119862

(C9)



119862= (119899

119862+119877minus Δ119899



⟨]119891Σ

119891sdot V sdot [119899

119862+119877minus Δ119899

119862+119877] sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

= 1 minus

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C10)


119877from those


120588

119877=

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C11)


⟨119899

119862 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

= Λ

119862sdot (1 minus 120588

119877) (C12)


119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877 then the sum-


⟨Δ119899

119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

=

⟨Δ119899

119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

sdot

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C13)


Λ

119877equiv

⟨Δ119899

119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C14)









119877

Λprt = Λ

119862sdot (1 minus 120588

119877) + Λ

119877sdot 120588

119877 (C15)


119877


Λ

119895+1

prt = Λ

119895


119895+1

119877) + Λ

119895+1

119877sdot 120588

119895+1

119877 (C16)



120588 =

Λ

119862

119879

+ int

Λmax

Λmin

Λ

1015840

119877sdot (119889120588

119877119889Λ

1015840

119877)

Λ

1015840

119877+ 119879

sdot 119889Λ

1015840

119877+

6

sum

119894=1

120573

119894

1 + 120582

119894sdot 119879

(D1)




119877(Λ














119889119899 (119905)

119889119905

=

120588

0minus 120573 minus 120588

119877(Λmax)

Λ

119862

sdot 119899 (119905)

+ int

Λmax

Λ119862

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

= 120573

119894sdot

119899 (119905)

Λ

119862

minus 120582

119894sdot 119862

119894(119905) 119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

=

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

minus

1

Λ

119877

sdot 119862

119877(Λ

119877 119905)

119889120588

119877

119889Λ

119877

=

119889120588

119877(119903)

119889119903

sdot

119889119903

119889Λ

119877

(D2)


119862asymp 10

minus7divide 10









0= 10




119889119899 (119905)

119889119905

=

120588

0minus 120573 minus [120588

119877(Λmax) minus 120588

119877(Λ

0)]

Λ

119862

sdot 119899 (119905)

+ int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

= 120573

119894sdot

119899 (119905)

Λ

119862

minus 120582

119894sdot 119862

119894(119905) 119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

=

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

minus

1

Λ

119877

sdot 119862

119877(Λ

119877 119905)

(D3)



120588 =

Λ

119862

119879

+ int

Λmax

Λ 0

Λ

1015840

119877sdot (119889120588

119877119889Λ

1015840

119877)

Λ

1015840

119877+ 119879

sdot 119889Λ

1015840

119877+

6

sum

119894=1

120573

119894

1 + 120582

119894sdot 119879

(D4)



119899 (119905)

Λ

119862

=

(minus1)

120588

0minus 120573 minus [120588

119877(Λmax) minus 120588

119877(Λ

0)]

sdot [int

Λmax

Λ 0

1

Λ

1015840

119877

119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894119862

119894(119905)]








119889119862

119894(119905)

119889119905

+ 120582

119894sdot 119862

119894(119905) = 120573

119894sdot

119899 (119905)

Λ

119862

119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

+

1

Λ

119877

sdot 119862

119877(Λ

119877 119905) =

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

(D5)





119889119910

119889119905

+ 120582 sdot 119910 = 119860 (119905)

119910 (119905) = 119910 (0) sdot 119890


119905

0

119889119905

1015840sdot 119890

minus120582(119905minus1199051015840)sdot 119860 (119905

1015840)

(D6)


119894(119905 = minus0) = 119862

119894(119905 =




119877(Λ 119905) must



119862

119877(Λ

119877 119905 = +0)

119862

119877(Λ

119877 119905 = minus0)

=

119899 (119905 = +0)

119899 (119905 = minus0)

(D7)






119889119899 (119905)

119889119905

=

120588

0minus 120573 minus [120588

119877(Λmax) minus 120588

119877(Λ

0)] minus 120572

119879int

119905

0119899 (119905

1015840) sdot 119889119905

1015840

Λ

119862

sdot 119899 (119905)

+ int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

+ 120582

119894sdot 119862

119894(119905) = 120573

119894sdot

119899 (119905)

Λ

119862

119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

+

1

Λ

119877

sdot 119862

119877(Λ

119877 119905) =

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

(D8)


0119899(119905

1015840)119889119905





119889119899 (119905)

119889119905

=

120588

0minus 120573 minus 120572

119879int

119905

0119899 (119905

1015840) sdot 119889119905

1015840

Λ

119862

sdot 119899 (119905) (D9)



120573 = 120573 + [120588

119877(Λmax) minus 120588

119877(Λ

0)] thanks





119910 (119905) equiv [120588

0minus

120573 minus 120572

119879int

119905

0

119899 (119905

1015840) sdot 119889119905

1015840]

119889119910 (119905)

119889119905

= minus120572

119879sdot 119899 (119905)

119910 (119905) sdot

119889119910 (119905)

119889119905

= minus (120572

119879sdot Λ

119862) sdot [

119910 (119905)

Λ

119862

sdot 119899 (119905)]

(D10)


119862) sdot 119899(119905)

[

119910 (119905)

Λ

119862

sdot 119899 (119905)]

= minusint

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

(D11)



120588

0lt

120573 = 120573 + [120588

119877(Λmax) minus 120588

119877(Λ

0)] (D12)


119910 (119905) sdot

119889119910 (119905)

119889119905

= + (120572

119879sdot Λ

119862)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

(D13)


1

2

119889119910

2

119889119905

= + (120572

119879sdot Λ

119862)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119910

2(119905) = 2 sdot (120572

119879sdot Λ

119862)

times int

119905

0

119889119905

1015840int

Λmax

Λ 0

1

Λ

1015840

119877

119862

119877(Λ

1015840

119877 119905

1015840) sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905

1015840) + const

(D14)


2(0) = (120588

0minus

120573)

2


0minus

120573)

2

Then

119910 (119905) = minus2 sdot (120572

119879sdot Λ

119862)

times int

119905

0

119889119905

1015840[int

Λmax

Λ 0

1

Λ

1015840

119877

119862

119877(Λ

1015840

119877 119905

1015840) 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894119862

119894(119905

1015840)] + (120588

0minus

120573)

2

12

(D15)


[

119899 (119905)

Λ

119862

] =

(minus1)

119910 (119905)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

+ 120582

119894sdot 119862

119894(119905) = 120573

119894sdot [

119899 (119905)

Λ

119862

] 119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

+

1

Λ

119877

sdot 119862

119877(Λ

119877 119905) =

119889120588

119877

119889Λ

119877

sdot [

119899 (119905)

Λ

119862

]

(D16)


[

119899 (119905)

Λ

119862

] =

(minus1)

119910 (119905)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119862

119894(119905) = 119862

119894(119905 = +0) sdot 119890

minus120582119894119905

+ 120573

119894sdot int

119905

0

119889119905

1015840sdot 119890

minus120582119894(119905minus1199051015840)sdot [

119899 (119905

1015840)

Λ

119862

]

119862

119877(Λ

119877 119905) = 119862

119877(Λ

119877 119905 = +0)

sdot 119890

minus119905Λ 119877+

119889120588

119877

119889Λ

119877

sdot int

119905

0

119889119905

1015840sdot 119890

minus(1Λ 119877)(119905minus1199051015840)

sdot [

119899 (119905

1015840)

Λ

119862

]

(D17)




References





































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of



















9 Conclusions








Appendices








396 6

371 5348 4

320 5

261 3

00 0

234 72

163 32

089 32

057 52

00 12

182 3

156 4

127 4

090 2

238 6

080 2

00 0 00 0

120 0

168 4

200 4

134 3

Ei Ii Ei IiEi Ii

Ei Ii

204 Pb 206 Pb207 Pb 208 Pb

M϶B M϶B M϶B

M϶B


0 2 4

Inel

astic

scat

terin

g cr

oss-

sect

ion

(b)

6 8 10


0

1

2

3

204 Pb

206 Pb207 Pb

208 Pb


D

01

Capt

ure c

ross

-sec

tion

(b)

10 1000

01

10

1E minus 3

1E minus 5

1E + 5 1E + 7

Neutron energy (eV)

204Pb

12C

206Pb

207Pb

208Pb

Pbnat










Nuclide 120590


(01MeVrarr 05 eV) 120590

th(119899120574)

(mb) 119877119868

119899120574+ 1119881 (mb)

1H 301 12 332 1492D 42 17 055 0259Be 65 59 85 3812C 49 77 39 1816O 40 102 019 016Pbnat 113 1269 174 95208Pb 115 1274 023 078



Moderatingratio

120585ΣΣ

119886

th


(cm2)



















2

119864=


119903

2






9 years) 997888rarr

208Pb238U 997888rarr 8120572 + 6120573 (46 sdot 10

9 years) 997888rarr

206Pb235U 997888rarr 7120572 + 4120573 (07 sdot 10

9 years) 997888rarr

207Pb

(B1)











+(119903 119864)

Λprt equiv⟨119899 119899

+⟩

⟨]119891Σ

119891sdot V sdot 119899 sdot 119899

+⟩

(C1)




С R

r

Core Reflector




119871119899 =

1

119870efsdot

119876119899

119871

+119899

+=

1

119870efsdot

119876

+119899

+

(C2)

where operator












tion 119899

+


119871

119862+119877119899

119862+119877=

1

119870efsdot

119876119899

119862+119877 (C3)

119871

+

119862+119877119899

+

119862+119877=

1

119870efsdot

119876

+119899

+

119862+119877 (C4)




119871

119862119899

119862=

1

119870efsdot

119876119899

119862+119877 (C5)


119862(119903 119864) characterizes




119862+119877= (119899

119862+119877minus 119899






119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C6)


119862+119877can be

replaced by 119899

119862+119877= 119899

119862+ Δ119899



C R

r

n C+R

n C

Δn C+R


119862+119877(119903) in the bare core 119899


Δ119899

119862+119877(119903)


Λprt =⟨(119899

119862+ Δ119899

119862+119877) 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

=

⟨119899

119862 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

+

⟨Δ119899

119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C7)


119891Σ

119891sdot V sdot 119899

119888sdot 119899

+

119862+119877⟩


the following form

⟨119899

119862 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

=

⟨119899

119862 119899

+

119862+119877⟩

119862

⟨]119891Σ

119891sdot V sdot 119899

119862sdot 119899

+

119862+119877⟩

119862

sdot

⟨]119891Σ

119891sdot V sdot 119899

119862sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C8)


119862(119903 119864) that is


Λ

119862equiv

⟨119899

119862 119899

+

119862+119877⟩

119862

⟨]119891Σ

119891sdot V sdot 119899

119862sdot 119899

+

119862+119877⟩

119862

(C9)



119862= (119899

119862+119877minus Δ119899



⟨]119891Σ

119891sdot V sdot [119899

119862+119877minus Δ119899

119862+119877] sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

= 1 minus

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C10)


119877from those


120588

119877=

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C11)


⟨119899

119862 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

= Λ

119862sdot (1 minus 120588

119877) (C12)


119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877 then the sum-


⟨Δ119899

119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

=

⟨Δ119899

119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

sdot

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C13)


Λ

119877equiv

⟨Δ119899

119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C14)









119877

Λprt = Λ

119862sdot (1 minus 120588

119877) + Λ

119877sdot 120588

119877 (C15)


119877


Λ

119895+1

prt = Λ

119895


119895+1

119877) + Λ

119895+1

119877sdot 120588

119895+1

119877 (C16)



120588 =

Λ

119862

119879

+ int

Λmax

Λmin

Λ

1015840

119877sdot (119889120588

119877119889Λ

1015840

119877)

Λ

1015840

119877+ 119879

sdot 119889Λ

1015840

119877+

6

sum

119894=1

120573

119894

1 + 120582

119894sdot 119879

(D1)




119877(Λ














119889119899 (119905)

119889119905

=

120588

0minus 120573 minus 120588

119877(Λmax)

Λ

119862

sdot 119899 (119905)

+ int

Λmax

Λ119862

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

= 120573

119894sdot

119899 (119905)

Λ

119862

minus 120582

119894sdot 119862

119894(119905) 119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

=

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

minus

1

Λ

119877

sdot 119862

119877(Λ

119877 119905)

119889120588

119877

119889Λ

119877

=

119889120588

119877(119903)

119889119903

sdot

119889119903

119889Λ

119877

(D2)


119862asymp 10

minus7divide 10









0= 10




119889119899 (119905)

119889119905

=

120588

0minus 120573 minus [120588

119877(Λmax) minus 120588

119877(Λ

0)]

Λ

119862

sdot 119899 (119905)

+ int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

= 120573

119894sdot

119899 (119905)

Λ

119862

minus 120582

119894sdot 119862

119894(119905) 119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

=

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

minus

1

Λ

119877

sdot 119862

119877(Λ

119877 119905)

(D3)



120588 =

Λ

119862

119879

+ int

Λmax

Λ 0

Λ

1015840

119877sdot (119889120588

119877119889Λ

1015840

119877)

Λ

1015840

119877+ 119879

sdot 119889Λ

1015840

119877+

6

sum

119894=1

120573

119894

1 + 120582

119894sdot 119879

(D4)



119899 (119905)

Λ

119862

=

(minus1)

120588

0minus 120573 minus [120588

119877(Λmax) minus 120588

119877(Λ

0)]

sdot [int

Λmax

Λ 0

1

Λ

1015840

119877

119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894119862

119894(119905)]








119889119862

119894(119905)

119889119905

+ 120582

119894sdot 119862

119894(119905) = 120573

119894sdot

119899 (119905)

Λ

119862

119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

+

1

Λ

119877

sdot 119862

119877(Λ

119877 119905) =

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

(D5)





119889119910

119889119905

+ 120582 sdot 119910 = 119860 (119905)

119910 (119905) = 119910 (0) sdot 119890


119905

0

119889119905

1015840sdot 119890

minus120582(119905minus1199051015840)sdot 119860 (119905

1015840)

(D6)


119894(119905 = minus0) = 119862

119894(119905 =




119877(Λ 119905) must



119862

119877(Λ

119877 119905 = +0)

119862

119877(Λ

119877 119905 = minus0)

=

119899 (119905 = +0)

119899 (119905 = minus0)

(D7)






119889119899 (119905)

119889119905

=

120588

0minus 120573 minus [120588

119877(Λmax) minus 120588

119877(Λ

0)] minus 120572

119879int

119905

0119899 (119905

1015840) sdot 119889119905

1015840

Λ

119862

sdot 119899 (119905)

+ int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

+ 120582

119894sdot 119862

119894(119905) = 120573

119894sdot

119899 (119905)

Λ

119862

119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

+

1

Λ

119877

sdot 119862

119877(Λ

119877 119905) =

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

(D8)


0119899(119905

1015840)119889119905





119889119899 (119905)

119889119905

=

120588

0minus 120573 minus 120572

119879int

119905

0119899 (119905

1015840) sdot 119889119905

1015840

Λ

119862

sdot 119899 (119905) (D9)



120573 = 120573 + [120588

119877(Λmax) minus 120588

119877(Λ

0)] thanks





119910 (119905) equiv [120588

0minus

120573 minus 120572

119879int

119905

0

119899 (119905

1015840) sdot 119889119905

1015840]

119889119910 (119905)

119889119905

= minus120572

119879sdot 119899 (119905)

119910 (119905) sdot

119889119910 (119905)

119889119905

= minus (120572

119879sdot Λ

119862) sdot [

119910 (119905)

Λ

119862

sdot 119899 (119905)]

(D10)


119862) sdot 119899(119905)

[

119910 (119905)

Λ

119862

sdot 119899 (119905)]

= minusint

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

(D11)



120588

0lt

120573 = 120573 + [120588

119877(Λmax) minus 120588

119877(Λ

0)] (D12)


119910 (119905) sdot

119889119910 (119905)

119889119905

= + (120572

119879sdot Λ

119862)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

(D13)


1

2

119889119910

2

119889119905

= + (120572

119879sdot Λ

119862)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119910

2(119905) = 2 sdot (120572

119879sdot Λ

119862)

times int

119905

0

119889119905

1015840int

Λmax

Λ 0

1

Λ

1015840

119877

119862

119877(Λ

1015840

119877 119905

1015840) sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905

1015840) + const

(D14)


2(0) = (120588

0minus

120573)

2


0minus

120573)

2

Then

119910 (119905) = minus2 sdot (120572

119879sdot Λ

119862)

times int

119905

0

119889119905

1015840[int

Λmax

Λ 0

1

Λ

1015840

119877

119862

119877(Λ

1015840

119877 119905

1015840) 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894119862

119894(119905

1015840)] + (120588

0minus

120573)

2

12

(D15)


[

119899 (119905)

Λ

119862

] =

(minus1)

119910 (119905)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

+ 120582

119894sdot 119862

119894(119905) = 120573

119894sdot [

119899 (119905)

Λ

119862

] 119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

+

1

Λ

119877

sdot 119862

119877(Λ

119877 119905) =

119889120588

119877

119889Λ

119877

sdot [

119899 (119905)

Λ

119862

]

(D16)


[

119899 (119905)

Λ

119862

] =

(minus1)

119910 (119905)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119862

119894(119905) = 119862

119894(119905 = +0) sdot 119890

minus120582119894119905

+ 120573

119894sdot int

119905

0

119889119905

1015840sdot 119890

minus120582119894(119905minus1199051015840)sdot [

119899 (119905

1015840)

Λ

119862

]

119862

119877(Λ

119877 119905) = 119862

119877(Λ

119877 119905 = +0)

sdot 119890

minus119905Λ 119877+

119889120588

119877

119889Λ

119877

sdot int

119905

0

119889119905

1015840sdot 119890

minus(1Λ 119877)(119905minus1199051015840)

sdot [

119899 (119905

1015840)

Λ

119862

]

(D17)




References





































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of


















396 6

371 5348 4

320 5

261 3

00 0

234 72

163 32

089 32

057 52

00 12

182 3

156 4

127 4

090 2

238 6

080 2

00 0 00 0

120 0

168 4

200 4

134 3

Ei Ii Ei IiEi Ii

Ei Ii

204 Pb 206 Pb207 Pb 208 Pb

M϶B M϶B M϶B

M϶B


0 2 4

Inel

astic

scat

terin

g cr

oss-

sect

ion

(b)

6 8 10


0

1

2

3

204 Pb

206 Pb207 Pb

208 Pb


D

01

Capt

ure c

ross

-sec

tion

(b)

10 1000

01

10

1E minus 3

1E minus 5

1E + 5 1E + 7

Neutron energy (eV)

204Pb

12C

206Pb

207Pb

208Pb

Pbnat










Nuclide 120590


(01MeVrarr 05 eV) 120590

th(119899120574)

(mb) 119877119868

119899120574+ 1119881 (mb)

1H 301 12 332 1492D 42 17 055 0259Be 65 59 85 3812C 49 77 39 1816O 40 102 019 016Pbnat 113 1269 174 95208Pb 115 1274 023 078



Moderatingratio

120585ΣΣ

119886

th


(cm2)



















2

119864=


119903

2






9 years) 997888rarr

208Pb238U 997888rarr 8120572 + 6120573 (46 sdot 10

9 years) 997888rarr

206Pb235U 997888rarr 7120572 + 4120573 (07 sdot 10

9 years) 997888rarr

207Pb

(B1)











+(119903 119864)

Λprt equiv⟨119899 119899

+⟩

⟨]119891Σ

119891sdot V sdot 119899 sdot 119899

+⟩

(C1)




С R

r

Core Reflector




119871119899 =

1

119870efsdot

119876119899

119871

+119899

+=

1

119870efsdot

119876

+119899

+

(C2)

where operator












tion 119899

+


119871

119862+119877119899

119862+119877=

1

119870efsdot

119876119899

119862+119877 (C3)

119871

+

119862+119877119899

+

119862+119877=

1

119870efsdot

119876

+119899

+

119862+119877 (C4)




119871

119862119899

119862=

1

119870efsdot

119876119899

119862+119877 (C5)


119862(119903 119864) characterizes




119862+119877= (119899

119862+119877minus 119899






119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C6)


119862+119877can be

replaced by 119899

119862+119877= 119899

119862+ Δ119899



C R

r

n C+R

n C

Δn C+R


119862+119877(119903) in the bare core 119899


Δ119899

119862+119877(119903)


Λprt =⟨(119899

119862+ Δ119899

119862+119877) 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

=

⟨119899

119862 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

+

⟨Δ119899

119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C7)


119891Σ

119891sdot V sdot 119899

119888sdot 119899

+

119862+119877⟩


the following form

⟨119899

119862 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

=

⟨119899

119862 119899

+

119862+119877⟩

119862

⟨]119891Σ

119891sdot V sdot 119899

119862sdot 119899

+

119862+119877⟩

119862

sdot

⟨]119891Σ

119891sdot V sdot 119899

119862sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C8)


119862(119903 119864) that is


Λ

119862equiv

⟨119899

119862 119899

+

119862+119877⟩

119862

⟨]119891Σ

119891sdot V sdot 119899

119862sdot 119899

+

119862+119877⟩

119862

(C9)



119862= (119899

119862+119877minus Δ119899



⟨]119891Σ

119891sdot V sdot [119899

119862+119877minus Δ119899

119862+119877] sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

= 1 minus

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C10)


119877from those


120588

119877=

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C11)


⟨119899

119862 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

= Λ

119862sdot (1 minus 120588

119877) (C12)


119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877 then the sum-


⟨Δ119899

119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

=

⟨Δ119899

119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

sdot

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C13)


Λ

119877equiv

⟨Δ119899

119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C14)









119877

Λprt = Λ

119862sdot (1 minus 120588

119877) + Λ

119877sdot 120588

119877 (C15)


119877


Λ

119895+1

prt = Λ

119895


119895+1

119877) + Λ

119895+1

119877sdot 120588

119895+1

119877 (C16)



120588 =

Λ

119862

119879

+ int

Λmax

Λmin

Λ

1015840

119877sdot (119889120588

119877119889Λ

1015840

119877)

Λ

1015840

119877+ 119879

sdot 119889Λ

1015840

119877+

6

sum

119894=1

120573

119894

1 + 120582

119894sdot 119879

(D1)




119877(Λ














119889119899 (119905)

119889119905

=

120588

0minus 120573 minus 120588

119877(Λmax)

Λ

119862

sdot 119899 (119905)

+ int

Λmax

Λ119862

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

= 120573

119894sdot

119899 (119905)

Λ

119862

minus 120582

119894sdot 119862

119894(119905) 119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

=

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

minus

1

Λ

119877

sdot 119862

119877(Λ

119877 119905)

119889120588

119877

119889Λ

119877

=

119889120588

119877(119903)

119889119903

sdot

119889119903

119889Λ

119877

(D2)


119862asymp 10

minus7divide 10









0= 10




119889119899 (119905)

119889119905

=

120588

0minus 120573 minus [120588

119877(Λmax) minus 120588

119877(Λ

0)]

Λ

119862

sdot 119899 (119905)

+ int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

= 120573

119894sdot

119899 (119905)

Λ

119862

minus 120582

119894sdot 119862

119894(119905) 119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

=

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

minus

1

Λ

119877

sdot 119862

119877(Λ

119877 119905)

(D3)



120588 =

Λ

119862

119879

+ int

Λmax

Λ 0

Λ

1015840

119877sdot (119889120588

119877119889Λ

1015840

119877)

Λ

1015840

119877+ 119879

sdot 119889Λ

1015840

119877+

6

sum

119894=1

120573

119894

1 + 120582

119894sdot 119879

(D4)



119899 (119905)

Λ

119862

=

(minus1)

120588

0minus 120573 minus [120588

119877(Λmax) minus 120588

119877(Λ

0)]

sdot [int

Λmax

Λ 0

1

Λ

1015840

119877

119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894119862

119894(119905)]








119889119862

119894(119905)

119889119905

+ 120582

119894sdot 119862

119894(119905) = 120573

119894sdot

119899 (119905)

Λ

119862

119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

+

1

Λ

119877

sdot 119862

119877(Λ

119877 119905) =

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

(D5)





119889119910

119889119905

+ 120582 sdot 119910 = 119860 (119905)

119910 (119905) = 119910 (0) sdot 119890


119905

0

119889119905

1015840sdot 119890

minus120582(119905minus1199051015840)sdot 119860 (119905

1015840)

(D6)


119894(119905 = minus0) = 119862

119894(119905 =




119877(Λ 119905) must



119862

119877(Λ

119877 119905 = +0)

119862

119877(Λ

119877 119905 = minus0)

=

119899 (119905 = +0)

119899 (119905 = minus0)

(D7)






119889119899 (119905)

119889119905

=

120588

0minus 120573 minus [120588

119877(Λmax) minus 120588

119877(Λ

0)] minus 120572

119879int

119905

0119899 (119905

1015840) sdot 119889119905

1015840

Λ

119862

sdot 119899 (119905)

+ int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

+ 120582

119894sdot 119862

119894(119905) = 120573

119894sdot

119899 (119905)

Λ

119862

119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

+

1

Λ

119877

sdot 119862

119877(Λ

119877 119905) =

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

(D8)


0119899(119905

1015840)119889119905





119889119899 (119905)

119889119905

=

120588

0minus 120573 minus 120572

119879int

119905

0119899 (119905

1015840) sdot 119889119905

1015840

Λ

119862

sdot 119899 (119905) (D9)



120573 = 120573 + [120588

119877(Λmax) minus 120588

119877(Λ

0)] thanks





119910 (119905) equiv [120588

0minus

120573 minus 120572

119879int

119905

0

119899 (119905

1015840) sdot 119889119905

1015840]

119889119910 (119905)

119889119905

= minus120572

119879sdot 119899 (119905)

119910 (119905) sdot

119889119910 (119905)

119889119905

= minus (120572

119879sdot Λ

119862) sdot [

119910 (119905)

Λ

119862

sdot 119899 (119905)]

(D10)


119862) sdot 119899(119905)

[

119910 (119905)

Λ

119862

sdot 119899 (119905)]

= minusint

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

(D11)



120588

0lt

120573 = 120573 + [120588

119877(Λmax) minus 120588

119877(Λ

0)] (D12)


119910 (119905) sdot

119889119910 (119905)

119889119905

= + (120572

119879sdot Λ

119862)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

(D13)


1

2

119889119910

2

119889119905

= + (120572

119879sdot Λ

119862)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119910

2(119905) = 2 sdot (120572

119879sdot Λ

119862)

times int

119905

0

119889119905

1015840int

Λmax

Λ 0

1

Λ

1015840

119877

119862

119877(Λ

1015840

119877 119905

1015840) sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905

1015840) + const

(D14)


2(0) = (120588

0minus

120573)

2


0minus

120573)

2

Then

119910 (119905) = minus2 sdot (120572

119879sdot Λ

119862)

times int

119905

0

119889119905

1015840[int

Λmax

Λ 0

1

Λ

1015840

119877

119862

119877(Λ

1015840

119877 119905

1015840) 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894119862

119894(119905

1015840)] + (120588

0minus

120573)

2

12

(D15)


[

119899 (119905)

Λ

119862

] =

(minus1)

119910 (119905)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

+ 120582

119894sdot 119862

119894(119905) = 120573

119894sdot [

119899 (119905)

Λ

119862

] 119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

+

1

Λ

119877

sdot 119862

119877(Λ

119877 119905) =

119889120588

119877

119889Λ

119877

sdot [

119899 (119905)

Λ

119862

]

(D16)


[

119899 (119905)

Λ

119862

] =

(minus1)

119910 (119905)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119862

119894(119905) = 119862

119894(119905 = +0) sdot 119890

minus120582119894119905

+ 120573

119894sdot int

119905

0

119889119905

1015840sdot 119890

minus120582119894(119905minus1199051015840)sdot [

119899 (119905

1015840)

Λ

119862

]

119862

119877(Λ

119877 119905) = 119862

119877(Λ

119877 119905 = +0)

sdot 119890

minus119905Λ 119877+

119889120588

119877

119889Λ

119877

sdot int

119905

0

119889119905

1015840sdot 119890

minus(1Λ 119877)(119905minus1199051015840)

sdot [

119899 (119905

1015840)

Λ

119862

]

(D17)




References





































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of



















Nuclide 120590


(01MeVrarr 05 eV) 120590

th(119899120574)

(mb) 119877119868

119899120574+ 1119881 (mb)

1H 301 12 332 1492D 42 17 055 0259Be 65 59 85 3812C 49 77 39 1816O 40 102 019 016Pbnat 113 1269 174 95208Pb 115 1274 023 078



Moderatingratio

120585ΣΣ

119886

th


(cm2)



















2

119864=


119903

2






9 years) 997888rarr

208Pb238U 997888rarr 8120572 + 6120573 (46 sdot 10

9 years) 997888rarr

206Pb235U 997888rarr 7120572 + 4120573 (07 sdot 10

9 years) 997888rarr

207Pb

(B1)











+(119903 119864)

Λprt equiv⟨119899 119899

+⟩

⟨]119891Σ

119891sdot V sdot 119899 sdot 119899

+⟩

(C1)




С R

r

Core Reflector




119871119899 =

1

119870efsdot

119876119899

119871

+119899

+=

1

119870efsdot

119876

+119899

+

(C2)

where operator












tion 119899

+


119871

119862+119877119899

119862+119877=

1

119870efsdot

119876119899

119862+119877 (C3)

119871

+

119862+119877119899

+

119862+119877=

1

119870efsdot

119876

+119899

+

119862+119877 (C4)




119871

119862119899

119862=

1

119870efsdot

119876119899

119862+119877 (C5)


119862(119903 119864) characterizes




119862+119877= (119899

119862+119877minus 119899






119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C6)


119862+119877can be

replaced by 119899

119862+119877= 119899

119862+ Δ119899



C R

r

n C+R

n C

Δn C+R


119862+119877(119903) in the bare core 119899


Δ119899

119862+119877(119903)


Λprt =⟨(119899

119862+ Δ119899

119862+119877) 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

=

⟨119899

119862 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

+

⟨Δ119899

119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C7)


119891Σ

119891sdot V sdot 119899

119888sdot 119899

+

119862+119877⟩


the following form

⟨119899

119862 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

=

⟨119899

119862 119899

+

119862+119877⟩

119862

⟨]119891Σ

119891sdot V sdot 119899

119862sdot 119899

+

119862+119877⟩

119862

sdot

⟨]119891Σ

119891sdot V sdot 119899

119862sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C8)


119862(119903 119864) that is


Λ

119862equiv

⟨119899

119862 119899

+

119862+119877⟩

119862

⟨]119891Σ

119891sdot V sdot 119899

119862sdot 119899

+

119862+119877⟩

119862

(C9)



119862= (119899

119862+119877minus Δ119899



⟨]119891Σ

119891sdot V sdot [119899

119862+119877minus Δ119899

119862+119877] sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

= 1 minus

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C10)


119877from those


120588

119877=

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C11)


⟨119899

119862 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

= Λ

119862sdot (1 minus 120588

119877) (C12)


119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877 then the sum-


⟨Δ119899

119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

=

⟨Δ119899

119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

sdot

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C13)


Λ

119877equiv

⟨Δ119899

119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C14)









119877

Λprt = Λ

119862sdot (1 minus 120588

119877) + Λ

119877sdot 120588

119877 (C15)


119877


Λ

119895+1

prt = Λ

119895


119895+1

119877) + Λ

119895+1

119877sdot 120588

119895+1

119877 (C16)



120588 =

Λ

119862

119879

+ int

Λmax

Λmin

Λ

1015840

119877sdot (119889120588

119877119889Λ

1015840

119877)

Λ

1015840

119877+ 119879

sdot 119889Λ

1015840

119877+

6

sum

119894=1

120573

119894

1 + 120582

119894sdot 119879

(D1)




119877(Λ














119889119899 (119905)

119889119905

=

120588

0minus 120573 minus 120588

119877(Λmax)

Λ

119862

sdot 119899 (119905)

+ int

Λmax

Λ119862

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

= 120573

119894sdot

119899 (119905)

Λ

119862

minus 120582

119894sdot 119862

119894(119905) 119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

=

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

minus

1

Λ

119877

sdot 119862

119877(Λ

119877 119905)

119889120588

119877

119889Λ

119877

=

119889120588

119877(119903)

119889119903

sdot

119889119903

119889Λ

119877

(D2)


119862asymp 10

minus7divide 10









0= 10




119889119899 (119905)

119889119905

=

120588

0minus 120573 minus [120588

119877(Λmax) minus 120588

119877(Λ

0)]

Λ

119862

sdot 119899 (119905)

+ int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

= 120573

119894sdot

119899 (119905)

Λ

119862

minus 120582

119894sdot 119862

119894(119905) 119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

=

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

minus

1

Λ

119877

sdot 119862

119877(Λ

119877 119905)

(D3)



120588 =

Λ

119862

119879

+ int

Λmax

Λ 0

Λ

1015840

119877sdot (119889120588

119877119889Λ

1015840

119877)

Λ

1015840

119877+ 119879

sdot 119889Λ

1015840

119877+

6

sum

119894=1

120573

119894

1 + 120582

119894sdot 119879

(D4)



119899 (119905)

Λ

119862

=

(minus1)

120588

0minus 120573 minus [120588

119877(Λmax) minus 120588

119877(Λ

0)]

sdot [int

Λmax

Λ 0

1

Λ

1015840

119877

119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894119862

119894(119905)]








119889119862

119894(119905)

119889119905

+ 120582

119894sdot 119862

119894(119905) = 120573

119894sdot

119899 (119905)

Λ

119862

119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

+

1

Λ

119877

sdot 119862

119877(Λ

119877 119905) =

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

(D5)





119889119910

119889119905

+ 120582 sdot 119910 = 119860 (119905)

119910 (119905) = 119910 (0) sdot 119890


119905

0

119889119905

1015840sdot 119890

minus120582(119905minus1199051015840)sdot 119860 (119905

1015840)

(D6)


119894(119905 = minus0) = 119862

119894(119905 =




119877(Λ 119905) must



119862

119877(Λ

119877 119905 = +0)

119862

119877(Λ

119877 119905 = minus0)

=

119899 (119905 = +0)

119899 (119905 = minus0)

(D7)






119889119899 (119905)

119889119905

=

120588

0minus 120573 minus [120588

119877(Λmax) minus 120588

119877(Λ

0)] minus 120572

119879int

119905

0119899 (119905

1015840) sdot 119889119905

1015840

Λ

119862

sdot 119899 (119905)

+ int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

+ 120582

119894sdot 119862

119894(119905) = 120573

119894sdot

119899 (119905)

Λ

119862

119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

+

1

Λ

119877

sdot 119862

119877(Λ

119877 119905) =

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

(D8)


0119899(119905

1015840)119889119905





119889119899 (119905)

119889119905

=

120588

0minus 120573 minus 120572

119879int

119905

0119899 (119905

1015840) sdot 119889119905

1015840

Λ

119862

sdot 119899 (119905) (D9)



120573 = 120573 + [120588

119877(Λmax) minus 120588

119877(Λ

0)] thanks





119910 (119905) equiv [120588

0minus

120573 minus 120572

119879int

119905

0

119899 (119905

1015840) sdot 119889119905

1015840]

119889119910 (119905)

119889119905

= minus120572

119879sdot 119899 (119905)

119910 (119905) sdot

119889119910 (119905)

119889119905

= minus (120572

119879sdot Λ

119862) sdot [

119910 (119905)

Λ

119862

sdot 119899 (119905)]

(D10)


119862) sdot 119899(119905)

[

119910 (119905)

Λ

119862

sdot 119899 (119905)]

= minusint

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

(D11)



120588

0lt

120573 = 120573 + [120588

119877(Λmax) minus 120588

119877(Λ

0)] (D12)


119910 (119905) sdot

119889119910 (119905)

119889119905

= + (120572

119879sdot Λ

119862)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

(D13)


1

2

119889119910

2

119889119905

= + (120572

119879sdot Λ

119862)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119910

2(119905) = 2 sdot (120572

119879sdot Λ

119862)

times int

119905

0

119889119905

1015840int

Λmax

Λ 0

1

Λ

1015840

119877

119862

119877(Λ

1015840

119877 119905

1015840) sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905

1015840) + const

(D14)


2(0) = (120588

0minus

120573)

2


0minus

120573)

2

Then

119910 (119905) = minus2 sdot (120572

119879sdot Λ

119862)

times int

119905

0

119889119905

1015840[int

Λmax

Λ 0

1

Λ

1015840

119877

119862

119877(Λ

1015840

119877 119905

1015840) 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894119862

119894(119905

1015840)] + (120588

0minus

120573)

2

12

(D15)


[

119899 (119905)

Λ

119862

] =

(minus1)

119910 (119905)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

+ 120582

119894sdot 119862

119894(119905) = 120573

119894sdot [

119899 (119905)

Λ

119862

] 119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

+

1

Λ

119877

sdot 119862

119877(Λ

119877 119905) =

119889120588

119877

119889Λ

119877

sdot [

119899 (119905)

Λ

119862

]

(D16)


[

119899 (119905)

Λ

119862

] =

(minus1)

119910 (119905)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119862

119894(119905) = 119862

119894(119905 = +0) sdot 119890

minus120582119894119905

+ 120573

119894sdot int

119905

0

119889119905

1015840sdot 119890

minus120582119894(119905minus1199051015840)sdot [

119899 (119905

1015840)

Λ

119862

]

119862

119877(Λ

119877 119905) = 119862

119877(Λ

119877 119905 = +0)

sdot 119890

minus119905Λ 119877+

119889120588

119877

119889Λ

119877

sdot int

119905

0

119889119905

1015840sdot 119890

minus(1Λ 119877)(119905minus1199051015840)

sdot [

119899 (119905

1015840)

Λ

119862

]

(D17)




References





































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of





















2

119864=


119903

2






9 years) 997888rarr

208Pb238U 997888rarr 8120572 + 6120573 (46 sdot 10

9 years) 997888rarr

206Pb235U 997888rarr 7120572 + 4120573 (07 sdot 10

9 years) 997888rarr

207Pb

(B1)











+(119903 119864)

Λprt equiv⟨119899 119899

+⟩

⟨]119891Σ

119891sdot V sdot 119899 sdot 119899

+⟩

(C1)




С R

r

Core Reflector




119871119899 =

1

119870efsdot

119876119899

119871

+119899

+=

1

119870efsdot

119876

+119899

+

(C2)

where operator












tion 119899

+


119871

119862+119877119899

119862+119877=

1

119870efsdot

119876119899

119862+119877 (C3)

119871

+

119862+119877119899

+

119862+119877=

1

119870efsdot

119876

+119899

+

119862+119877 (C4)




119871

119862119899

119862=

1

119870efsdot

119876119899

119862+119877 (C5)


119862(119903 119864) characterizes




119862+119877= (119899

119862+119877minus 119899






119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C6)


119862+119877can be

replaced by 119899

119862+119877= 119899

119862+ Δ119899



C R

r

n C+R

n C

Δn C+R


119862+119877(119903) in the bare core 119899


Δ119899

119862+119877(119903)


Λprt =⟨(119899

119862+ Δ119899

119862+119877) 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

=

⟨119899

119862 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

+

⟨Δ119899

119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C7)


119891Σ

119891sdot V sdot 119899

119888sdot 119899

+

119862+119877⟩


the following form

⟨119899

119862 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

=

⟨119899

119862 119899

+

119862+119877⟩

119862

⟨]119891Σ

119891sdot V sdot 119899

119862sdot 119899

+

119862+119877⟩

119862

sdot

⟨]119891Σ

119891sdot V sdot 119899

119862sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C8)


119862(119903 119864) that is


Λ

119862equiv

⟨119899

119862 119899

+

119862+119877⟩

119862

⟨]119891Σ

119891sdot V sdot 119899

119862sdot 119899

+

119862+119877⟩

119862

(C9)



119862= (119899

119862+119877minus Δ119899



⟨]119891Σ

119891sdot V sdot [119899

119862+119877minus Δ119899

119862+119877] sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

= 1 minus

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C10)


119877from those


120588

119877=

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C11)


⟨119899

119862 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

= Λ

119862sdot (1 minus 120588

119877) (C12)


119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877 then the sum-


⟨Δ119899

119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

=

⟨Δ119899

119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

sdot

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C13)


Λ

119877equiv

⟨Δ119899

119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C14)









119877

Λprt = Λ

119862sdot (1 minus 120588

119877) + Λ

119877sdot 120588

119877 (C15)


119877


Λ

119895+1

prt = Λ

119895


119895+1

119877) + Λ

119895+1

119877sdot 120588

119895+1

119877 (C16)



120588 =

Λ

119862

119879

+ int

Λmax

Λmin

Λ

1015840

119877sdot (119889120588

119877119889Λ

1015840

119877)

Λ

1015840

119877+ 119879

sdot 119889Λ

1015840

119877+

6

sum

119894=1

120573

119894

1 + 120582

119894sdot 119879

(D1)




119877(Λ














119889119899 (119905)

119889119905

=

120588

0minus 120573 minus 120588

119877(Λmax)

Λ

119862

sdot 119899 (119905)

+ int

Λmax

Λ119862

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

= 120573

119894sdot

119899 (119905)

Λ

119862

minus 120582

119894sdot 119862

119894(119905) 119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

=

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

minus

1

Λ

119877

sdot 119862

119877(Λ

119877 119905)

119889120588

119877

119889Λ

119877

=

119889120588

119877(119903)

119889119903

sdot

119889119903

119889Λ

119877

(D2)


119862asymp 10

minus7divide 10









0= 10




119889119899 (119905)

119889119905

=

120588

0minus 120573 minus [120588

119877(Λmax) minus 120588

119877(Λ

0)]

Λ

119862

sdot 119899 (119905)

+ int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

= 120573

119894sdot

119899 (119905)

Λ

119862

minus 120582

119894sdot 119862

119894(119905) 119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

=

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

minus

1

Λ

119877

sdot 119862

119877(Λ

119877 119905)

(D3)



120588 =

Λ

119862

119879

+ int

Λmax

Λ 0

Λ

1015840

119877sdot (119889120588

119877119889Λ

1015840

119877)

Λ

1015840

119877+ 119879

sdot 119889Λ

1015840

119877+

6

sum

119894=1

120573

119894

1 + 120582

119894sdot 119879

(D4)



119899 (119905)

Λ

119862

=

(minus1)

120588

0minus 120573 minus [120588

119877(Λmax) minus 120588

119877(Λ

0)]

sdot [int

Λmax

Λ 0

1

Λ

1015840

119877

119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894119862

119894(119905)]








119889119862

119894(119905)

119889119905

+ 120582

119894sdot 119862

119894(119905) = 120573

119894sdot

119899 (119905)

Λ

119862

119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

+

1

Λ

119877

sdot 119862

119877(Λ

119877 119905) =

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

(D5)





119889119910

119889119905

+ 120582 sdot 119910 = 119860 (119905)

119910 (119905) = 119910 (0) sdot 119890


119905

0

119889119905

1015840sdot 119890

minus120582(119905minus1199051015840)sdot 119860 (119905

1015840)

(D6)


119894(119905 = minus0) = 119862

119894(119905 =




119877(Λ 119905) must



119862

119877(Λ

119877 119905 = +0)

119862

119877(Λ

119877 119905 = minus0)

=

119899 (119905 = +0)

119899 (119905 = minus0)

(D7)






119889119899 (119905)

119889119905

=

120588

0minus 120573 minus [120588

119877(Λmax) minus 120588

119877(Λ

0)] minus 120572

119879int

119905

0119899 (119905

1015840) sdot 119889119905

1015840

Λ

119862

sdot 119899 (119905)

+ int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

+ 120582

119894sdot 119862

119894(119905) = 120573

119894sdot

119899 (119905)

Λ

119862

119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

+

1

Λ

119877

sdot 119862

119877(Λ

119877 119905) =

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

(D8)


0119899(119905

1015840)119889119905





119889119899 (119905)

119889119905

=

120588

0minus 120573 minus 120572

119879int

119905

0119899 (119905

1015840) sdot 119889119905

1015840

Λ

119862

sdot 119899 (119905) (D9)



120573 = 120573 + [120588

119877(Λmax) minus 120588

119877(Λ

0)] thanks





119910 (119905) equiv [120588

0minus

120573 minus 120572

119879int

119905

0

119899 (119905

1015840) sdot 119889119905

1015840]

119889119910 (119905)

119889119905

= minus120572

119879sdot 119899 (119905)

119910 (119905) sdot

119889119910 (119905)

119889119905

= minus (120572

119879sdot Λ

119862) sdot [

119910 (119905)

Λ

119862

sdot 119899 (119905)]

(D10)


119862) sdot 119899(119905)

[

119910 (119905)

Λ

119862

sdot 119899 (119905)]

= minusint

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

(D11)



120588

0lt

120573 = 120573 + [120588

119877(Λmax) minus 120588

119877(Λ

0)] (D12)


119910 (119905) sdot

119889119910 (119905)

119889119905

= + (120572

119879sdot Λ

119862)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

(D13)


1

2

119889119910

2

119889119905

= + (120572

119879sdot Λ

119862)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119910

2(119905) = 2 sdot (120572

119879sdot Λ

119862)

times int

119905

0

119889119905

1015840int

Λmax

Λ 0

1

Λ

1015840

119877

119862

119877(Λ

1015840

119877 119905

1015840) sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905

1015840) + const

(D14)


2(0) = (120588

0minus

120573)

2


0minus

120573)

2

Then

119910 (119905) = minus2 sdot (120572

119879sdot Λ

119862)

times int

119905

0

119889119905

1015840[int

Λmax

Λ 0

1

Λ

1015840

119877

119862

119877(Λ

1015840

119877 119905

1015840) 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894119862

119894(119905

1015840)] + (120588

0minus

120573)

2

12

(D15)


[

119899 (119905)

Λ

119862

] =

(minus1)

119910 (119905)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

+ 120582

119894sdot 119862

119894(119905) = 120573

119894sdot [

119899 (119905)

Λ

119862

] 119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

+

1

Λ

119877

sdot 119862

119877(Λ

119877 119905) =

119889120588

119877

119889Λ

119877

sdot [

119899 (119905)

Λ

119862

]

(D16)


[

119899 (119905)

Λ

119862

] =

(minus1)

119910 (119905)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119862

119894(119905) = 119862

119894(119905 = +0) sdot 119890

minus120582119894119905

+ 120573

119894sdot int

119905

0

119889119905

1015840sdot 119890

minus120582119894(119905minus1199051015840)sdot [

119899 (119905

1015840)

Λ

119862

]

119862

119877(Λ

119877 119905) = 119862

119877(Λ

119877 119905 = +0)

sdot 119890

minus119905Λ 119877+

119889120588

119877

119889Λ

119877

sdot int

119905

0

119889119905

1015840sdot 119890

minus(1Λ 119877)(119905minus1199051015840)

sdot [

119899 (119905

1015840)

Λ

119862

]

(D17)




References





































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of




















С R

r

Core Reflector




119871119899 =

1

119870efsdot

119876119899

119871

+119899

+=

1

119870efsdot

119876

+119899

+

(C2)

where operator












tion 119899

+


119871

119862+119877119899

119862+119877=

1

119870efsdot

119876119899

119862+119877 (C3)

119871

+

119862+119877119899

+

119862+119877=

1

119870efsdot

119876

+119899

+

119862+119877 (C4)




119871

119862119899

119862=

1

119870efsdot

119876119899

119862+119877 (C5)


119862(119903 119864) characterizes




119862+119877= (119899

119862+119877minus 119899






119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C6)


119862+119877can be

replaced by 119899

119862+119877= 119899

119862+ Δ119899



C R

r

n C+R

n C

Δn C+R


119862+119877(119903) in the bare core 119899


Δ119899

119862+119877(119903)


Λprt =⟨(119899

119862+ Δ119899

119862+119877) 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

=

⟨119899

119862 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

+

⟨Δ119899

119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C7)


119891Σ

119891sdot V sdot 119899

119888sdot 119899

+

119862+119877⟩


the following form

⟨119899

119862 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

=

⟨119899

119862 119899

+

119862+119877⟩

119862

⟨]119891Σ

119891sdot V sdot 119899

119862sdot 119899

+

119862+119877⟩

119862

sdot

⟨]119891Σ

119891sdot V sdot 119899

119862sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C8)


119862(119903 119864) that is


Λ

119862equiv

⟨119899

119862 119899

+

119862+119877⟩

119862

⟨]119891Σ

119891sdot V sdot 119899

119862sdot 119899

+

119862+119877⟩

119862

(C9)



119862= (119899

119862+119877minus Δ119899



⟨]119891Σ

119891sdot V sdot [119899

119862+119877minus Δ119899

119862+119877] sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

= 1 minus

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C10)


119877from those


120588

119877=

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C11)


⟨119899

119862 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

= Λ

119862sdot (1 minus 120588

119877) (C12)


119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877 then the sum-


⟨Δ119899

119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

=

⟨Δ119899

119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

sdot

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C13)


Λ

119877equiv

⟨Δ119899

119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C14)









119877

Λprt = Λ

119862sdot (1 minus 120588

119877) + Λ

119877sdot 120588

119877 (C15)


119877


Λ

119895+1

prt = Λ

119895


119895+1

119877) + Λ

119895+1

119877sdot 120588

119895+1

119877 (C16)



120588 =

Λ

119862

119879

+ int

Λmax

Λmin

Λ

1015840

119877sdot (119889120588

119877119889Λ

1015840

119877)

Λ

1015840

119877+ 119879

sdot 119889Λ

1015840

119877+

6

sum

119894=1

120573

119894

1 + 120582

119894sdot 119879

(D1)




119877(Λ














119889119899 (119905)

119889119905

=

120588

0minus 120573 minus 120588

119877(Λmax)

Λ

119862

sdot 119899 (119905)

+ int

Λmax

Λ119862

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

= 120573

119894sdot

119899 (119905)

Λ

119862

minus 120582

119894sdot 119862

119894(119905) 119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

=

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

minus

1

Λ

119877

sdot 119862

119877(Λ

119877 119905)

119889120588

119877

119889Λ

119877

=

119889120588

119877(119903)

119889119903

sdot

119889119903

119889Λ

119877

(D2)


119862asymp 10

minus7divide 10









0= 10




119889119899 (119905)

119889119905

=

120588

0minus 120573 minus [120588

119877(Λmax) minus 120588

119877(Λ

0)]

Λ

119862

sdot 119899 (119905)

+ int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

= 120573

119894sdot

119899 (119905)

Λ

119862

minus 120582

119894sdot 119862

119894(119905) 119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

=

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

minus

1

Λ

119877

sdot 119862

119877(Λ

119877 119905)

(D3)



120588 =

Λ

119862

119879

+ int

Λmax

Λ 0

Λ

1015840

119877sdot (119889120588

119877119889Λ

1015840

119877)

Λ

1015840

119877+ 119879

sdot 119889Λ

1015840

119877+

6

sum

119894=1

120573

119894

1 + 120582

119894sdot 119879

(D4)



119899 (119905)

Λ

119862

=

(minus1)

120588

0minus 120573 minus [120588

119877(Λmax) minus 120588

119877(Λ

0)]

sdot [int

Λmax

Λ 0

1

Λ

1015840

119877

119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894119862

119894(119905)]








119889119862

119894(119905)

119889119905

+ 120582

119894sdot 119862

119894(119905) = 120573

119894sdot

119899 (119905)

Λ

119862

119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

+

1

Λ

119877

sdot 119862

119877(Λ

119877 119905) =

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

(D5)





119889119910

119889119905

+ 120582 sdot 119910 = 119860 (119905)

119910 (119905) = 119910 (0) sdot 119890


119905

0

119889119905

1015840sdot 119890

minus120582(119905minus1199051015840)sdot 119860 (119905

1015840)

(D6)


119894(119905 = minus0) = 119862

119894(119905 =




119877(Λ 119905) must



119862

119877(Λ

119877 119905 = +0)

119862

119877(Λ

119877 119905 = minus0)

=

119899 (119905 = +0)

119899 (119905 = minus0)

(D7)






119889119899 (119905)

119889119905

=

120588

0minus 120573 minus [120588

119877(Λmax) minus 120588

119877(Λ

0)] minus 120572

119879int

119905

0119899 (119905

1015840) sdot 119889119905

1015840

Λ

119862

sdot 119899 (119905)

+ int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

+ 120582

119894sdot 119862

119894(119905) = 120573

119894sdot

119899 (119905)

Λ

119862

119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

+

1

Λ

119877

sdot 119862

119877(Λ

119877 119905) =

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

(D8)


0119899(119905

1015840)119889119905





119889119899 (119905)

119889119905

=

120588

0minus 120573 minus 120572

119879int

119905

0119899 (119905

1015840) sdot 119889119905

1015840

Λ

119862

sdot 119899 (119905) (D9)



120573 = 120573 + [120588

119877(Λmax) minus 120588

119877(Λ

0)] thanks





119910 (119905) equiv [120588

0minus

120573 minus 120572

119879int

119905

0

119899 (119905

1015840) sdot 119889119905

1015840]

119889119910 (119905)

119889119905

= minus120572

119879sdot 119899 (119905)

119910 (119905) sdot

119889119910 (119905)

119889119905

= minus (120572

119879sdot Λ

119862) sdot [

119910 (119905)

Λ

119862

sdot 119899 (119905)]

(D10)


119862) sdot 119899(119905)

[

119910 (119905)

Λ

119862

sdot 119899 (119905)]

= minusint

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

(D11)



120588

0lt

120573 = 120573 + [120588

119877(Λmax) minus 120588

119877(Λ

0)] (D12)


119910 (119905) sdot

119889119910 (119905)

119889119905

= + (120572

119879sdot Λ

119862)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

(D13)


1

2

119889119910

2

119889119905

= + (120572

119879sdot Λ

119862)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119910

2(119905) = 2 sdot (120572

119879sdot Λ

119862)

times int

119905

0

119889119905

1015840int

Λmax

Λ 0

1

Λ

1015840

119877

119862

119877(Λ

1015840

119877 119905

1015840) sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905

1015840) + const

(D14)


2(0) = (120588

0minus

120573)

2


0minus

120573)

2

Then

119910 (119905) = minus2 sdot (120572

119879sdot Λ

119862)

times int

119905

0

119889119905

1015840[int

Λmax

Λ 0

1

Λ

1015840

119877

119862

119877(Λ

1015840

119877 119905

1015840) 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894119862

119894(119905

1015840)] + (120588

0minus

120573)

2

12

(D15)


[

119899 (119905)

Λ

119862

] =

(minus1)

119910 (119905)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

+ 120582

119894sdot 119862

119894(119905) = 120573

119894sdot [

119899 (119905)

Λ

119862

] 119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

+

1

Λ

119877

sdot 119862

119877(Λ

119877 119905) =

119889120588

119877

119889Λ

119877

sdot [

119899 (119905)

Λ

119862

]

(D16)


[

119899 (119905)

Λ

119862

] =

(minus1)

119910 (119905)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119862

119894(119905) = 119862

119894(119905 = +0) sdot 119890

minus120582119894119905

+ 120573

119894sdot int

119905

0

119889119905

1015840sdot 119890

minus120582119894(119905minus1199051015840)sdot [

119899 (119905

1015840)

Λ

119862

]

119862

119877(Λ

119877 119905) = 119862

119877(Λ

119877 119905 = +0)

sdot 119890

minus119905Λ 119877+

119889120588

119877

119889Λ

119877

sdot int

119905

0

119889119905

1015840sdot 119890

minus(1Λ 119877)(119905minus1199051015840)

sdot [

119899 (119905

1015840)

Λ

119862

]

(D17)




References





































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of


















C R

r

n C+R

n C

Δn C+R


119862+119877(119903) in the bare core 119899


Δ119899

119862+119877(119903)


Λprt =⟨(119899

119862+ Δ119899

119862+119877) 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

=

⟨119899

119862 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

+

⟨Δ119899

119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C7)


119891Σ

119891sdot V sdot 119899

119888sdot 119899

+

119862+119877⟩


the following form

⟨119899

119862 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

=

⟨119899

119862 119899

+

119862+119877⟩

119862

⟨]119891Σ

119891sdot V sdot 119899

119862sdot 119899

+

119862+119877⟩

119862

sdot

⟨]119891Σ

119891sdot V sdot 119899

119862sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C8)


119862(119903 119864) that is


Λ

119862equiv

⟨119899

119862 119899

+

119862+119877⟩

119862

⟨]119891Σ

119891sdot V sdot 119899

119862sdot 119899

+

119862+119877⟩

119862

(C9)



119862= (119899

119862+119877minus Δ119899



⟨]119891Σ

119891sdot V sdot [119899

119862+119877minus Δ119899

119862+119877] sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

= 1 minus

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C10)


119877from those


120588

119877=

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C11)


⟨119899

119862 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

= Λ

119862sdot (1 minus 120588

119877) (C12)


119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877 then the sum-


⟨Δ119899

119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

=

⟨Δ119899

119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

sdot

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ

119891sdot V sdot 119899

119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C13)


Λ

119877equiv

⟨Δ119899

119862+119877 119899

+

119862+119877⟩

119862+119877

⟨]119891Σ


119862+119877sdot 119899

+

119862+119877⟩

119862+119877

(C14)









119877

Λprt = Λ

119862sdot (1 minus 120588

119877) + Λ

119877sdot 120588

119877 (C15)


119877


Λ

119895+1

prt = Λ

119895


119895+1

119877) + Λ

119895+1

119877sdot 120588

119895+1

119877 (C16)



120588 =

Λ

119862

119879

+ int

Λmax

Λmin

Λ

1015840

119877sdot (119889120588

119877119889Λ

1015840

119877)

Λ

1015840

119877+ 119879

sdot 119889Λ

1015840

119877+

6

sum

119894=1

120573

119894

1 + 120582

119894sdot 119879

(D1)




119877(Λ














119889119899 (119905)

119889119905

=

120588

0minus 120573 minus 120588

119877(Λmax)

Λ

119862

sdot 119899 (119905)

+ int

Λmax

Λ119862

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

= 120573

119894sdot

119899 (119905)

Λ

119862

minus 120582

119894sdot 119862

119894(119905) 119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

=

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

minus

1

Λ

119877

sdot 119862

119877(Λ

119877 119905)

119889120588

119877

119889Λ

119877

=

119889120588

119877(119903)

119889119903

sdot

119889119903

119889Λ

119877

(D2)


119862asymp 10

minus7divide 10









0= 10




119889119899 (119905)

119889119905

=

120588

0minus 120573 minus [120588

119877(Λmax) minus 120588

119877(Λ

0)]

Λ

119862

sdot 119899 (119905)

+ int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

= 120573

119894sdot

119899 (119905)

Λ

119862

minus 120582

119894sdot 119862

119894(119905) 119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

=

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

minus

1

Λ

119877

sdot 119862

119877(Λ

119877 119905)

(D3)



120588 =

Λ

119862

119879

+ int

Λmax

Λ 0

Λ

1015840

119877sdot (119889120588

119877119889Λ

1015840

119877)

Λ

1015840

119877+ 119879

sdot 119889Λ

1015840

119877+

6

sum

119894=1

120573

119894

1 + 120582

119894sdot 119879

(D4)



119899 (119905)

Λ

119862

=

(minus1)

120588

0minus 120573 minus [120588

119877(Λmax) minus 120588

119877(Λ

0)]

sdot [int

Λmax

Λ 0

1

Λ

1015840

119877

119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894119862

119894(119905)]








119889119862

119894(119905)

119889119905

+ 120582

119894sdot 119862

119894(119905) = 120573

119894sdot

119899 (119905)

Λ

119862

119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

+

1

Λ

119877

sdot 119862

119877(Λ

119877 119905) =

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

(D5)





119889119910

119889119905

+ 120582 sdot 119910 = 119860 (119905)

119910 (119905) = 119910 (0) sdot 119890


119905

0

119889119905

1015840sdot 119890

minus120582(119905minus1199051015840)sdot 119860 (119905

1015840)

(D6)


119894(119905 = minus0) = 119862

119894(119905 =




119877(Λ 119905) must



119862

119877(Λ

119877 119905 = +0)

119862

119877(Λ

119877 119905 = minus0)

=

119899 (119905 = +0)

119899 (119905 = minus0)

(D7)






119889119899 (119905)

119889119905

=

120588

0minus 120573 minus [120588

119877(Λmax) minus 120588

119877(Λ

0)] minus 120572

119879int

119905

0119899 (119905

1015840) sdot 119889119905

1015840

Λ

119862

sdot 119899 (119905)

+ int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

+ 120582

119894sdot 119862

119894(119905) = 120573

119894sdot

119899 (119905)

Λ

119862

119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

+

1

Λ

119877

sdot 119862

119877(Λ

119877 119905) =

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

(D8)


0119899(119905

1015840)119889119905





119889119899 (119905)

119889119905

=

120588

0minus 120573 minus 120572

119879int

119905

0119899 (119905

1015840) sdot 119889119905

1015840

Λ

119862

sdot 119899 (119905) (D9)



120573 = 120573 + [120588

119877(Λmax) minus 120588

119877(Λ

0)] thanks





119910 (119905) equiv [120588

0minus

120573 minus 120572

119879int

119905

0

119899 (119905

1015840) sdot 119889119905

1015840]

119889119910 (119905)

119889119905

= minus120572

119879sdot 119899 (119905)

119910 (119905) sdot

119889119910 (119905)

119889119905

= minus (120572

119879sdot Λ

119862) sdot [

119910 (119905)

Λ

119862

sdot 119899 (119905)]

(D10)


119862) sdot 119899(119905)

[

119910 (119905)

Λ

119862

sdot 119899 (119905)]

= minusint

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

(D11)



120588

0lt

120573 = 120573 + [120588

119877(Λmax) minus 120588

119877(Λ

0)] (D12)


119910 (119905) sdot

119889119910 (119905)

119889119905

= + (120572

119879sdot Λ

119862)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

(D13)


1

2

119889119910

2

119889119905

= + (120572

119879sdot Λ

119862)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119910

2(119905) = 2 sdot (120572

119879sdot Λ

119862)

times int

119905

0

119889119905

1015840int

Λmax

Λ 0

1

Λ

1015840

119877

119862

119877(Λ

1015840

119877 119905

1015840) sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905

1015840) + const

(D14)


2(0) = (120588

0minus

120573)

2


0minus

120573)

2

Then

119910 (119905) = minus2 sdot (120572

119879sdot Λ

119862)

times int

119905

0

119889119905

1015840[int

Λmax

Λ 0

1

Λ

1015840

119877

119862

119877(Λ

1015840

119877 119905

1015840) 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894119862

119894(119905

1015840)] + (120588

0minus

120573)

2

12

(D15)


[

119899 (119905)

Λ

119862

] =

(minus1)

119910 (119905)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

+ 120582

119894sdot 119862

119894(119905) = 120573

119894sdot [

119899 (119905)

Λ

119862

] 119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

+

1

Λ

119877

sdot 119862

119877(Λ

119877 119905) =

119889120588

119877

119889Λ

119877

sdot [

119899 (119905)

Λ

119862

]

(D16)


[

119899 (119905)

Λ

119862

] =

(minus1)

119910 (119905)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119862

119894(119905) = 119862

119894(119905 = +0) sdot 119890

minus120582119894119905

+ 120573

119894sdot int

119905

0

119889119905

1015840sdot 119890

minus120582119894(119905minus1199051015840)sdot [

119899 (119905

1015840)

Λ

119862

]

119862

119877(Λ

119877 119905) = 119862

119877(Λ

119877 119905 = +0)

sdot 119890

minus119905Λ 119877+

119889120588

119877

119889Λ

119877

sdot int

119905

0

119889119905

1015840sdot 119890

minus(1Λ 119877)(119905minus1199051015840)

sdot [

119899 (119905

1015840)

Λ

119862

]

(D17)




References





































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of























119877

Λprt = Λ

119862sdot (1 minus 120588

119877) + Λ

119877sdot 120588

119877 (C15)


119877


Λ

119895+1

prt = Λ

119895


119895+1

119877) + Λ

119895+1

119877sdot 120588

119895+1

119877 (C16)



120588 =

Λ

119862

119879

+ int

Λmax

Λmin

Λ

1015840

119877sdot (119889120588

119877119889Λ

1015840

119877)

Λ

1015840

119877+ 119879

sdot 119889Λ

1015840

119877+

6

sum

119894=1

120573

119894

1 + 120582

119894sdot 119879

(D1)




119877(Λ














119889119899 (119905)

119889119905

=

120588

0minus 120573 minus 120588

119877(Λmax)

Λ

119862

sdot 119899 (119905)

+ int

Λmax

Λ119862

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

= 120573

119894sdot

119899 (119905)

Λ

119862

minus 120582

119894sdot 119862

119894(119905) 119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

=

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

minus

1

Λ

119877

sdot 119862

119877(Λ

119877 119905)

119889120588

119877

119889Λ

119877

=

119889120588

119877(119903)

119889119903

sdot

119889119903

119889Λ

119877

(D2)


119862asymp 10

minus7divide 10









0= 10




119889119899 (119905)

119889119905

=

120588

0minus 120573 minus [120588

119877(Λmax) minus 120588

119877(Λ

0)]

Λ

119862

sdot 119899 (119905)

+ int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

= 120573

119894sdot

119899 (119905)

Λ

119862

minus 120582

119894sdot 119862

119894(119905) 119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

=

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

minus

1

Λ

119877

sdot 119862

119877(Λ

119877 119905)

(D3)



120588 =

Λ

119862

119879

+ int

Λmax

Λ 0

Λ

1015840

119877sdot (119889120588

119877119889Λ

1015840

119877)

Λ

1015840

119877+ 119879

sdot 119889Λ

1015840

119877+

6

sum

119894=1

120573

119894

1 + 120582

119894sdot 119879

(D4)



119899 (119905)

Λ

119862

=

(minus1)

120588

0minus 120573 minus [120588

119877(Λmax) minus 120588

119877(Λ

0)]

sdot [int

Λmax

Λ 0

1

Λ

1015840

119877

119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894119862

119894(119905)]








119889119862

119894(119905)

119889119905

+ 120582

119894sdot 119862

119894(119905) = 120573

119894sdot

119899 (119905)

Λ

119862

119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

+

1

Λ

119877

sdot 119862

119877(Λ

119877 119905) =

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

(D5)





119889119910

119889119905

+ 120582 sdot 119910 = 119860 (119905)

119910 (119905) = 119910 (0) sdot 119890


119905

0

119889119905

1015840sdot 119890

minus120582(119905minus1199051015840)sdot 119860 (119905

1015840)

(D6)


119894(119905 = minus0) = 119862

119894(119905 =




119877(Λ 119905) must



119862

119877(Λ

119877 119905 = +0)

119862

119877(Λ

119877 119905 = minus0)

=

119899 (119905 = +0)

119899 (119905 = minus0)

(D7)






119889119899 (119905)

119889119905

=

120588

0minus 120573 minus [120588

119877(Λmax) minus 120588

119877(Λ

0)] minus 120572

119879int

119905

0119899 (119905

1015840) sdot 119889119905

1015840

Λ

119862

sdot 119899 (119905)

+ int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

+ 120582

119894sdot 119862

119894(119905) = 120573

119894sdot

119899 (119905)

Λ

119862

119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

+

1

Λ

119877

sdot 119862

119877(Λ

119877 119905) =

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

(D8)


0119899(119905

1015840)119889119905





119889119899 (119905)

119889119905

=

120588

0minus 120573 minus 120572

119879int

119905

0119899 (119905

1015840) sdot 119889119905

1015840

Λ

119862

sdot 119899 (119905) (D9)



120573 = 120573 + [120588

119877(Λmax) minus 120588

119877(Λ

0)] thanks





119910 (119905) equiv [120588

0minus

120573 minus 120572

119879int

119905

0

119899 (119905

1015840) sdot 119889119905

1015840]

119889119910 (119905)

119889119905

= minus120572

119879sdot 119899 (119905)

119910 (119905) sdot

119889119910 (119905)

119889119905

= minus (120572

119879sdot Λ

119862) sdot [

119910 (119905)

Λ

119862

sdot 119899 (119905)]

(D10)


119862) sdot 119899(119905)

[

119910 (119905)

Λ

119862

sdot 119899 (119905)]

= minusint

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

(D11)



120588

0lt

120573 = 120573 + [120588

119877(Λmax) minus 120588

119877(Λ

0)] (D12)


119910 (119905) sdot

119889119910 (119905)

119889119905

= + (120572

119879sdot Λ

119862)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

(D13)


1

2

119889119910

2

119889119905

= + (120572

119879sdot Λ

119862)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119910

2(119905) = 2 sdot (120572

119879sdot Λ

119862)

times int

119905

0

119889119905

1015840int

Λmax

Λ 0

1

Λ

1015840

119877

119862

119877(Λ

1015840

119877 119905

1015840) sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905

1015840) + const

(D14)


2(0) = (120588

0minus

120573)

2


0minus

120573)

2

Then

119910 (119905) = minus2 sdot (120572

119879sdot Λ

119862)

times int

119905

0

119889119905

1015840[int

Λmax

Λ 0

1

Λ

1015840

119877

119862

119877(Λ

1015840

119877 119905

1015840) 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894119862

119894(119905

1015840)] + (120588

0minus

120573)

2

12

(D15)


[

119899 (119905)

Λ

119862

] =

(minus1)

119910 (119905)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

+ 120582

119894sdot 119862

119894(119905) = 120573

119894sdot [

119899 (119905)

Λ

119862

] 119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

+

1

Λ

119877

sdot 119862

119877(Λ

119877 119905) =

119889120588

119877

119889Λ

119877

sdot [

119899 (119905)

Λ

119862

]

(D16)


[

119899 (119905)

Λ

119862

] =

(minus1)

119910 (119905)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119862

119894(119905) = 119862

119894(119905 = +0) sdot 119890

minus120582119894119905

+ 120573

119894sdot int

119905

0

119889119905

1015840sdot 119890

minus120582119894(119905minus1199051015840)sdot [

119899 (119905

1015840)

Λ

119862

]

119862

119877(Λ

119877 119905) = 119862

119877(Λ

119877 119905 = +0)

sdot 119890

minus119905Λ 119877+

119889120588

119877

119889Λ

119877

sdot int

119905

0

119889119905

1015840sdot 119890

minus(1Λ 119877)(119905minus1199051015840)

sdot [

119899 (119905

1015840)

Λ

119862

]

(D17)




References





































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of






















0= 10




119889119899 (119905)

119889119905

=

120588

0minus 120573 minus [120588

119877(Λmax) minus 120588

119877(Λ

0)]

Λ

119862

sdot 119899 (119905)

+ int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

= 120573

119894sdot

119899 (119905)

Λ

119862

minus 120582

119894sdot 119862

119894(119905) 119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

=

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

minus

1

Λ

119877

sdot 119862

119877(Λ

119877 119905)

(D3)



120588 =

Λ

119862

119879

+ int

Λmax

Λ 0

Λ

1015840

119877sdot (119889120588

119877119889Λ

1015840

119877)

Λ

1015840

119877+ 119879

sdot 119889Λ

1015840

119877+

6

sum

119894=1

120573

119894

1 + 120582

119894sdot 119879

(D4)



119899 (119905)

Λ

119862

=

(minus1)

120588

0minus 120573 minus [120588

119877(Λmax) minus 120588

119877(Λ

0)]

sdot [int

Λmax

Λ 0

1

Λ

1015840

119877

119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894119862

119894(119905)]








119889119862

119894(119905)

119889119905

+ 120582

119894sdot 119862

119894(119905) = 120573

119894sdot

119899 (119905)

Λ

119862

119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

+

1

Λ

119877

sdot 119862

119877(Λ

119877 119905) =

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

(D5)





119889119910

119889119905

+ 120582 sdot 119910 = 119860 (119905)

119910 (119905) = 119910 (0) sdot 119890


119905

0

119889119905

1015840sdot 119890

minus120582(119905minus1199051015840)sdot 119860 (119905

1015840)

(D6)


119894(119905 = minus0) = 119862

119894(119905 =




119877(Λ 119905) must



119862

119877(Λ

119877 119905 = +0)

119862

119877(Λ

119877 119905 = minus0)

=

119899 (119905 = +0)

119899 (119905 = minus0)

(D7)






119889119899 (119905)

119889119905

=

120588

0minus 120573 minus [120588

119877(Λmax) minus 120588

119877(Λ

0)] minus 120572

119879int

119905

0119899 (119905

1015840) sdot 119889119905

1015840

Λ

119862

sdot 119899 (119905)

+ int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

+ 120582

119894sdot 119862

119894(119905) = 120573

119894sdot

119899 (119905)

Λ

119862

119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

+

1

Λ

119877

sdot 119862

119877(Λ

119877 119905) =

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

(D8)


0119899(119905

1015840)119889119905





119889119899 (119905)

119889119905

=

120588

0minus 120573 minus 120572

119879int

119905

0119899 (119905

1015840) sdot 119889119905

1015840

Λ

119862

sdot 119899 (119905) (D9)



120573 = 120573 + [120588

119877(Λmax) minus 120588

119877(Λ

0)] thanks





119910 (119905) equiv [120588

0minus

120573 minus 120572

119879int

119905

0

119899 (119905

1015840) sdot 119889119905

1015840]

119889119910 (119905)

119889119905

= minus120572

119879sdot 119899 (119905)

119910 (119905) sdot

119889119910 (119905)

119889119905

= minus (120572

119879sdot Λ

119862) sdot [

119910 (119905)

Λ

119862

sdot 119899 (119905)]

(D10)


119862) sdot 119899(119905)

[

119910 (119905)

Λ

119862

sdot 119899 (119905)]

= minusint

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

(D11)



120588

0lt

120573 = 120573 + [120588

119877(Λmax) minus 120588

119877(Λ

0)] (D12)


119910 (119905) sdot

119889119910 (119905)

119889119905

= + (120572

119879sdot Λ

119862)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

(D13)


1

2

119889119910

2

119889119905

= + (120572

119879sdot Λ

119862)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119910

2(119905) = 2 sdot (120572

119879sdot Λ

119862)

times int

119905

0

119889119905

1015840int

Λmax

Λ 0

1

Λ

1015840

119877

119862

119877(Λ

1015840

119877 119905

1015840) sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905

1015840) + const

(D14)


2(0) = (120588

0minus

120573)

2


0minus

120573)

2

Then

119910 (119905) = minus2 sdot (120572

119879sdot Λ

119862)

times int

119905

0

119889119905

1015840[int

Λmax

Λ 0

1

Λ

1015840

119877

119862

119877(Λ

1015840

119877 119905

1015840) 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894119862

119894(119905

1015840)] + (120588

0minus

120573)

2

12

(D15)


[

119899 (119905)

Λ

119862

] =

(minus1)

119910 (119905)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

+ 120582

119894sdot 119862

119894(119905) = 120573

119894sdot [

119899 (119905)

Λ

119862

] 119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

+

1

Λ

119877

sdot 119862

119877(Λ

119877 119905) =

119889120588

119877

119889Λ

119877

sdot [

119899 (119905)

Λ

119862

]

(D16)


[

119899 (119905)

Λ

119862

] =

(minus1)

119910 (119905)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119862

119894(119905) = 119862

119894(119905 = +0) sdot 119890

minus120582119894119905

+ 120573

119894sdot int

119905

0

119889119905

1015840sdot 119890

minus120582119894(119905minus1199051015840)sdot [

119899 (119905

1015840)

Λ

119862

]

119862

119877(Λ

119877 119905) = 119862

119877(Λ

119877 119905 = +0)

sdot 119890

minus119905Λ 119877+

119889120588

119877

119889Λ

119877

sdot int

119905

0

119889119905

1015840sdot 119890

minus(1Λ 119877)(119905minus1199051015840)

sdot [

119899 (119905

1015840)

Λ

119862

]

(D17)




References





































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of



















119862

119877(Λ

119877 119905 = +0)

119862

119877(Λ

119877 119905 = minus0)

=

119899 (119905 = +0)

119899 (119905 = minus0)

(D7)






119889119899 (119905)

119889119905

=

120588

0minus 120573 minus [120588

119877(Λmax) minus 120588

119877(Λ

0)] minus 120572

119879int

119905

0119899 (119905

1015840) sdot 119889119905

1015840

Λ

119862

sdot 119899 (119905)

+ int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

+ 120582

119894sdot 119862

119894(119905) = 120573

119894sdot

119899 (119905)

Λ

119862

119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

+

1

Λ

119877

sdot 119862

119877(Λ

119877 119905) =

119889120588

119877

119889Λ

119877

sdot

119899 (119905)

Λ

119862

(D8)


0119899(119905

1015840)119889119905





119889119899 (119905)

119889119905

=

120588

0minus 120573 minus 120572

119879int

119905

0119899 (119905

1015840) sdot 119889119905

1015840

Λ

119862

sdot 119899 (119905) (D9)



120573 = 120573 + [120588

119877(Λmax) minus 120588

119877(Λ

0)] thanks





119910 (119905) equiv [120588

0minus

120573 minus 120572

119879int

119905

0

119899 (119905

1015840) sdot 119889119905

1015840]

119889119910 (119905)

119889119905

= minus120572

119879sdot 119899 (119905)

119910 (119905) sdot

119889119910 (119905)

119889119905

= minus (120572

119879sdot Λ

119862) sdot [

119910 (119905)

Λ

119862

sdot 119899 (119905)]

(D10)


119862) sdot 119899(119905)

[

119910 (119905)

Λ

119862

sdot 119899 (119905)]

= minusint

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

(D11)



120588

0lt

120573 = 120573 + [120588

119877(Λmax) minus 120588

119877(Λ

0)] (D12)


119910 (119905) sdot

119889119910 (119905)

119889119905

= + (120572

119879sdot Λ

119862)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

(D13)


1

2

119889119910

2

119889119905

= + (120572

119879sdot Λ

119862)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119910

2(119905) = 2 sdot (120572

119879sdot Λ

119862)

times int

119905

0

119889119905

1015840int

Λmax

Λ 0

1

Λ

1015840

119877

119862

119877(Λ

1015840

119877 119905

1015840) sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905

1015840) + const

(D14)


2(0) = (120588

0minus

120573)

2


0minus

120573)

2

Then

119910 (119905) = minus2 sdot (120572

119879sdot Λ

119862)

times int

119905

0

119889119905

1015840[int

Λmax

Λ 0

1

Λ

1015840

119877

119862

119877(Λ

1015840

119877 119905

1015840) 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894119862

119894(119905

1015840)] + (120588

0minus

120573)

2

12

(D15)


[

119899 (119905)

Λ

119862

] =

(minus1)

119910 (119905)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

+ 120582

119894sdot 119862

119894(119905) = 120573

119894sdot [

119899 (119905)

Λ

119862

] 119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

+

1

Λ

119877

sdot 119862

119877(Λ

119877 119905) =

119889120588

119877

119889Λ

119877

sdot [

119899 (119905)

Λ

119862

]

(D16)


[

119899 (119905)

Λ

119862

] =

(minus1)

119910 (119905)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119862

119894(119905) = 119862

119894(119905 = +0) sdot 119890

minus120582119894119905

+ 120573

119894sdot int

119905

0

119889119905

1015840sdot 119890

minus120582119894(119905minus1199051015840)sdot [

119899 (119905

1015840)

Λ

119862

]

119862

119877(Λ

119877 119905) = 119862

119877(Λ

119877 119905 = +0)

sdot 119890

minus119905Λ 119877+

119889120588

119877

119889Λ

119877

sdot int

119905

0

119889119905

1015840sdot 119890

minus(1Λ 119877)(119905minus1199051015840)

sdot [

119899 (119905

1015840)

Λ

119862

]

(D17)




References





































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of



















120588

0lt

120573 = 120573 + [120588

119877(Λmax) minus 120588

119877(Λ

0)] (D12)


119910 (119905) sdot

119889119910 (119905)

119889119905

= + (120572

119879sdot Λ

119862)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

(D13)


1

2

119889119910

2

119889119905

= + (120572

119879sdot Λ

119862)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119910

2(119905) = 2 sdot (120572

119879sdot Λ

119862)

times int

119905

0

119889119905

1015840int

Λmax

Λ 0

1

Λ

1015840

119877

119862

119877(Λ

1015840

119877 119905

1015840) sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905

1015840) + const

(D14)


2(0) = (120588

0minus

120573)

2


0minus

120573)

2

Then

119910 (119905) = minus2 sdot (120572

119879sdot Λ

119862)

times int

119905

0

119889119905

1015840[int

Λmax

Λ 0

1

Λ

1015840

119877

119862

119877(Λ

1015840

119877 119905

1015840) 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894119862

119894(119905

1015840)] + (120588

0minus

120573)

2

12

(D15)


[

119899 (119905)

Λ

119862

] =

(minus1)

119910 (119905)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119889119862

119894(119905)

119889119905

+ 120582

119894sdot 119862

119894(119905) = 120573

119894sdot [

119899 (119905)

Λ

119862

] 119894 = 1 6

120597119862

119877(Λ

119877 119905)

120597119905

+

1

Λ

119877

sdot 119862

119877(Λ

119877 119905) =

119889120588

119877

119889Λ

119877

sdot [

119899 (119905)

Λ

119862

]

(D16)


[

119899 (119905)

Λ

119862

] =

(minus1)

119910 (119905)

sdot int

Λmax

Λ 0

1

Λ

1015840

119877

sdot 119862

119877(Λ

1015840

119877 119905) sdot 119889Λ

1015840

119877

+

6

sum

119894=1

120582

119894sdot 119862

119894(119905)

119862

119894(119905) = 119862

119894(119905 = +0) sdot 119890

minus120582119894119905

+ 120573

119894sdot int

119905

0

119889119905

1015840sdot 119890

minus120582119894(119905minus1199051015840)sdot [

119899 (119905

1015840)

Λ

119862

]

119862

119877(Λ

119877 119905) = 119862

119877(Λ

119877 119905 = +0)

sdot 119890

minus119905Λ 119877+

119889120588

119877

119889Λ

119877

sdot int

119905

0

119889119905

1015840sdot 119890

minus(1Λ 119877)(119905minus1199051015840)

sdot [

119899 (119905

1015840)

Λ

119862

]

(D17)




References





































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of


















































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of





















FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of

















Documents

Research Article Improving Nuclear Safety of Fast Reactors by … · 2019. 5. 12. · =1 1+/ + 6 =1 1+ , +1 prt = 1 +1 prt + +1 +1, ( ) where is the number of annular layers in neutron