Research Article On Some Fundamental Peculiarities of the ...downloads.hindawi.com/journals/stni/2015/703069.pdf · Science and Technology of Nuclear Installations 4 3.5 3 2.5 2 1.5

Research ArticleOn Some Fundamental Peculiarities ofthe Traveling Wave Reactor

V D Rusov1 V A Tarasov1 I V Sharph1 V N Vashchenko2 E P Linnik1

T N Zelentsova1 M E Beglaryan1 S A Chernegenko1 S I Kosenko1 and V P Smolyar1

1Odessa National Polytechnic University Shevchenko Avenue 1 Odessa 65044 Ukraine2State Ecological Academy for Postgraduate Education and Management 35 Mytropolyt Vasyl Lypkivskyi Street Kyiv 03035 Ukraine

Correspondence should be addressed to V D Rusov siiistenetua

Received 24 October 2014 Accepted 5 February 2015

Academic Editor Michael I Ojovan

Copyright copy 2015 V D Rusov 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

On the basis of the condition for nuclear burning wave existence in the neutron-multiplying media (U-Pu and Th-U cycles) weshow the possibility of surmounting the so-called dpa-parameter problem and suggest an algorithm of the optimal nuclear burningwave mode adjustment which is supposed to yield the wave parameters (fluenceneutron flux width and speed of nuclear burningwave) that satisfy the dpa-condition associated with the tolerable level of the reactor materials radioactive stability in particularthat of the cladding materials It is shown for the first time that the capture and fission cross sections of 238U and 239Pu increasewith temperature within 1000ndash3000K range which under certain conditions may lead to a global loss of the nuclear burning wavestability Some variants of the possible stability loss due to the so-called blow-up modes (anomalous nuclear fuel temperature andneutron flow evolution) are discussed and are found to possibly become a reason for a trivial violation of the traveling wave reactorinternal safety

1 Introduction

Despite the obvious and unique effectiveness of nuclearenergy of the new generation there are difficulties of itsunderstanding related to the nontrivial properties of an idealnuclear reactor of the future

First nuclear fuel should be natural that is nonenricheduranium or thorium Second traditional control rods shouldbe absolutely absent in reactor active zone control systemThird despite the absence of the control rods the reactormust exhibit the so-called internal safety This means thatunder any circumstances the reactor active zone must stayat a critical state that is sustain a normal operation modeautomatically with no operator actions through physicalcauses and laws that naturally prevent the explosion-typechain reaction Figuratively speaking the reactors with inter-nal safety are ldquothe nuclear devices that never exploderdquo [1]

Surprisingly reactors that meet such unusual require-ments are really possible The idea of such self-regulating fast

reactor was expressed for the first time in a general form(the so-called breed-and-burn mode) by Russian physicistsFeynberg and Kunegin during the II Geneva conferencein 1958 [2] and was relatively recently ldquoreanimatedrdquo in aform of the self-regulating fast reactor in traveling nuclearburning wave mode by Russian physicist Feoktistov [3] andindependently by American physicists Teller et al [4]

The main idea of the reactor with internal safety is thatthe fuel components are chosen in such a way that firstthe characteristic time 120591

120573of the active fuel component (the

fissile component) nuclear burning is significantly largerthan the time of the delayed neutrons appearance andsecond all the self-regulation conditions are sustained in theoperation mode Particularly the equilibrium concentration119899fis of the active fuel component according to Feoktistovrsquoscondition of the wave mode existence is greater than itscritical concentration (concentrations of the active element(239Pu and 233U in cycles (1) and (2)) are called equilibriumor critical when an equal number of the active element nuclei

Hindawi Publishing CorporationScience and Technology of Nuclear InstallationsVolume 2015 Article ID 703069 23 pageshttpdxdoiorg1011552015703069

2 Science and Technology of Nuclear Installations

or neutrons respectively is born and destroyed at the sametime during the nuclear cycle) 119899crit [3] These conditions arevery important though they are almost always practicallyimplementable in case when the nuclear transformationschain of Feoktistovrsquos uranium-plutonium cycle type [3] issignificant among other reactions in the reactor

238U (119899 120574) 997888rarr239U

120573minus

997888rarr239

119873119901

120573minus

997888rarr239Pu (119899 fission) (1)

The same is also true for the Teller-Ishikawa-Woodthorium-uranium cycle type [4]

232Th (119899 120574) 997888rarr233Th

120573minus

997888rarr233Pa

120573minus

997888rarr233U (119899 fission) (2)

In these cases the fissionable isotopes form (239Pu in (1) or233U in (2)) which are the active components of the nuclearfuel The characteristic time of such reaction depends onthe time of the corresponding 120573-decays and is approximatelyequal to 120591

120573= 23 ln 2 asymp 33 days in case (1) and 120591

120573asymp 395

days in case (2) which is many orders of magnitude higherthan the corresponding time for the delayed neutrons

The effect of the nuclear burning process self-regulation isprovided by the fact that the system being left by itself cannotsurpass the critical state and enter the uncontrolled reactorrunaway mode because the critical concentration is limitedfrom above by a finite value of the active fuel componentequilibrium concentration (plutonium in (1) or uranium in(2)) 119899fis gt 119899crit (the Feoktistovrsquos wave existence condition [3])

Phenomenologically the process of the nuclear burningself-regulation is as follows Any increase in neutron flowleads to a quick burnout of the active fuel component(plutonium in (1) or uranium in (2)) that is to a reductionof their concentration and neutron flow meanwhile theformation of the new nuclei by the corresponding active fuelcomponent proceeds with the prior rate during the time 120591

120573

On the other hand if the neutron flow drops due to someexternal impact the burnout speed reduces and the activecomponent nuclei generation rate increases followed by theincrease of a number of neutrons generated in the reactorduring approximately the same time 120591

120573

The system of kinetic equations for nuclei (the compo-nents of nuclear fuel) and neutrons (in diffuse approxima-tion) in such chains is rather simple They differ only by thedepth of description of all the possible active fuel componentsand nonburnable poison (here by poisonwemean the oxygennuclei or other elements chemically bound to heavy nuclidesconstruction materials coolant and the poison itself thatis the nuclei added to the initial reactor composition inorder to control the neutron balance) Figure 1 shows thecharacteristic solutions for such problem (equations (3)ndash(9)in [5]) in a form of the soliton-like waves of the nuclearfuel components and neutrons concentrations for uranium-plutonium cycle in a cylindrical geometry case Within thetheory of the soliton-like fast reactors it is easy to show thatin general case the phase speed 119906 of soliton-like neutron wave

of nuclear burning is defined by the following approximateequality [5]

119906120591120573

2119871

≃ (

8

3radic120587

)

6

exp (minus 64

9120587

1198862)

1198862=

1205872

4

sdot

119899crit119899fis minus 119899crit

(3)

where 119899fis and 119899crit are the equilibrium and critical concen-tration of the active (fissile) isotope 119871 is the mean neutrondiffusion length and 120591

120573is the delay time associated with the

birth of the active (fissile) isotope and equal to an effective120573-decay period of the intermediate nuclei in Feoktistovrsquosuranium-plutonium cycle (1) or in Teller-Ishikawa-Woodthorium-uranium cycle (2)

Let us note that expression (3) automatically incorporatesa condition of nuclear burning process self-regulation sincethe fact of a wave existence is obviously predetermined by theinequality 119899fis gt 119899crit In other words the expression (3) is anecessary physical condition of the soliton-like neutron waveexistence Let us note for comparison that the maximal valueof the nuclear burning wave phase velocity as follows from(3) is characterized for both uranium and thorium cyclesby the equal average diffusion length (119871 sim 5 cm) of the fastneutrons (1MeV) and is equal to 370 cmday for uranium-plutonium cycle (4) and 031 cmday for thorium-uraniumcycle (2)

Generalizing the results of a wide range of numericalexperiments [5ndash23] we can positively affirm that the princi-pal possibility of themain stationarywave parameters controlwas reliably established within the theory of a self-regulatingfast reactor in traveling wave mode or in other words thetraveling wave reactor (TWR) It is possible both to increasethe speed the thermal power and the final fluence and todecrease them Obviously according to (3) it is achieved byvarying the equilibrium and critical concentrations of theactive fuel component that is by the purposeful change ofthe initial nuclear fuel composition

The technological problems of TWR are actively dis-cussed in science nowadays The essence of these problemsusually comes to a principal impossibility of such projectrealization and is defined by the following insurmountableflaws

(i) high degree of nuclear fuel burnup (over 20 in aver-age) leading to the following adverse consequences

(a) high damaging dose of fast neutrons acting atat the constructional materials (sim500 dpa) (forcomparison the claimed parameters for theRussian FN-800 reactor are 93 dpa at the sametime it is known that one of the main tasks ofthe construction materials radioactive stabilityinvestigations conducted at the BochvarHi-techInstitute for nonorganic materials (Moscow) isto achieve 133 to 164 dpa by 2020)

(b) high gas release which requires an increased gascavity length on top of a long fuel rod as it is

Science and Technology of Nuclear Installations 3

4353

252

151

050

140120

10080

6040

200 0

200 400600

8001000

1200

n

r (cm)z (cm)

times109

4

5

35

45

3252

151

050

140120

10080

6040

200 0

200400

600800 1000

1200

r (cm)z (cm)

times1022

238U

25

2

15

1

05

0140

120100

8060

4020

0 0200

400600 800

10001200

r (cm)z (cm)

times1021

239U

14

12

10

8

6

4

2

0140

120100

8060

4020

0 0200

400600

8001000

1200

r (cm)z (cm)

times1020

239Pu

Figure 1 Kinetics of the neutrons 238U 239U and 239Pu concentrations in the core of a cylindrical reactor with radius of 125 cm and 1000 cmlong at the time of 240 days Here 119903 is the transverse spatial coordinate axis (cylinder radius) 119911 is the longitudinal spatial coordinate axis(cylinder length) Temporal step of the numerical calculations is 01 s adopted from [5]

(ii) long active zone requiring the correspondingly longfuel rods whichmakes their parameters unacceptablefrom the technological use point of view and forinstance this is the case for the parameters charac-terized by a significant increase in

(a) the value of a positive void coefficient of reactiv-ity

(b) hydraulic resistance(c) energy consumption for the coolant circulation

through the reactor

(iii) the problem of nuclear waste associated with theunburned plutonium reprocessing and nuclear wasteutilization

Themain goal of the present paper is to solve the specifiedtechnological problems of the TWRon the basis of a technicalconcept which makes it impossible for the damaging dose ofthe fast neutrons in the reactor (fuel rods jackets reflectionshield and reactor pit) to exceed the sim200 dpa level Theessence of this technical concept is to provide the givenneutron flux on in-reactor devices by defining the speed ofthe fuelmovement relative to the nuclear burningwave speedThe neutron flux wave speed and fuel movement speed are

in their turn predetermined by the chosen parameters (equi-librium and critical concentrations of the active componentin the initial nuclear fuel composition)

Section 1 of this paper is dedicated to a brief analysis ofthe state-of-the-art idea of a self-regulating fast reactor intraveling wave mode Based on this analysis we formulatethe problem statement and chalk out the possible ways tosolve it Section 2 considers the analytical solution for a non-stationary 1D reactor equation in one-group approximationwith negative reactivity feedback (1D van Dam [8] model)It yields the expressions for the amplitude 120593

119898 phase 120572 and

phase speed 119906 as well as the dispersion (FWHM) of thesoliton-like burning wave Knowing the FWHM we mayfurther estimate the spatial distribution of the neutron fluxand thus a final neutron fluence Section 3 is dedicated toa description of the nontrivial neutron fluence dependenceon phase velocity of the solitary burnup waves in case of thefissible and nonfissible absorbents It reveals a possibility ofthe purposeful (in terms of the required neutron fluence andnuclear burning wave speed values) variation of the initialnuclear fuel composition Section 4 analyzes the dependenceof the damaging dose on neutron fluence phase velocityand dispersion of the solitary burnup waves Section 5considers the possible causes of the TWR internal safetyviolation caused by ldquoFukushima plutoniumrdquo effect or in


other words the temperature blowup modes driven eitherby temperature or neutron flux Section 6 is dedicated toanalysis of the practical examples of the temperature blowupmodes in neutron-multiplying mediaThe idea of an impulsethermonuclear TWR is also proposed The conclusion of thepaper is presented in Section 7

2 On Entropy and Dispersion of SolitaryBurnup Waves

Let us discuss the physical causes defining the main charac-teristics of the soliton-like propagation of ldquocriticalityrdquo wave inthe initially undercritical environment characterized by theinfinitemultiplication factor 119896

infinless than unit Obviously the

supercritical area (119896infin

gt 1) must be created by some externalneutron source (eg by an accelerator or another supercriti-cal area) In the general case the supercritical area is a result ofthe breeding effects in fast nuclear systems or the burning ofthe fissible absorbents (fuel components) in thermal nuclearsystems (Seifritz (1995) was the first to find theoretically anuclear solitary burnup wave in opaque neutron absorbers[24] the supercriticality waves in thermal nuclear reactorsare investigated and analysed in the papers by Akhiezeret al [25ndash28] where they show the possibility of both fast[25ndash27] and slow [28] modes of nuclear burning distribution(ie the supercriticality waves) in the framework of diffuseapproximation) Due to the gradual burn-out of the neutron-multiplying medium in the supercritical area this area loosesits supercritical properties since 119896

infinbecomes less than unit

The wave would have to stop and diminish at this point inan ordinary case However because of the neutrons appear-ing during breeding and diffusely ldquoinfectingrdquo the nearbyareas this ldquovirginrdquo area before the wavefront is forced toobtain the properties of supercriticality and the wave movesforward in this direction Apparently the stable movementof such soliton-like wave requires some kind of stabilizingmechanism for example the negative self-catalysis or anyother negative feedback This is called the negative reactivityfeedback in traditional nuclear reactors (according to vanDam [8] the procedure of the reactivity introduction intothe 1D nonstationary equation of the reactor in one-groupapproximation though implicitly takes the kinetic equationsof the burnout into account particularly the production ofplutonium in U-Pu cycle or uranium in Th-U cycle) Letus therefore consider such an example qualitatively which ismentioned below

For this purpose let us write down a 1D nonstationaryequation of the reactor in one-group approximation [29ndash31]with negative reactivity feedback

119863

1205972120593

1205971199092+ (119896infin

minus 1 + 120574120593) Σ119886120593 =

1

V120597120593

120597119905

(4)

where 120593 is the neutron flux [cmminus2 sminus1] 119863 is the diffusioncoefficient [cm] 120574120593 is the reactivity dimensionless value Σ

119886

is the total macroscopic absorption cross section [cmminus1] andV is the neutron speed [cmsdotsminus1] In this case the negativefeedback 120574 is defined mainly by the fact that the infinite

multiplication factor is used in (4) and therefore the fluxdensity must be corrected

Let us search for the solution in an autowave form

120593 (119909 119905) = 120593 (119909 minus 119906119905) equiv 120593 (120585) (5)

where 119906 is the wave phase velocity and 120585 is the coordinate ina coordinate system which moves with phase speed In suchcase consider the following

1

V120597120593

120597119905

= minus

119906

V120597120593

120597119905

(6)

As is known [8] the relation 119906V by order of magnitude isequal to 10minus13 and 10minus11 for fast and thermal nuclear systemsrespectively Therefore the partial time derivative in (4) maybe neglected without loss of generality Further taking intoaccount (5) equation (4) may be presented in the followingform

1198712 1205972120593

1205971205852+ [119896infin(120595) minus 1 + 120574120593] 120593 = 0 (7)

where 119871 = (119863Σ119886)12 is the neutron diffusion length and 120595 is

the so-called neutron fluence function

120595 (119909 119905) = int

119905

0

120593 (119909 1199051015840) 1198891199051015840 (8)

In order to find a physically sensible analytic solutionof (7) by substituting (8) we need to define some realisticform of the function 119896

infin(120595) (usually referred to as the burnup

function) Since the real burnup function 119896infin(120595) has a form

of some asymmetric bell-shaped dependence on fluencenormalized to its maximal value 120595max (Figure 1) following[8] let us define it in a form of parabolic dependency withoutlosing generality

119896infin

= 119896max + (1198960minus 119896max) (

120595

120595maxminus 1)

2

(9)

where 119896max and 1198960are the maximal and initial neutron

multiplication factors Substituting (9) into (7) we obtain thefollowing

1198712120593120585120585+ 120588max120593 + 120574

01205932minus 120575[

[

int

infin

120585120593119889120585

119906120595119898

minus 1]

]

2

120593 = 0 (10)

where 120588max = 119896max minus 1 120575 = 119896max minus 1198960 and 120574

0equiv 120574

Suppose we are searching for a partial solution of (10) Letus rewrite it in the following form

1198712 1198892120593

1198891205852+(120588max + 120574

0120593 minus 120575(

int

+infin

120585120593119889120585

119906120595119898

minus 1)

2

)120593 = 0

(11)

Let us introduce a new unknown function

120594 (120585) = int

+infin

120585

120593119889120585 997904rArr 120593 (120585) = minus

119889120594 (120585)

119889120585

(12)


that due to its nonnegativity on the interval 120585 isin [0infin] mustsatisfy the following boundary conditions120593 = 0 for 120585 = 0infin

The equation will take the form

11987121198893120594 (120585)

1198891205853

+ (120588max minus 1205740

119889120594 (120585)

119889120585

minus 120575(

120594 (120585)

119906120595119898

minus 1)

2

)

sdot

119889120594 (120585)

119889120585

= 0

(13)

In order to find a partial solution of (13) we require thefollowing additional condition to hold

120588max minus 1205740

119889120594 (120585)

119889120585

minus 120575(

120594(120585)

119906120595119898

minus 1)

2

= 119891 (120594 (120585)) (14)

where 119891(120585) is an arbitrary function the exact form of whichwill be defined later The condition (14) is chosen because itmakes it possible to integrate (13) Really if (14) is true (5)takes the following form

11987121198893120594 (120585)

1198891205853

+ 119891 (120594 (120585))

119889120594 (120585)

119889120585

= 0 (15)

That allows us to reduce the order of the equation

11987121198892120594 (120585)

1198891205852

+ 1198651(120594 (120585)) = 119862 (16)

Here 1198651(120594) denotes a primitive of 119891(120594) and 119862 is an

arbitrary integration constant The order of (16) may befurther reduced by multiplying both sides of the equation by119889120594(120585)119889120585

1198712

2

(

119889120594(120585)

119889120585

)

2

+ 1198652(120594 (120585)) minus 119862120594 (120585) = 119861 (17)

1198652(120594) here denotes a primitive of 119865

1(120594) that is ldquothe

second primitiverdquo of the function 119891(120594) introduced in (5) and119861 is a new integration constant

The obtained equation (17) is a separable equation andmay be rewritten in the following form

119889120585 = plusmn

119889120594

radic(21198712) (119861 minus 119865

2(120594 (120585)) + 119862120594 (120585))

(18)

On the other hand (14) may also be considered aseparable equation relative to 120594(120585)Then separating variablesin (14) we obtain the following

119889120585 = 119889120594(

120588max1205740

minus

120575

1205740

(

120594(120585)

119906120595119898

minus 1)

2

minus

1

1205740

119891 (120594(120585)))

minus1

(19)

Since (18) and (19) are for the same function 120594(120585) bycomparing them we derive that the following relation musthold

plusmn radic2

1198712(119861 minus 119865

2(120594 (120585)) + 119862120594 (120585))

=

120588max1205740

minus

120575

1205740

(

120594(120585)

119906120595119898

minus 1)

2

minus

1

1205740

119891 (120594 (120585))

(20)

In order to simplify (18) and (19) let us choose 119891(120594) in apolynomial form of 120594The order of this polynomial is 119899Then1198652(120585) obtained by double integration of119891(120594) is a polynomial

of order (119899 + 2) Taking the square root according to (20)should also lead to a polynomial of the order 119899 Therefore 119899+2 = 2119899 rArr 119899 = 2

Consequently the function 119891(120594) may only be a second-order polynomial under the assumptions made above Con-sider the following

119891 (120594) = 11990421205942+ 1199041120594 + 1199040 (21)

where 1199040 1199041 and 119904

2are the polynomial coefficients

Double integration of (21) leads to the following

1198652(120594) =

1199042

12

1205944+

1199041

6

1205943+

1199040

2

1205942+ 1198881120594 + 1198882 (22)

where 1198881and 1198882are the integration constants

Substituting (21) and (22) into (20) we get the following

(

120588max1205740

minus

120575

1205740

(

120594

119906120595119898

minus 1)

2

minus

1

1205740

(11990421205942+ 1199041120594 + 1199040))

2

=

2

1198712(119861 minus (

1199042

12

1205944+

1199041

6

1205943+

1199040

2

1205942+ 1198881120594 + 1198882) + 119862120594)

(23)

Further in (23) we set the coefficients at the same ordersof 120594 equal the following

(

120575

120574011990621205952

119898

+

1199042

1205740

)

2

= minus

1199042

61198712 (24)

minus2(

120575

120574011990621205952

119898

+

1199042

1205740

)(

2120575

1205740119906120595119898

minus

1199041

1205740

) = minus

1199041

31198712 (25)

(

2120575

1205740119906120595119898

minus

1199041

1205740

)

2

minus 2(

120575

120574011990621205952

119898

+

1199042

1205740

)

sdot (

120588max1205740

minus

120575

1205740

minus

1199040

1205740

) = minus

1199040

1198712

(26)

2 (

2120575

1205740119906120595119898

minus

1199041

1205740

)(

120588max1205740

minus

120575

1205740

minus

1199040

1205740

)

= minus(

21198881

1198712+

2119862

1198712)

(27)

(

120588max1205740

minus

120575

1205740

minus

1199040

1205740

)

2

=

2

1198712119861 minus

21198882

1198712 (28)

Note that the first three equations are enough to find thecoefficients 119904

0 1199041 and 119904

2 and the remaining two equations

may be satisfied with the appropriate constants 119861 119862 1198881 and

1198882as follows

1199040= 120588max minus 120575

1199041=

2120575

119906120595119898

minus

1205740

119871

radic120575 minus 120588max

1199042=

1205751205740

3119871119906120595119898radic120575 minus 120588max

minus

1205742

0

61198712minus

120575

11990621205952

119898

(29)


After finding 1199040 1199041 and 119904

2from this system we may

consider (19) in more detail which takes the form

119889120585 = 119889120594(minus(

120575

120574011990621205952

119898

+

1199042

1205740

)1205942+ (

2120575

1205740119906120595119898

+

1199041

1205740

)120594

+ (

120588max1205740

minus

120575

1205740

minus

1199040

1205740

))

minus1

(30)

Solving this equation yields

119889120594

(120594 minus 119870)2

minus1198722

= minus119873119889120585 (31)

where

119870 =

1205751205740119906120595119898minus 119904121205740

120575120574011990621205952

119898+ 11990421205740

1198722= (

1205751205740119906120595119898minus 119904121205740

120575120574011990621205952

119898+ 11990421205740

) +

(120588max1205740 minus 1205751205740minus 11990401205740)

(120575120574011990621205952

119898+ 11990421205740)

119873 =

120575

120574011990621205952

119898

+

1199042

1205740

(32)

Let us introduce a new variable1205941into (31) by substituting

the following

120594 minus 119870 = 1198721205941 119889120594 = 119872119889120594

1 (33)

Then (31) will take the following form

1198891205941

(1205941)2

minus 1

= minus119872119873119889120585 (34)

Hence

1205941= minus tanh (119872119873120585 minus 119863) (35)

where119863 is the integration constant Taking into account that120594 minus 119870 = 119872120594

1

120594 = 119870 minus119872 tanh (119872119873120585 minus 119863) (36)

Considering (12) we obtain the soliton-like solution inthe form

120593 (120585) = 1198722119873sech2 (119872119873120585 minus 119863) (37)

Let us remember that together with introducing a newunknown function 120594(120585) (see (12)) we obtained an obviouscondition for this function

lim120585rarrinfin

120594 (120585) = 0 (38)

Let us show that this condition eventually leads toan autowave existence condition Obviously condition (38)along with (36)

lim120585rarrinfin

120594 (120585) = lim120585rarrinfin

[119870 minus119872 tanh (119872119873120585 minus 119863)] = 0 (39)

leads to

119870 = 119872 (40)

This relation lets us define the amplitude 120593119898 phase 120572 and

phase velocity 119906 of the soliton-like wave

120572 = 119872119873 =

radic120575 minus 120588max2119871

(41)

120593119898= 1198722119873 =

120575 minus 3120588max21205740

=

3120588max minus 120575

210038161003816100381610038161205740

1003816100381610038161003816

(42)

119906 =

120593119898

120572120595119898

(43)

From the condition of nonnegative width (41) and ampli-tude (42) of the nuclear burning wave the following is thecondition of 1D autowave existence or the so-called ldquoignitionconditionrdquo by van Dam [8]

3120588max minus 120575 = 2119896max + 1198960minus 3 ⩾ 0 where 1 minus 119896

0gt 0 (44)

It is noteworthy that the analogous results for a nonlinearone-group diffusion 1D-model (4) with explicit feedback andburnup effects were first obtained by van Dam [8] The sameresults (see (41)ndash(43)) were obtained by Chen and Maschek[11] while investigating the 3D-model by van Dam usingthe perturbation method The only difference is that thevalue of neutron fluence associated with the maximum ofburnup parameter 119896

infinwas adapted to the transverse buckling

(Figure 2) In other words they considered the transversegeometric buckling mode as a basis for perturbation Hencethey introduced a geometric multiplication factor 119896GB dueto transverse buckling into two-dimensional equation (4)which led to a change in some initial parameters (120588max =

119896max minus 119896GB 120575 = 119896max minus 1198960) and consequently to a change

in the conditions of the autowave existence in 3D case

3120588max minus 120575 = 2119896max + 1198960minus 3119896GB ⩾ 0 where 119896GB minus 119896

0gt 0

(45)

that in the case of 119896GB = 1 is exactly the same as the so-calledignition condition by van Dam [8]

From the point of view of the more detailed Feoktistovmodel [3] analysis thoroughly considered in [5] and relatedto a concept of the nuclear systems internal safety thecondition (45) is necessary but not sufficient On the otherhand it is an implicit form of the necessary condition of wave


115

111

107

103

099

09500 05 10 15 20

kinfin

Neutron fluence (in units of 1020 cmminus2)

Figure 2 Asymmetric burnup function as characteristic for realisticburnup function adapted from [8]

existence according to Feoktistov where the equilibriumconcentration 119899fis of the active fuel component must begreater than its critical concentration 119899crit (119899fis gt 119899crit) [35] The physics of such hidden but simple relation will beexplained below (see Section 3)

Returning to a 1D reactor equation solution (10) in one-group approximation with negative reactivity feedback let uswrite it in a more convenient form for analysis

120593 = 120593119898sech2 (120572120585) = 120593

119898sech2 [120572 (119909 minus 119906119905)] (46)

where 120593119898is the amplitude of the neutron flux and 1120572 is the

characteristic length proportional to the soliton wave widthwhich is a full width at half-maximum (FWHM)by definitionand is equal to

Δ12

= FWHM = 2 ln (1 + radic2) 120572minus1

[cm] (47)

Apparently integrating (45) yields the area under suchsoliton

119860area =2120593119898

120572

[cm2] (48)

In order to estimate the extent of the found parametersinfluence on the dynamics of the soliton-like nuclear burn-ing wave stability let us invoke an information-probabilityapproach developed by Seifritz [32] For this purpose let uswrite down the expression for the mean value of informationor more precisely the entropy of the studied process

119878 = minus119896119861int

infin

minusinfin

119901 (119909) ln119901 (119909) 119889119909 (49)

where 119901(119909) is the function of probability density relative to adimensionless variable 119909 ln 1119901(119909) is the mean informationvalue 119896

119861is the Boltzmann constant

Substituting the soliton-like solution (45) into (49) weobtain the following

119878 = minus2119896119861int

infin

0

sech2 (120572120585) ln [sech2 (120572120585)] 119889( 119909

119906120591120573

)

= 4119896119861int

infin

0

ln cosh (120573119910)cosh2 (120573119910)

119889119910

(50)

where 120573 = 120572(119906120591120573) is another (dimensionless) scaling factor

and 120591120573is the proper 120573-decay time of the active component of

the nuclear fuel Let us point out the procedure of making the119909 argument dimensionless in (50) which takes into accountthe fact that the neutron flux amplitude is proportional to thephase velocity of the nuclear burning wave (see (43)) that is120593119898

sim 119906 Calculating the integral (50) leads to the followingquite simple expression for the entropy

119878 =

4 (1 minus ln 2)120573

= 4 (1 minus ln 2)119896119861

120572119906120591120573

=

4119896119861(1 minus ln 2)

radic119896GB minus 1198960

2119871

119906120591120573

(51)

which in the case of

119878 sim 119896119861

2119871

119906120591120573

= const (52)

points to an isoentropic transport of the nuclear burningwave

It is interesting to note here that if the width Δ12

rarr 0then due to isoentropicity of the process (52) the form of thesoliton becomes similar to the so-called Dirac 120575-functionIntroducing two characteristic sizes or two length scales (119897

1=

1120572 and 1198972

= 120593119898) it is possible to see that when the first

of them is small the second one increases and vice versa Ithappens because the area 119860area (41) under the soliton mustremain constant since 119860area prop 119897

11198972 In this case the soliton

entropy tends to zero because the entropy is proportional tothe ratio of these values (119878 prop 119897

11198972rarr 0) These features are

the consequence of the fact that the scale 1198971is a characteristic

of a dispersion of the process while another scale 1198972is a

characteristic of the soliton nonlinearity If 1198971≪ 1198972 then the

process is weakly dispersed (Figure 3(a)) If 1198971

≫ 1198972 then

the process is strongly dispersed (Figure 3(b)) In the lattercase the soliton amplitude becomes relatively large (a case of120575-function) And finally if 119897

1= l2 then the soliton speed

119906 prop (1198972

11198972

2)12 (see (44)) is proportional to the geometrical

mean of the dispersion and nonlinearity parametersOn the other hand it is clear that according to (41)

and (43) the burning wave width Δ12

is a parameter thatparticipates in formation of the time for neutron fluenceaccumulation 120591

120573on the internal surface of the TWR long fuel

rod cladding material

120595119898sim

Δ12

119906

120593119898= 120591120593sdot 120593119898 (53)


005

1 0

02040608

0

1

005

01

01502120601

minus1

minus05

r(m

)

z (m)05m

(a)

02

46

810 0

05115

200025005007501

120601

r(m

)

z (m)

22m

(b)

Figure 3 (a) Weakly dispersive wave pattern obtained by Chen andMaschek [11] investigating a 3D-model by van Dam using perturbationmethod Example with the following parameters 119871 = 002m 119896GB = 104 119896

0= 102 119896max = 106 120601

119898= 1017mminus2 sminus1 and 119906 = 0244 cmday

(b) Strongly dispersive wave pattern obtained by Chen et al [19] within a 3D-model of traveling wave reactor Example with the followingparameters 119871 = 0017m 119896GB = 100030 119896

0= 099955 120601

119898= 3 sdot 10

15mminus2 sminus1 and 119906 = 005 cmday

jex

r

Z

Ignition zone Breeding zone

Figure 4 Schematic sketch of the two-zone cylindrical TWR

And finally there is one more important conclusionFrom the analysis of (41) and (42) it is clear that the initialparameter 119896

0for burning zone (Figure 4) is predefined solely

by the nuclear burning wave burnup conditions that is bythe parameters of an external neutron source and burn-up area composition (Figure 4) In other words it meansthat by tuning the corresponding burnup conditions fora given nuclear fuel composition we can set the certainvalue of the nuclear burning front width Moreover byselecting the corresponding equilibrium 119899fis and critical 119899critconcentrations of the active nuclear fuel component wecan define the required value of the nuclear burning wavespeed 119906 Hence it is an obvious way for us to control thecorresponding neutron fluence 120591

120593accumulation time in the

cladding material of the TWR fuel rodHence we can make an important conclusion that the

realization of the TWR with inherent safety requires theknowledge about the physics of nuclear burningwave burnupand the interrelation between the speed of nuclear burningwave and the fuel composition As shown in [5] the prop-erties of the fuel are completely defined by the equilibrium119899fis and critical 119899crit (see (3)) concentrations of the activenuclear fuel component We examine this in more detailbelow

3 Control Parameter andCondition of Existence of StationaryWave of Nuclear Burning

The above stated raises a natural question ldquoWhat does thenuclear burning wave speed in uranium-plutonium (1) andthorium-uranium (2) cycles mainly depend onrdquoThe answeris rather simple and obvious The nuclear burning wavespeed in both cycles (far away from the burnup source) iscompletely characterized by its equilibrium 119899fis and critical119899crit concentrations of the active fuel component

First of all this is determined by a significant fact that theequilibrium 119899fis and critical 119899crit concentrations of the activefuel component completely identify the neutron-multiplyingproperties of the fuel environment They are the conjugatepair of the integral parameters which due to their physicalcontent fully and adequately characterize all the physics ofthe nuclear transformations predefined by the initial fuelcompositionThis is also easy to see from a simple analysis ofthe kinetics equations solutions for the neutrons and nucleiused in differentmodels [5ndash23] Itmeans that regardless of thenuclear cycle type and initial fuel composition the nuclearburningwave speed is defined by the equilibrium 119899fis and crit-ical 119899crit concentrations of the active fuel component throughthe so-called para-parameter 119886 (see (3)) Consequently as thenumerical simulation results show (Figure 5 [5]) it followsthe Wigner statistics

It is important to note that each of these concentrationsvaries during the nuclear burning but their ratio

119886 =

120587

2

radic


(54)

is a characteristic constant value for the given nuclear burningprocess [5] In addition to that this para-parameter alsodetermines (and it is extremely important) the conditions forthe nuclear burning wave existence (3) the neutron nuclear

Science and Technology of Nuclear Installations 9Λ(alowast)

alowast

25

2

15

1

05

0

0 05 1 15 2 25 3

U-Pu fuel cycleFeoktistovSekimotoFominErshovRusov

Th-U fuel cycleTellerRusov

Figure 5 The theoretical (solid line) and calculated (points)dependence of Λ(119886

lowast) = 119906120591

1205732119871 on the parameter 119886 [5]

burning wave speed (see (3)) and the dimensionless width(54) of the supercritical area in the burning wave of the activenuclear fuel component

Therefore when Pavlovich et al [14ndash17] state that theycontrol the values of the parameterswith purposeful variationof the initial reactor composition it actually means thatchanging the effective concentration of the absorbent theypurposefully and definitely change (by definition (see [35])) the equilibrium and critical concentrations of the activecomponent that is the para-parameter (54) of the nuclearTWR-burning process

It is necessary to note that the adequate understanding ofthe control parameter determination problem is not a simpleor even a scholastic task but is extremely important for theeffective solution of another problem (themajor one in fact)related to investigation of the nuclear burning wave stabilityconditions In our opinion the specified condition for thestationary nuclear burning wave existence (3) from [5] aswell as the one obtained by Pavlovich group reveals the pathto a sensible application of the so-called direct Lyapunovmethod [33 34] (the base theory for the movement stability)and thus the path to a reliable justification of the Lyapunovfunctionalminimumexistence (if it does exist) [33ndash37] Somevariants of a possible solution stability loss due to anomalousevolution of the nuclear fuel temperature are considered inSection 5

At the same time one may conclude that the ldquodifferen-tialrdquo [5] and ldquointegralrdquo [14ndash16] conditions for the stationarynuclear burning wave existence provide a complete descrip-tion of the wave reactor physics and can become a basis forthe future engineering calculations of a contemporary TWRproject with the optimal or preset wave properties

4 On the Dependence of the Damaging Doseon Neutron Fluence Phase Velocity andDispersion of the Solitary Burnup Waves

As follows from the expression for the soliton-like solution(46) it is defined by three parameters the maximal neutronflux120595

119898 the phase120572 and the speed 119906 of nuclear burningwave

And even if we can control them it is still unclear whichcondition determines the optimal values of these parametersLet us try to answer this question shortly

It is known that high cost effectiveness and competitive-ness of the fast reactors including TWR may be achievedonly in case of a high nuclear fuel burnup (since the maxi-mum burnup of the FN-600 reactor is currently sim10 [38]the burnup degree of sim20 for the TWR may be consideredmore than acceptable) As the experience of the fast reactorsoperation shows the main hindrance in achieving the highnuclear fuel burnup is the insufficient radiation resistance ofthe fuel rod shells Therefore the main task of the radiationmaterial science (along with the study of the physics behindthe process) is to create a material (or select among theexisting materials) which would keep the required level ofperformance characteristics being exposed to the neutronirradiation One of the most significant phenomena leadingto a premature fuel rods destruction is the void swellingof the shell material [39ndash42] Moreover the absence of theswelling saturation at an acceptable level and its accelerationwith the damaging dose increase lead to a significant swelling(volume change up to 30 and more) and subsequently toa significant increase of the active zone elements size Theconsequences of such effect are amplified by the fact that thehigh sensitivity of the swelling to temperature and irradia-tion damaging dose leads to distortions of the active zonecomponents form because of the temperature and irradiationgradients And finally one more aggravating consequence ofthe high swelling is almost complete embrittlement of theconstructionmaterials at certain level of swelling (it is knownthat the fuel rod shell diameter increase due to swelling isaccompanied by an anomalously high corrosion damage ofthe shell by the fuel [38]) Consequently in order to estimatethe possible amount of swelling a damaging dose (measuredin dpa) initiated by the fast neutrons for example in the fuelrod shells must be calculated (Figure 6)

Usually in order to evaluate the displacements per atom(DPA) created by the spallation residues the so-called modi-fiedNRTmethod is applied [44 45] which takes into accountthe known Lindhard correction [46] Within such modifiedNRT method the total number of displacements producedby the residues created by spallation reactions in the energeticwindowcanbe calculated as the addition of the displacementsproduced by each of these residues (119885119860) that in its turn leadsto the following expression for the displacements per atom

119899dpa = 119905 sdot

119873

sum

119894

⟨120590119894

dpa⟩int119864119894

119864119894

minus1

Φ(119864119894) 119889119864119894=

119873

sum

119894=1

⟨120590119894

dpa⟩120593119894119905 (55)

where

⟨120590119894

dpa⟩ (119864119894) = ⟨120590119889(119864119894 119885 119860)⟩ sdot 119889 (119864

119889(119885 119860)) (56)

10 Science and Technology of Nuclear Installations(

)

10

9

8

7

6

5

4

3

2

1

060 70 80 90 100 110 120 130 140 150 160 170 180 190 200

Dose (dpa)

Average316Ti

Average 1515Ti Best lot of1515Ti

Embrittlement limit

Ferritic-martensitic (FM)steels ODS included

Figure 6 Swelling of austenitic Phenix cladding compared withferritic-martensitic materials ODS included adopted from [43]

while 120590119889(119864119894 119885 119860) is the displacement cross section of recoil

atom (119885119860) produced at incident particle energy 119864119894Φ(119864) is

the energy-dependent flux of incident particles during time119905 and 119889(119864

119889(119885 119860)) is the number of displacements created at

threshold displacement energy 119864119889of recoil atom (119885119860) or its

so-called damage functionActually all the recoil energy of the residue 119864

119903is not

going to be useful to produce displacements because a partof it lost inelastic scattering with electrons in the mediumAn estimation of the damage energy of the residue can becalculated using the Lindhard factor 120585 [46]

119864dam = 119864119903120585 (57)

The number of displacements created by a residue (119885119860)is calculated using this damage energy (57) and the NRTformula

119889 (119864119889(119885 119860)) = 120578

119864dam2 ⟨119864119889⟩

(58)

where ⟨119864119889⟩ is the average threshold displacement energy of

an atom to its lattice site 120578 = 08 [44 45]Consequently the condition of the maximal damaging

dose for the cladding materials of the fast neutron reactorstaking into account the metrological data of IAEA [47](see Figure 7) and contemporary estimates by Pukari andWallenius (see yellow inset at Figure 6) takes the form

119899dpa ≃ ⟨120590dpa⟩ sdot 120593 sdot

2Δ12

119906

⩽ 200 [dpa] (59)

In this case the selection strategy for the required waveparameters and allowed values of the neutron fluence for thefuture TWR project must take into account the condition ofthe maximal damaging dose (58) for cladding materials infast neutron reactors and therefore must comply with thefollowing dpa-relation

⟨120590dpa⟩ sdot 120593 sdot

Δ12

119906

⩽ 100 [dpa] (60)

The question here is whether or not the parameters ofthe wave and neutron fluence which can provide the burnup

1000

750

500

250

01 1 10 100 1000

Radiation dose (dpa)

Ope

ratin

g te

mpe

ratu

re (∘

C)

HTR materialsFusion

Fast reactormaterials

materials

Thermal

materials

AusteniticPH ferriticmartensitic

ODS1000

950

900

850 100 150 200 250

reactor

Figure 7 Operating condition for core structural materials indifferent power reactors [47] The upper yellow inset represents thedata of Pukari and Wallenius [43]

Discretized ENDFB-VI-based iron displacement cross section104

103

102

10

1

10minus1

10minus2

10minus10 10minus8 10minus6 10minus4 10minus2 1

Neutron energy (MeV)

120590D(E)

(bar

n)

Figure 8 Discretized displacement cross section for stainless steelbased on the Lindhard model and ENDFB scattering cross sectionadopted from [48]

level of the active nuclear fuel component in TWR-type fastreactor of at least 20 are possible Since we are interested inthe cladding materials resistible to the fast neutron damagingdose we will assume that the displacement cross section forthe stainless steel according to Mascitti and Madariaga [48]for neutrons with average energy 2MeV equals ⟨120590dpa⟩ asymp

1000 dpa (Figure 8)The analysis of the nuclear burning wave parameters in

some authorsrsquo models of TWR presented in Table 1 showsthat in the case of U-Pu cycle none of the considered modelssatisfy the dpa-parameter while two Th-U cycle models byTeller et al [4] and Seifritz [6] groups correspond well to themajor requirements to wave reactors


Table 1 Results of the numerical experiments of the wave mode parameters based on U-Pu andTh-U cycles

Δ12

[cm] 119906 [cmday] 120593 [cmminus2 sminus1] 120595 [cmminus2] ⟨120590dpa⟩ [barn] 119899dpa200 Fuel burn-up SolutionU-Pu cycle

Sekimoto and Udagawa [12] 90 0008 325 sdot 1015 32 sdot 1023 1000 32 sim43 NoRusov et al [5] 200 277 1018 62 sdot 1024 1000 62 sim60 NoPavlovich et al [16] mdash 0003 mdash 17 sdot 1024 1000 17 sim30 NoFomin et al [49] 100 007 2 sdot 1016 25 sdot 1024 1000 25 sim30 NoFomin et al [50] 125 17 5 sdot 1017 32 sdot 1024 1000 32 sim40 NoChen et al [18] 216 0046 3 sdot 1015 12 sdot 1024 1000 12 sim30 NoWeaver et al [23] mdash mdash mdash mdash mdash 175 sim20 No

Th-U cycleTeller et al [4] 70 014 sim2 sdot 1015 86 sdot 1022 1000 096 sim50 YesSeifritz [6] 100 0096 1015 90 sdot 1022 1000 090 sim30 YesMelnik et al [51] 100 00055 05 sdot 1016 79 sdot 1024 1000 sim80 sim50 No

U-Pu (+ moderator)Example 100 0234 25 sdot 1015 92 sdot 1023 100 092 sim20 YesIdeal TWR mdash mdash mdash 1024 100 10 ⩾20 Yes

Table 2 Moderating and absorbing properties of some substances moderator layer width estimate for moderating neutron from 119864fuel =

10MeV to 119864 mod = 01MeV

Moderator Mass number 119860 Mean logarithmicenergy 120585 Density 120588 gcm 3

Impacts numberrequired formoderating 119899

Neutron mean freepath 120582

Moderator layerwidth 119877 mod cm

Be 9 021 185 11 139 153C 12 0158 160 15 356 534H2O 18 0924 10 25 167 416

H2O + B 25 100 250

He 4 0425 018 541 112 607

On the other hand these authors apparently did not takethe problem of dpa-parameter in cladding materials intoaccount since they were mainly interested in the fact of thewave mode of nuclear burning existence in U-Pu and Th-Ucycles at the time

However as the analysis of Table 1 shows the procedureaccounting for the dpa-parameter is not problematic but itleads to unsatisfactory results relative to the burnout of themain fissile material It follows that when ⟨120590dpa⟩ asymp 1000 dpathe dpa-condition considered above for the maximum possi-ble damaging dose for cladding materials of the fast neutronreactors

1205951000

=

Δ12

119906

120593 ≃ 1023

[cmminus2] (61)

is not met by any example in the table Here 1205951000

is theneutron fluence in case ⟨120590dpa⟩ asymp 1000 dpa 120593 is the neutronflux and Δ

12and 119906 are the width and speed of the soliton-

like nuclear burning waveSo on the one hand the neutron fluence must be

increased by an order of magnitude to increase the burnuplevel significantly and on the other hand the maximumdamaging dose for the cladding materials must also bereduced by an order of magnitude Such a controversialcondition may be fulfilled considering that the reduction of

the fuel rod shell radioactive damage for a given amount maybe achieved by reducing the neutron flux density and energy(see Figure 8) The latter is achieved by placing a speciallyselected substance between fissile medium and fuel rod shellwhich has the suitable characteristics of neutron moderatorand absorbent

At the same time it is known from the reactor neutronphysics [29 52] that themoderator layerwidth estimate119877 modis

119877 mod ≃

1

Σ119878+ Σ119886

sdot

1

120585

ln119864fuel119864 mod

(62)

where Σ119878

asymp ⟨120590119878⟩119873mod and Σ

119886asymp ⟨120590

119886⟩119873mod are the

macroscopic neutron scattering and absorption cross sec-tions respectively ⟨120590

119878⟩ and ⟨120590

119886⟩ are the microscopic neu-

tron scattering and absorption cross sections respectivelyaveraged by energy interval of the moderating neutrons from119864fuel = 2MeV to 119864 mod = 01MeV 119873mod is the moderatornuclei density and 120585 = 1 + (119860 + 1)

2 ln[(119860 minus 1)119860 + 1]2119860

is the neutron energy decrement of its moderation in themoderator-absorbent medium with atomic number 119860

It is clearly seen that the process of neutron moderationfrom 20MeV to 01MeV energy in moderator-absorbent of agivenwidth (see Table 2) creates a new but satisfactory level ofmaximumpossible damaging dose for the claddingmaterials


corresponding to ⟨120590dpa⟩ asymp 100 dpa (Figure 8) Thereforeif we are satisfied with the main fissile material burnoutlevel around sim20 then analyzing Tables 1 and 2 theconditions accounting for the dpa-parameter problem andcontemporary level of the radioactive material science willhave the following form

120595100

=

Δ12

119906

120593 ≃ 1024

[cmminus2] (63)

where 120595100

is the neutron fluence (with 01MeV energy) oncladding materials surface in case ⟨120590dpa⟩ asymp 100 dpa (seeFigure 8 and ldquoidealrdquo case in Table 1)

And finally one can make the following intermediateconclusion As shown above in Section 3 the algorithm fordetermining the parameters (61) is mainly defined by para-parameter that plays a role of a ldquoresponse functionrdquo to allthe physics of nuclear transformations predefined by initialfuel composition It is also very important that this param-eter unequivocally determines the conditions of the nuclearburningwave existence (3) the neutronnuclear burningwavespeed (see (3)) and the dimensionlesswidth (54) of the super-critical area in the wave of the active component burning

Based on the para-parameter ideology [5] and Pavlovychgroup results [14ndash16] we managed to pick up a mode for thenuclear burning wave in U-Pu cycle having the parametersshown in Table 1 satisfyied (61) The latter means that theproblem of dpa-parameter in claddingmaterials in the TWR-project is currently not an insurmountable technical problemand can be successfully solved

In our opinion the major problem of TWR is the so-called temperature blow-up modes that take place due tocoolant loss as observed during Fukushima nuclear accidentTherefore below we will consider the possible causes of theTWR inherent safety breach due to temperature blow-upmode

5 Possible Causes of the TWR InherentSafety Failure Fukushima Plutonium Effectand the Temperature Blowup Mode

It is known that with loss of coolant at three nuclear reactorsduring the Fukushima nuclear accident its nuclear fuelmelted It means that the temperature in the active zonereached the melting point of uranium-oxide fuel at somemoments (note that the third block partially used MOX-fuelenriched with plutonium) that is sim3000K

Surprisingly enough in scientific literature today thereare absolutely no either experimental or even theoreticallycalculated data on behavior of the 238U and 239Pu capturecross sections depending on temperature at least in 1000ndash3000K range At the same time there are serious reasonsto suppose that the cross section values of the specifiedelements increasewith temperatureWemay at least point outqualitative estimates by Ukraintsev [53] Obninsk Institute ofAtomic Energetics (Russia) that confirm the possibility of thecross sectionsrsquo growth for 239Pu in 300ndash1500K range

Obviously such anomalous temperature dependency ofcapture and fission cross sections of 238U and 239Pu may

change the neutron and thermal kinetics of a nuclear reactordrastically including the perspective fast uranium-plutoniumnew generation reactors (reactors of Feoktistov (1) and Teller(2) type) which we classify as fast TWR reactors Hence it isvery important to know the anomalous temperature behaviorof 238U and 239Pu capture and fission cross sections as well astheir influence on the heat transfer kinetics because it mayturn into a reason of the positive feedback (positive feedbackis a type of feedback when a change in the output signal leadsto such a change in the input signal which leads to evengreater deviation of the output signal from its original valueIn other words PF leads to the instability and appearanceof qualitatively new (often self-oscillation) systems) (PF)with the neutron kinetics leading to an undesirable solutionstability loss (the nuclear burning wave) and consequentlyto a trivial reactor runaway with a subsequent nontrivialcatastrophe

A special case of the PF is a nonlinear PF which leadsto the system evolution in the so-called blowup mode [54ndash59] or in other words in such a dynamic mode when one orseveral modeled values (eg temperature and neutron flux)grow to infinity at a finite time In reality instead of theinfinite values a phase transition is observed in this casewhich can become a first stage or a precursor of the futuretechnogenic disaster

Investigation of the temperature dependency of 238Uand 239Pu capture and fission cross sections in 300ndash3000Krange and the corresponding kinetics of heat transfer and itsinfluence on neutron kinetics in TWR is the main goal of thesection

Heat transfer equation for uranium-plutonium fissilemedium is as follows

120588 ( 119903 119879 119905) sdot 119888 ( 119903 119879 119905) sdot

120597119879 ( 119903 119905)

120597119905

= alefsym ( 119903 119879 119905) sdot Δ119879 ( 119903 119905) + nablaalefsym ( 119903 119879 119905) sdot nabla119879 (119903 119905)

+ 119902119891

119879( 119903 119879 119905)

(64)

where the effective substance density is

120588 ( 119903 119879 119905) = sum

119894

119873119894( 119903 119879 119905) sdot 120588

119894 (65)

120588119894are tabulated values and 119873

119894( 119903 119879 119905) are the components

concentrations in the medium while the effective specificheat capacity (accounting for the medium components heatcapacity values 119888

119894) and fissile material heat conductivity

coefficient (accounting for the medium components heatconductivity coefficients alefsym

119894(119879)) respectively are

119888 ( 119903 119879 119905) = sum

119894

119888119894(119879)119873

119894( 119903 119879 119905) (66)

alefsym ( 119903 119879 119905) = sum

119894

alefsym119894(119879)119873

119894( 119903 119879 119905) (67)

Here 119902119891119879( 119903 119879 119905) is the heat source density generated by the

nuclear fissions 119873119894of fissile metal components that vary in

time


Theoretical temperature dependency of heat capacity 119888(119879)formetals is known at low temperatures 119888(119905) sim 119879

3 and at hightemperatures 119888(119879) rarr const and the constant value (const asymp6Cal(molsdotdeg)) is determined by Dulong-Petit law At thesame time it is known that the thermal expansion coefficientis small for metals therefore the specific heat capacity atconstant volume 119888V is almost equal to the specific heat capacityat constant pressure 119888

119901 On the other hand the theoretical

dependency of heat conductivity alefsym119894(119879) at high temperature

of ldquofissilerdquo metals is not known while it is experimentallydetermined that the heat conductivity coefficient alefsym(119879) offissile medium is a nonlinear function of temperature (egsee [60] where the heat conductivity coefficient is given for120572-uranium238 and formetallic plutonium239 and also [61])

While solving the heat conduction equations we used thefollowing initial and boundary conditions

119879 (119903 119905 = 0) = 300K 119895119899= alefsym [119879 (119903 isin R 119905) minus 119879

0] (68)

where 119895119899is the normal (to the fissile medium boundary) heat

flux density component alefsym(119879) is the thermal conductivitycoefficient R is the fissile medium boundary and 119879

0is the

temperature of the medium adjacent to the active zoneObviously if the cross sections of some fissile nuclides

increase then due to the nuclei fission reaction exother-micity the direct consequence of the significantly nonlinearkinetics of the parental and child nuclides in the nuclearreactor is an autocatalyst increase of generated heat similar toautocatalyst processes of the exothermic chemical reactionsIn this case the heat flux density 119902119891

119879( 119903 Φ 119879 119905) that character-

izes the generated heat amount will be

119902119891

119879( 119903 Φ 119879 119905)

= Φ ( 119903 119879 119905)sum

119894

119876119891

119894120590119894

119891( 119903 119879 119905)119873

119894( 119903 119879 119905) [Wcm3]

(69)

where

Φ ( 119903 119879 119905) = int

119864max119899

0

Φ ( 119903 119864 119879 119905) 119889119864 (70)

is the full neutron flux densityΦ( 119903 119864 119879 119905) is the neutron fluxdensity with energy 119864119876119891

119894is the mean generated heat emitted

due to fission of one nucleus of the 119894th nuclide

120590119894

119891( 119903 119879 119905) = int

119864max119899

0

120590119894

119891(119864 119879) 120588 ( 119903 119864 119879 119905) 119889119864 (71)

is the fission cross section of the 119894th nuclide averaged over theneutron spectrum

120588 ( 119903 119864 119879 119905) =

Φ ( 119903 119864 119879 119905)

Φ ( 119903 119879 119905)

(72)

is the neutron energy distribution probability density func-tion 120590119894

119891(119864 119879) is the microscopic fission cross section of the

119894th nuclide that as known depends on the neutron energyand fissile medium temperature (Doppler effect [29]) and119873119894( 119903 119879 119905) is the density of the 119894th nuclide nuclei

35

3

25

2

15

1

05

0

0 1 2 3 4 5 6

0

E (eV)

3000

2000

1000 T(K

)

(times100

barn

)

Figure 9 Calculated dependency of radioactive neutron capturecross section on the energy for 235

92U at different temperatures within

300K to 3000K

As follows from (69) in order to build the thermal sourcedensity function it is necessary to derive the theoreticaldependency of the cross sections 120590119894

119891( 119903 119879 119905) averaged over

the neutron spectrum on the reactor fuel temperature Asis known the influence of the nuclei thermal motion onthe medium comes to a broadening and height reductionof the resonances By optical analogy this phenomenon isreferred to as Doppler effect [29] Since the resonance levelsare observed only for heavy nuclei in the low energy areathen Doppler effect is notable only during the interaction ofneutrons with such nuclei And the higher the environmenttemperature is the stronger the effect is

Therefore a program was developed using MicrosoftFortran Power Station 40 (MFPS 40) that allows at the firststage the cross sections of the resonance neutron reactionsto be calculated depending on neutron energy taking intoaccount the Doppler effect The cross sections dependencyon neutron energy for reactor nuclides from ENDFB-VIIdatabase [62] corresponding to 300K environment temper-ature was taken as the input data for the calculations Forexample the results of radioactive neutron capture crosssections dependency on neutron energy for 235U are givenin Figure 9 for different temperatures of the fissile mediumin 300Kndash3000K temperature range Using this programthe dependency of scattering fission and radioactive neutroncapture cross sections for the major reactor fuel nuclides235

92U 23892U 23992U and 239

94Pu for different temperatures in range

300K to 3000K was obtainedAt the second stage a programwas developed to obtain the

calculated dependency of the cross sections 120590119894119891( 119903 119879 119905) aver-

aged over the neutron spectrum formain reactor nuclides andfor main neutron reactions for the specified temperaturesThe averaging of the neutron cross sections for the Maxwelldistribution was performed using the following expression

⟨120590 (119864lim 119879)⟩ =

int

119864lim

011986412

119890minus119864119896119879

120590 (119864 119879) 119889119864

int

119864lim

011986412

119890minus119864119896119879

119889119864

(73)


where 119864lim is the upper limit of the neutrons thermalizationwhile for the procedure of neutron cross sections averagingover the Fermi spectrum the following expression was used

⟨120590 (119864lim 119879)⟩ =

int

infin

119864lim120590 (119864 119879) 119864

minus1119889119864

int

infin

119864lim119864minus1119889119864

(74)

During further calculations in our programs we used theresults obtained at the first stage that is the dependency ofreaction cross sections on neutron energy and environmenttemperature (Doppler effect) The neutron spectrum wasspecified in a combined way by Maxwell spectrum Φ

119872(119864119899)

below the limit of thermalization 119864lim by Fermi spectrumΦ119865(119864) for a moderating medium with absorption above

119864lim but below 119864119865(upper limit for Fermi neutron energy

spectrum) and by 239Pu fission spectrum [15 21] above119864119865but below the maximal neutron energy 119864

max119899

Here theneutron gas temperature for Maxwell distribution was givenby (75) described in [29] According to this approach [29]the drawbacks of the standard slowing-down theory forthermalization area may be formally reduced if a variable120585(119909) = 120585(1 minus 2119911) is introduced instead of the averagelogarithmic energy loss 120585 which is almost independent ofthe neutron energy (as known the statement 120585 asymp 2119860 istrue for the environment consisting of nuclei with 119860 gt 10)Here 119911 = 119864

119899119896119879 where 119864

119899is the neutron energy and 119879 is

the environment temperature Then the following expressionmay be used for the neutron gas temperature in Maxwellspectrum of thermal neutrons (a very interesting expressionrevealing hidden connection between the temperature of aneutron gas and the environment (fuel) temperature)

119879119899= 1198790[1 + 120578 sdot

Σ119886(1198961198790)

⟨120585⟩ Σ119878

] (75)

where 1198790is the fuel environment temperature Σ

119886(1198961198790) is

an absorption cross section for energy 1198961198790 120578 = 18 is the

dimensionless constant and ⟨120585⟩ is averaged over the wholeenergy interval of Maxwell spectrum 120585(119911) at 119896119879 = 1 eV

Fermi neutron spectrum for a moderating medium withabsorption (we considered carbon as a moderator and 238U239U and 239Pu as the absorbers) was set in the form [29 52]

ΦFermi (119864 119864119865) =119878

⟨120585⟩ Σ119905119864

exp[minusint

119864119891

119864lim

Σ119886(1198641015840) 1198891198641015840

⟨120585⟩ Σ119905(1198641015840) 1198641015840]

(76)

where 119878 is the total volume neutron generation rate ⟨120585⟩ =

sum119894(120585119894Σ119894

119878)Σ119878 120585119894is the average logarithmic decrement of

energy loss Σ119894119878is the macroscopic scattering cross section

of the 119894th nuclide Σ119905= sum119894Σ119894

119878+ Σ119894

119886is the total macroscopic

cross section of the fissile material Σ119878= sum119894Σ119894

119878is the total

macroscopic scattering cross section of the fissile materialΣ119886is the macroscopic absorption cross section and 119864

119865is the

upper neutron energy for Fermi spectrumThe upper limit of neutron thermalization119864lim in our cal-

culation was considered a free parameter setting the neutron

fluxes of Maxwell and Fermi spectra at a common energylimit 119864lim equal to

ΦMaxwell (119864lim) = ΦFermi (119864lim) (77)

The high energy neutron spectrum part (119864 gt 119864119865) was

defined by the fission spectrum [52 63 64] in our calcu-lations Therefore the following expression may be writtenfor the total volume neutron generation rate 119878 in the Fermispectrum (76)

119878 ( 119903 119879 119905)

= int

119864max119899

119864119865

( 119903 119864 119879 119905)

sdot [sum

119894

]119894(119864) sdot Φ ( 119903 119864 119879 119905) sdot 120590

119894

119891(119864 119879) sdot 119873

119894( 119903 119879 119905)] 119889119864

(78)

where 119864max119899

is the maximum energy of the neutron fissionspectrum (usually taken as119864max

119899asymp 10MeV)119864

119865is the neutron

energy below which the moderating neutrons spectrumis described as Fermi spectrum (usually taken as 119864

119865asymp

02MeV) and ( 119903 119864 119879 119905) is the probability of neutron notleaving the boundaries of the fissile medium which dependson the fissile material geometry and the conditions at itsborder (eg presence of a reflector)

The obtained calculation results show that the crosssections averaged over the spectrum may increase (Figure 10for 239Pu and Figure 12 for 238U) as well as decrease (Figure 11for 235U) with fissile medium temperature As follows fromthe obtained results the arbitrariness in selection of thelimit energy for joining the Maxwell and Fermi spectra doesnot significantly alter the character of these dependenciesevolution

This can be justified by the fact that 239Pu resonance areastarts from significantly lower energies than that of 235U andwith fuel temperature increase the neutron gas temperaturealso increases producing Maxwellrsquos neutron distributionmaximum shift to the higher neutron energies that is theneutron gas spectrum hardening when more neutrons fitinto resonance area of 239Pu is the cause of the averagedcross sections growth

This process is not as significant for 235U because itsresonance area is located at higher energies As a resultthe 235U neutron gas spectrum hardening related to thefuel temperature increase (in the considered interval) doesnot result in a significant increase of a number of neutronsfitting into the resonance area Therefore according to theknown expressions of 235Udetermining the neutron reactionscross sections behaviour depending on their energy 119864

119899for

nonresonance areas we observe dependency for the averagedcross sections 120590

119899119887sim 1radic119864

119899

The data on the averaged fission and capture crosssections of 238Upresented in Figure 12 show that the averagedfission cross section for 238U is almost insensitive to theneutron spectrum hardening caused by the fuel temperature


2000

1500

1000

500

00 1000 2000 3000 4000 5000 6000

T (K)

120590f

(bar

n)

3kT35kT4kT

5kT6kT

(a)

2000

1500

1000

500

00 1000 2000 3000 4000 5000 6000

T (K)

3kT35kT4kT

5kT6kT

120590c

(bar

n)(b)

Figure 10 Temperature dependencies for the fission cross section (a) and radioactive capture cross section (b) for 239Pu averaged over theMaxwell spectrum on the Maxwell and Fermi spectra joining energy and 120578 = 18 (see (75))

200

180

160

140

120

100

80

60

400 1000 2000 3000 4000 5000 6000

T (K)

120590f

(bar

n)

3kT35 kT4kT

5kT6kT

(a)

140

120

100

80

60

0

20

40

0 1000 2000 3000 4000 5000 6000

T (K)

3kT35kT4kT

5kT6kT

120590c

(bar

n)

(b)

Figure 11 Temperature dependencies for the fission cross section (a) and radioactive capture cross section (b) for 235U averaged over theMaxwell spectrum on the Maxwell and Fermi spectra joining energy and 120578 = 18 (see (75))

increase due to a high fission threshold sim1MeV (see Fig-ure 12(a)) At the same time they confirm the capture crosssection dependence on temperature since its resonance areais located as low as for 239Pu Obviously in this case thefuel enrichment with 235U makes no difference because theaveraged cross sections for 235U as described above behavein a standard way

And finally we performed a computer estimate of the heatsource density dependence 119902119891

119879( 119903 Φ 119879 119905) (69) on temperature

for the different compositions of the uranium-plutoniumfissilemediumwith a constant neutron flux density presentedat Figure 13 We used the dependencies presented above atFigures 10ndash12 for these calculations Let us note that our pre-liminary calculations were made without taking into account


036

034

032

03

028

026

0240 1000 2000 3000 4000 5000 6000

T (K)

120590f

(bar

n)

3kT35 kT4kT

5kT6kT

(a)

800

700

600

500

400

300

200

120590c

(bar

n)

0 1000 2000 3000 4000 5000 6000

T (K)

3kT35kT4kT

5kT6kT

(b)

Figure 12 Temperature dependencies for the fission cross section (a) and radioactive capture cross section (b) for 23892U averaged over the

combined Maxwell and Fermi spectra depending on the Maxwell and Fermi spectra joining energy and 120578 = 18 (see (75))

25

20

15

10

05

0600 800 1000 1200 1400

T (K)

qf T(rarrrΦTN

it)

(times10

20eV

)

1

2

3

Figure 13 Dependence of the heat source density 119902119891

119879(119903 Φ 119879119873

119894

119905) [eV] on the fissile medium temperature (300ndash1400K) for severalcompositions of uranium-plutonium medium (1 10 Pu 2 5Pu and 3 1 Pu) at the constant neutron flux density Φ =

1013 n(cm2sdots)

the change in the composition and density of the fissileuranium-plutonium medium which is a direct consequenceof the constant neutron flux assumption

The necessity of such assumption is caused by the fol-lowingThe reasonable description of the heat source density119902119891

119879( 119903 Φ 119879 119905) (69) temperature dependence requires the solu-

tion of a system of three equations two of them correspondto the neutron kinetics equation (flux and fluence) and tothe system of equations for kinetics of the parental and childnuclides nuclear density (eg see [5 16]) while the thirdone corresponds to a heat transfer equation of (64) typeHowever some serious difficulties arise here associated withthe computational capabilities available And here is why

One of the principal physical peculiarities of the TWR isthe fact [14] that fluctuation residuals of plutonium (or 233UinTh-U cycle) over its critical concentration burn out for thetime comparable with reactor lifetime of a neutron 120591

119899(119909 119905)

(not considering the delayed neutrons) or at least comparablewith the reactor period (the reactor period by definition isequal to 119879(119909 119905) = 120591

119899(119909 119905)120588(119909 119905) that is a ratio of the

reactor neutron lifetime to reactivity) 119879(119909 119905) (consideringthe delayed neutrons) Meanwhile the new plutonium (or233U in Th-U cycle) is formed in a few days (or a month)and not immediately This means [14] that the numericalcalculation must be performed with a temporal step about10minus6ndash10minus7 in case of not taking into account the delayed

neutrons and sim10minus1ndash100 otherwise At first glance taking intoaccount the delayed neutrons according to [14] really ldquosavesthe dayrdquo however it is not always true If the heat transferequation contains a significantly nonlinear source then in thecase of a blowup mode the temperature may grow extremelyfast under some conditions and in 10ndash20 steps (with timestep 10minus6ndash10minus7 s) reaches the critical amplitude that may leadto (at least) a solution stability loss or (as maximum) to ablowup bifurcation of the phase state almost unnoticeablewith a rough time step

According to these remarks and considering the goaland format of this paper we did not aim at findingthe exact solution of some specific system of three jointequations described above Instead we found it impor-tant to illustrate at the qualitative level the consequencesof the possible blowup modes in case of a nonlinearheat source presence in the heat transfer equation Assaid above we made some estimate computer calculationsof the heat source density 119902

119891

119879( 119903 Φ 119879 119905) (69) temperature

dependence in 300ndash1400K range for some compositions of


uranium-plutoniumfissilemedium at a constant neutron flux(Figure 13)

The obtained dependencies for the heat source density119902119891

119879( 119903 Φ 119879 119905) were successfully approximated by a power

function of temperature with an exponent of 4 (Figure 13)In other words we obtained a heat transfer equation with asignificantly nonlinear heat source in the following form

119902119879(119879) = const sdot 119879(1+120575) (79)

where 120575 gt 1 in case of nonlinear thermal conductivitydependence on temperature [54ndash58] The latter means thatthe solutions of the heat transfer equation (64) describethe so-called Kurdyumov blow-up modes [54ndash59] that issuch dynamic modes when one of the modeled values (egtemperature) turns into infinity for a finite time As notedbefore in reality instead of reaching the infinite values aphase transition is observed (a final phase of the parabolictemperature growth) which requires a separate model and isa basis for an entirely new problem

Mathematical modeling of the blowup modes was per-formed mainly using Mathematica 52ndash60 Maple 10 MAT-LAB 70 utilizing multiprocessor calculations for effectiveapplication A RungeKutta method of 8-9th order and thenumerical methods of lines [65] were applied for the calcu-lations The numerical error estimate did not exceed 001The coordinate and temporal steps were variable and chosenby the program in order to fit the given error at every step ofthe calculation

Below we give the solutions for the heat transfer equation(64) with nonlinear exponential heat source (79) in uranium-plutonium fissile medium for boundary and initial parame-ters corresponding to the industrial reactorsThe calculationswere done for a cube of the fissile material with differentsizes boundary and initial temperature values Since thetemperature dependencies of the heat source density wereobtained without accounting for the changing compositionand density of the uranium-plutonium fissile medium dif-ferent blow-upmodes can take place (HS-mode S-mode LS-mode) depending on the ratio between the exponents of theheat conductivity and heat source temperature dependencesaccording to [54ndash59] Therefore we considered the casesfor 1st 2nd and 4th temperature order sources Here thepower of the source also varied by varying the proportionalityfactor in (79) (const = 100 J(cm3sdotssdotK) for the 1st tem-perature order source 010 J(cm3sdotssdotK2) 015 J(cm3sdotssdotK2)and 100 J(cm3sdotssdotK2) for the 2nd temperature order source100 J(cm3sdotssdotK4) for the 4th temperature order source)

During the calculations of the heat capacity 119888119901

(Figure 14(a)) and heat conductivity alefsym (Figure 14(b)) ofa fissile medium dependence on temperature in 300ndash1400Krange the specified parameters were given by analyticexpressions obtained by approximation of experimental datafor 238U based on polynomial progression

119888119901(119879) asymp minus 7206 + 064119879 minus 00047119879

2+ 00000126119879

3

+ 2004 sdot 10minus81198794minus 160 sdot 10

minus101198795

minus 215 sdot 10minus13

1198796

(80)

25

20

15

10

5

00 200 400 600 800 1000 1200 1400

T (K)

c p(J

molmiddotK

)

alefsym(W

mmiddotK

)

40

375

35

325

30

275

25

225

Figure 14 Temperature dependence of the heat capacity 119888119875and

heat conductivity 120594 of the fissile material Points represent theexperimental values for the heat capacity and heat conductivity of238U

alefsym (119879) asymp 21575 + 00152661119879 (81)

And finally the heat transfer equation (64) solution wasobtained for the constant heat conductivity (275W(msdotK))and heat capacity (115 J(Ksdotmol)) presented in Figure 15(a)and also the solutions of the heat transfer equation consider-ing their temperature dependencies (Figures 15(b)ndash15(d))

These results directly point out the possibility of the localuranium-plutoniumfissilemediummelting with themeltingtemperature almost identical to that of 238U which is 1400K(Figures 14(a)ndash14(d)) Moreover these regions of the localmelting are not the areas of the so-called thermal peaks [66]and probably are the anomalous areas of uranium surfacemelting observed by Ershler and Lapteva [67] that were alsomentioned in [68] More detailed analysis of the probabletemperature scenarios associated with the blowup modes isdiscussed below

6 The Blowup Modes in Neutron-MultiplyingMedia and the Pulse Thermonuclear TWR

Earlier we noted the fact that due to a coolant loss at thenuclear reactors during the Fukushima nuclear accident thefuel was melted which means that the temperature insidethe active zone reached the melting temperature of uranium-oxide fuel at some moment that is sim3000K

On the other hand we already know that the coolantloss may become a cause of the nonlinear heat sourceformation inside the nuclear fuel and therefore become acause of the temperature and neutron flux blowup modeonset A natural question arises of whether it is possible touse such blowup mode (temperature and neutron flux) forthe initiation of certain controlled physical conditions underwhich the nuclear burningwave would regularly ldquoexperiencerdquothe so-called ldquocontrolled blow-uprdquomode It is quite difficult toanswer this question definitely because such fast process hasa number of important physical vagueness problems any ofwhich can become experimentally insurmountable for suchprocess control

Nevertheless such process is very elegant and beautifulfrom the physics point of view and therefore requires a more


T (K)x (120583m)

t (s)

0

1

2

3 0

025

05

075

10

0

1025

05

075

103

(d)

0

1

2

3 0

025

05

075

10

1025

05

075

(b)

0

1

2

3 0

025

05

075

10

0

1

05

075

(c)

0

1

2

3 0

025

05

075

10

0

1025

05

075

(a)

4 middot 104

2 middot 103

6 middot 104

Figure 15 Heat transfer equation (64) solution for 3D case (crystal sizes 0001 times 0001 times 0001mm initial and boundary temperatures equalto 100K) (a) The source is proportional to the 4th order of temperature const = 100 J(cm3sdotssdotK4) heat capacity and heat conductivity areconstant and equal to 115 J(Ksdotmol) and 275W(msdotK) respectively (b) the source is proportional to the 4th order of temperature const =100 J(cm3sdotssdotK4) (c) the source is proportional to the 2nd order of temperature const = 100 J(cm3sdotssdotK2) (d) the source is proportional tothe 2nd order of temperature const = 010 J(cm3sdotssdotK2) Note in the cases (b)ndash(d) the heat capacity and heat conductivity were determinedby (80) and (81) respectively

detailed phenomenological description Let us try to make itin short

As we can see from the plots of the capture and fissioncross sections evolution for 239Pu (Figure 10) the blowupmode may develop rapidly at sim1000ndash2000K (depending onthe real value of the Fermi and Maxwell spectra joiningboundary) but at the temperatures over 2500ndash3000K thecross sections return almost to the initial values If someeffective heat sink is turned on at that point the fuel mayreturn to its initial temperature However while the blowupmode develops the fast neutrons already penetrate to theadjacent fuel areas where the new fissile material startsaccumulating and so on (see cycles (1) and (2)) After sometime the similar blowup mode starts developing in thisadjacent area and everything starts over again In otherwords

such hysteresis blowup mode closely time-conjugated to aheat takeoff procedure will appear on the background of astationary nuclear burning wave in a form of the periodicimpulse bursts

In order to demonstrate the marvelous power of suchprocess we investigated the heat transfer equation with non-linear exponential heat source in uranium-plutonium fissilemedium with boundary and initial parameters emulatingthe heat takeoff process In other words we investigated theblowup modes in the Feoktistov-type uranium-plutoniumreactor (1) where the temperature inside and at the boundarywas deliberately fixed at 6000K which corresponds to themodel of the georeactor (let us note that ourmodel georeactoris not a fast reactor the possibility of the nuclear wave burn-ing for a reactor other than the fast one is examined in our


next paper [69]) [70] Expression (75) for the neutron gastemperature used for the calculation of the cross sectionsaveraged over the neutron spectrum transforms in this caseto the following

119879119899asymp [1 + 18

80 sdot 1198702

⟨120585⟩ sdot 45

] (82)

This equation is obtained for the supposed fissile mediumcomposition of the uranium and plutonium dicarbides [70ndash74] where the 238U was the major absorber (its microscopicabsorption cross section for the thermalization temperatureswas set at1205908

119886= 80 barn) and the 12Cwas themajormoderator

(its microscopic scattering cross section was set at 12059012119904

= 45

barn)The 238U and 12C nuclei concentrations ratio was set tothe characteristic level for the dicarbides

1198702=

119873238

11987312

= 05 (83)

The Fermi spectrum for the neutrons in moderating andabsorbing medium of the georeactor (carbon played a role ofthe moderator and the 238U) 239U and 239Pu played the roleof the absorbers was taken in the same form (76)

As an example Figure 16 shows the calculated temperaturedependences of the 235U and 239Pu fission cross sectionsaveraged over the neutron spectrum

The temperature choice is conditioned by the followingimportant consideration ldquoIs it possible to obtain a solution(ie a spatiotemporal temperature distribution) in a formof the stationary solitary wave with a limited amplitudeinstead of a 120575-function at some local spatial area undersuch conditions (6000K) emulating the time-conjugated heattakeoff (see Figure 10)rdquo As shown below such approachreally works

Below we present some calculation characteristics andparameters During these calculations we used the followingexpression for dependence of the heat conductivity coeffi-cient

alefsym = 018 sdot 10minus4

sdot 119879 (84)

which was obtained using the WiedemannFranz law andthe data on electric conductivity of metals at temperature6000K [75] Specific heat capacity at constant pressure wasdetermined by value 119888

119901asymp 6 cal(molsdotdeg) according to

Dulong and Petit lawThe fissile uranium-plutonium medium was modeled as

a cube with dimensions 100 times 100 times 100m (Figure 17)Here for heat source we used the 2nd order temperaturedependence (see (79))

And finally Figures 17(a)ndash17(d) present a set of solutionsof heat transfer equation (64) with nonlinear exponentialheat source (79) in uranium-plutonium fissile medium withboundary and initial conditions emulating such process ofheat takeoff in which initial and boundary temperaturesremain constant and equal to 6000K

It is important to note here that the solution set presentedat Figure 17 demonstrates the solution tendency towards its

0

50

100

150

200

250

300

350

400

600

700

800

900

1000

1100

1200

1300

1400

0 1000 2000 3000 4000 5000 6000

0 1000 2000 3000 4000 5000 6000

T (K)

239Pu

fiss

ion

cros

s sec

tion

(bar

n)

235U

fiss

ion

cros

s sec

tion

(bar

n)

239Pu

235U

Figure 16 The temperature dependences of the 239Pu fission crosssection averaged over the neutron spectrum for the limit energy forthe Fermi and Maxwell spectra joining equal to 3 kTThe analogousdependency for the 235U is also shown

ldquostationaryrdquo state quite clearly This is achieved using the so-called ldquomagnifying glassrdquo approach when the solutions ofthe same problem are deliberately investigated at differenttimescales For example Figure 17(a) shows the solution atthe time scale 119905 isin [0 10

minus6 s] while Figure 17(b) describes thespatial solution of the problem (temperature field) for 119905 =

10minus6 s Figures 17(c) and 17(d) present the solution (spatial

temperature distribution) at 119905 = 05 s and 119905 = 50 sAs one can see the solution (Figure 17(d)) is completely

identical to the previous one (Figure 17(c)) that is thedistribution established in the medium in 05 seconds whichallowed us to make a conclusion on the temperature fieldstability starting from somemoment It is interesting that theestablished temperature field creates the conditions suitablefor the thermonuclear synthesis reaction that is reaching108 K and such temperature field lifetime is not less than50 s These conditions are highly favorable for a stable ther-monuclear burning according to a known Lawson criterionproviding the necessary nuclei concentration entering thethermonuclear synthesis reaction

One should keep in mind though that the results ofthis section are for the purpose of demonstration only sincetheir accuracy is rather uncertain and requires a carefulinvestigationwith application of the necessary computationalresources Nevertheless the qualitative peculiarities of thesesolutions should attract the researchersrsquo attention to thenontrivial properties of the blowup modes at least withrespect to the obvious problem of the inherent TWR safetyviolation

7 Conclusions

Let us give some short conclusions stimulated by the follow-ing significant problems


(d)

250

500

750

1000

0250

500750

6000

250

500

750

250500

750

(b)

0

250

500

750

1000

0

250500

7501000

01000 01000

250

500

750

0

250500

750

6000

(c)

0250

500

7500

250

500

750

6000

250

500

750

1000

(a)

0

250

500

750

1000

60000

250

500

750

10

0

y (cm)

x (cm)

t (s)t (s)

t (s) t (s)

T (K)

x(cm

)

x(cm

)

x(cm

)

x(cm

)

3 middot 10104 middot 1011

3 middot 1011

2 middot 1011

4 middot 1012

2 middot 1012

4 middot 1012

2 middot 1012


1 middot 1010

2 middot 10minus7

4 middot 10minus7

6 middot 10minus7

8 middot 10minus7

1 middot 10minus6

Figure 17 Heat transfer equation solution for a model georeactor (source sim 2nd order temperature dependence const = 419 J(cm3sdotssdotK2))initial and boundary temperatures equal to 6000K fissile medium is a cube 10 times 10 times 10mThe presented results correspond to the followingtimes of temperature field evolution (a) (1ndash10) sdot 10minus7 s (b) 10minus6 s (c) 05 s and (d) 50 s

(1) TWR and the Problem of dpa-Parameter in CladdingMaterials A possibility to surmount the so-called problem ofdpa-parameter based on the conditions of nuclear burningwave existence in U-Pu and Th-U cycles is shown In otherwords it is possible to find a nuclear burning wave modewhose parameters (fluenceneutron flux width and speed ofthe wave) satisfy the dpa-condition (61) of the reactor mate-rials radiation resistance particularly that of the claddingmaterials It can be done using the joined application of theldquodifferentialrdquo [5] and ldquointegralrdquo [14ndash16] conditions for nuclearburning wave existence The latter means that at the presenttime the problem of dpa-parameter in cladding materials inthe TWR project is not an insurmountable technical problemand can be satisfactorily solved

Herewemay add that this algorithmof an optimal nuclearburning wave mode selection predetermines a satisfactorysolution of other technical problems mentioned in Introduc-tion For example the fuel rod length in the proposed TWRvariant (see the ldquoidealrdquo case in Table 1) is predetermined bythe nuclear burning wave speed which in a given case isequal to 0254 cmday equiv 85 cmyear that is 20 years of TWR

operation requires the fuel rod length sim17m On the otherhand it is known [76] that for a twisted fuel rod form withtwo- or four-bladed symmetry the tension emerging fromthe fuel rod surface cooling is 30 lower than that of a roundrod with the same diameter other conditions are equal Thesame reduction effect applies to the hydraulic resistance incomparison with a round rod of the same diameter

Another problem associated with the reactor materialsswelling is also solved rather simply It is pertinent to notethat if a ferritic-martensitic material is chosen as a claddingmaterial (Figure 6 [43]) then the swelling effect at the endof operation will be only sim05 [43] We could discuss otherdrawbacks mentioned in the introduction as well but inour opinion the rest of the problems are not the superobstacles for the contemporary level of nuclear engineeringas compared with the main problem of dpa-condition andcan be solved in a traditional way

(2) The Consequences of the Anomalous 238119880 and 239119875119906 CrossSections Behavior with Temperature It is shown that thecapture and fission cross sections of 238U and 239Pu manifest


a monotonous growth in 1000ndash3000K range Obviouslysuch anomalous temperature dependence of 238U and 239Pucross sections changes the neutron and heat kinetics of thenuclear reactors drastically It becomes essential to know theirinfluence on kinetics of heat transfer because it may becomethe cause of a positive feedback with neutron kinetics whichmay lead not only to undesirable loss of the nuclear burningwave stability but also to a reactor runaway with a subsequentdisaster

(3) Blowup Modes and the Problem of the Nuclear BurningWave Stability One of the causes of possible fuel temperaturegrowth is a deliberate or spontaneous coolant loss similar toFukushima nuclear accident As shown above the coolantloss may become a cause of the nonlinear heat source for-mation in the nuclear fuel and the corresponding mode withtemperature and neutron flux blowup In our opinion thepreliminary results of heat transfer equation with nonlinearheat source investigations point out an extremely importantphenomenon of the anomalous behaviour of the heat andneutron flux blowup modes This result poses a naturalnontrivial problem of the fundamental nuclear burning wavestability and correspondingly of a physically reasonableapplication of the Lyapunov method to this problem

It is shown that some variants of the solution stability lossare caused by anomalous nuclear fuel temperature evolutionThey can lead not only to the TWR inherent safety loss butalso through a bifurcation of states (and this is very impor-tant) to a new stable mode when the nuclear burning waveperiodically ldquoexperiencesrdquo the so-called ldquocontrolled blowuprdquomode At the same time it is noted that such fast (blowupregime) process has a number of physical uncertaintieswhich may happen to be experimentally insurmountable forthe purposes of such process control

(4) On-Line Remote Neutrino Diagnostics of the IntrareactorProcesses The high-power TWR or a nuclear fuel transmuta-tion reactor is the project with the single-load fuel burn-upand the subsequent burial of the reactor apparatus Hence itis necessary to perform a remote neutrino monitoring of thenuclear burning wave during the normal operation and theneutron kinetics during the emergency situation The detailsand peculiarities of the isotope composition spatiotemporaldistribution calculation in the active zone of the TWR arepresented in [70 74 77] in detail within the inverse problemof the intrareactor processes neutrino diagnostics

Conflict of Interests

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

References

[1] L Feoktistov From Past towards the Future From the Hopesabout Bomb to Safe Reactor RFNC-ANRISPh Snezhinsk Rus-sia 1998

[2] S Feinberg ldquoDiscussion content in Record of ProceedingsSession B-10rdquo Proceedings of the International Conference on

the Peaceful Uses for Atomic Energy Vol 9 no 2 p 447UnitedNations Geneva Switzerland 1958

[3] L Feoktistov ldquoNeutron-fission waverdquo Doklady Akademii NaukSSSR no 309 pp 4ndash7 1989

[4] E TellerM Ishikawa LWood RHyde and J Nuckolls ldquoCom-pletely automated nuclear reactors for long-term operation IItoward a concept-level point-design of a high temperaturegascooled central power station system part IIrdquo in Proceedingsof the International Conference on Emerging Nuclear Energy Sys-tems (ICENES rsquo96) pp 123ndash127 Lawrence Livermore NationalLaboratory Obninsk Russia 1996

[5] V D Rusov E P Linnik V A Tarasov et al ldquoTravelingwavereactor and condition of existence of nuclear burning soliton-like wave in neutron-multiplying mediardquo Energies vol 4 no 9pp 1337ndash1361 2011

[6] W Seifritz ldquoOn the burn-up theory of fast soliton reactorsrdquoInternational Journal of Hydrogen Energy vol 23 no 2 pp 77ndash82 1998

[7] W Seifritz ldquoSolitary burn-up waves in multiplying mediumrdquoKerntechnik vol 65 no 5-6 pp 261ndash264 2000

[8] H van Dam ldquoSelf-stabilizing criticality wavesrdquo Annals ofNuclear Energy vol 27 no 16 pp 1505ndash1521 2000

[9] H Sekimoto K Ryu and Y Yoshimura ldquoCANDLE the newburnup strategyrdquo Nuclear Science and Engineering vol 139 no3 pp 306ndash317 2001

[10] N Khizhnyak ldquoOn the theory of the initial stage of slow nuclearburningrdquo Problems of Atomic Science and Technology vol 6 pp279ndash282 2001

[11] X-N Chen and W Maschek ldquoTransverse buckling effects onsolitary burn-up wavesrdquo Annals of Nuclear Energy vol 32 no12 pp 1377ndash1390 2005

[12] H Sekimoto and Y Udagawa ldquoEffects of fuel and coolanttemperatures and neutron fluence on CANDLE burnup calcu-lationrdquo Journal of Nuclear Science and Technology vol 43 no 2pp 189ndash197 2006

[13] V N Pavlovich E N Khotyaintseva V D Rusov V NKhotyaintsev and A S Yurchenko ldquoReactor operating on aslow wave of nuclear fissionrdquo Atomic Energy vol 102 no 3 pp181ndash189 2007

[14] V Pavlovich V Khotyaintsev and E Khotyaintseva ldquoPhysicalbasics of the nuclear burning wave reactor 1rdquo Nuclear Physicsand Energetics vol 3 no 2 pp 39ndash48 2008

[15] V Pavlovich V Khotyaintsev and E Khotyaintseva ldquoPhysicalbasics of the nuclear burning wave reactor 2 specific modelsrdquoNuclear Physics and Energetics no 3 pp 39ndash48 2008

[16] V M Pavlovich V M Khotyayintsev and O M Khotyayint-seva ldquoNuclear burning wave reactor wave parameter controlrdquoNuclear Physics andAtomic Energy vol 11 no 1 pp 49ndash56 2010

[17] V Khotyaintsev V Pavlovich and E Khotyaintseva ldquoTravelingwave reactor velocity formation mechanismsrdquo in Proceedingsof the International Conference on the Physics of Reactors(PHYSOR rsquo10) Advances in Reactor Physics to Power the NuclearRenaissance Pittsburgh Pa USA May 2010

[18] X-N Chen W Maschek A Rineiski and E KiefhaberldquoSolitary burn-up wave solution in a multi-group diffusion-burnup coupled systemrdquo in Proceedings of the 13th InternationalConference on Emerging Nuclear Energy Systems (ICENES rsquo07)pp 236ndash245 Istanbul Turkey June 2007

[19] X-N Chen E Kiefhaber and W Maschek ldquoNeutronic modeland its solitary wave solutions for a candle reactorrdquo in Proceed-ings of the 12th International Conference on Emerging Nuclear


Energy Systems (ICENES rsquo05) pp 742ndash767 Brussels BelgiumAugust 2005

[20] Y Ohoka P H Liem and H Sekimoto ldquoLong life small candle-htgrs with thoriumrdquo Annals of Nuclear Energy vol 34 no 1-2pp 120ndash129 2007

[21] R Hyde M Ishikawa N Myhrvold J Nuckolls and L WoodldquoNuclear fission power for 21st century needs enabling tech-nologies for large-scale low-risk affordable nuclear electricityrdquoProgress in Nuclear Energy vol 50 no 2ndash6 pp 32ndash91 2008

[22] X-N Chen E Kiefhaber and W Maschek ldquoFundamentalburn-upmode in a pebble-bed type reactorrdquo Progress in NuclearEnergy vol 50 no 2ndash6 pp 219ndash224 2008 Proceedings ofthe 2nd COE-INES International Symposium (INES-2 rsquo06)November 26ndash30 2006 Yokohama Japan

[23] K D Weaver J Gilleland C Ahlfeld C Whitmer and GZimmerman ldquoA once-through fuel cycle for fast reactorsrdquoJournal of Engineering for Gas Turbines and Power vol 132 no10 Article ID 102917 2010

[24] W Seifritz ldquoNon-linear burn-up waves in opaque neutronabsorbersrdquo Kerntechnik vol 60 no 4 pp 185ndash188 1995

[25] A Akhiezer D Belozorov F Rofe-Beketov L Davydov andZ Spolnik ldquoOn the theory of propagation of chain nuclearreaction in diffusion approximationrdquo Yadernaya Fizika vol 62no 9 pp 1567ndash1575 1999

[26] A I Akhiezer D P Belozorov F S Rofe-Beketov L NDavydov and Z A Spolnik ldquoOn the theory of propagation ofchain nuclear reactionrdquo Physica A Statistical Mechanics and itsApplications vol 273 no 3-4 pp 272ndash285 1999

[27] A Akhiezer D Belozorov F Rofe-Beketov L Davydov and ZSpolnik ldquoThe velocity of slow nuclear burning in two-groupapproximationrdquo Problems of Atomic Science and Technology no6 pp 276ndash278 2001

[28] A Akhiezer N Khizhnyak N Shulga V Pilipenko and LDavydov ldquoSlow nuclear burningrdquo Problems of Atomic Scienceand Technology no 6 pp 272ndash275 2001

[29] G Bartolomey G Batrsquo V Babaykov and M Altukhov BasicTheory and Methods of Nuclear Power Installations CalculationEnergoatomizdat Moscow Russia 1989

[30] W Stacey Nuclear Reactors Physics John Wiley amp Sons 2ndedition 2007

[31] E Lewis Fundamentals of Nuclear Reactors Physics AcademicPress New York NY USA 2008

[32] W Seifritz ldquoWhat is sustainable development An attempt tointerpret it as a soliton-like phenomenonrdquo Chaos Solitons andFractals vol 7 no 12 pp 2007ndash2018 1996

[33] A LyapunovThe general problem of the stability of motion [Doc-toral dissertation] (Russian) University of Kharkov Englishtranslations (1) Stability of Motion Academic Press New YorkNY USA 1966

[34] The General Problem of the Stability of Motion A T Fullertranslation Taylor amp Francis London UK 1992 Includedis a biography by Smirnov and an extensive bibliography ofLyapunovrsquos work 1892

[35] N Chetaev Stability of Motion Gostekhizdat Moscow Russia1955

[36] A Letov Stability of Nonlinear Control Systems GostekhizdatMoscow Russia 1955 (Russian) English Translation by Prince-ton University Press 1961

[37] J-J E Slotine and W Li Applied Nonlinear Control PrenticeHall Englewood Cliffs NJ USA 1991

[38] S Porolo Swelling and microstructure of the cladding steels ldquoei-847rdquo ldquoep-172rdquo and ldquochs-68rdquo after fuel elements using in bn-600[PhD thesis] 2008

[39] V Zelensky I Nekludov and Y Chernayaeva Radiation Defectand Swelling of Metals Naukova Dumka Kiev Ukraine 1998

[40] I Akhiezer and L Davydov Introduction to Theoretical Radi-ation Physics of Metals and Alloys Naukova dumka KievUkraine 1985

[41] A V Kozlov ldquoThe effect of neutron irradiation onmetals underdifferent temperatures and the opportunity of self-organizationof processes occurring in themrdquo Physics of Elementary Particlesand Atomic Nuclei vol 37 pp 1109ndash1150 2006

[42] G S Was Fundamentals of Radiation Materials Science Metalsand Alloys Springer Berlin Germany 2007

[43] M Pukari and J Wallenius Cladding Materials and RadiationDamage Department of Reactor Physics of Kungliga TekniskaHogskolan 2010

[44] M J Norgett M T Robinson and I M Torrens ldquoA proposedmethod of calculating displacement dose ratesrdquo Nuclear Engi-neering and Design vol 33 no 1 pp 50ndash54 1975

[45] M T Robinson ldquoBasic physics of radiation damage produc-tionrdquo Journal of Nuclear Materials vol 216 pp 1ndash28 1994

[46] J Lindhard M Scharff and H Schioslashtt ldquoRange concepts andheavy ion ranges (notes on atomic collisions II)rdquoMatematiske-Fysiske Meddelelser Udgivet af Det Kongelige Danske Vidensk-abernes Selskab vol 33 no 14 pp 1ndash42 1963

[47] International Atomic Energy Agency ldquo51st IAEA GeneralConference development of radiation resistant reactor corestructural materialsrdquo Tech Rep NTR2007 IAEA ViennaAustria 2007

[48] J A Mascitti and M Madariaga ldquoMethod for the calculationof DPA in the reactor pressure vessel of Atucha IIrdquo Science andTechnology of Nuclear Installations vol 2011 Article ID 5346896 pages 2011

[49] S Fomin YMelnik V Pilipenko andN Shulga ldquoSelf-sustainedregime of nuclear burning wave in u-pu fast neutron reactorwith pb-ni coolantrdquo Problems in Atomic Science and Technologyvol 3 pp 156ndash163 2007

[50] S Fomin YMelrsquonik V Pilipenko andN Shulrsquoga ldquoInvestigationof self-organization of the non-linear nuclear burning regime infast neutron reactorsrdquo Annals of Nuclear Energy vol 32 no 13pp 1435ndash1456 2005

[51] Y Melnik V Pilipenko A Fomin S Fomin and N ShulgaldquoStudy of a self-regulated nuclear burn wave regime in a fastreactor based on a thorium-uranium cyclerdquoAtomic Energy vol107 pp 49ndash56 2009

[52] S Shirokov Nuclear Reactor Physics Naukova dumka KievRussia 1992 (Russian)

[53] V Ukraintsev Reactivity Effects in Energetic InstallationsHandbook Obninsk Institute for Nuclear Power EngineeringObninsk Russia 2000 (Russian)

[54] T Akhromeeva S Kurdyumov G Malinetskii and ASamarskii Non-Stationary Structures and Diffusive ChaosNauka Moscow Russia 1992 (Russian)

[55] A A Samarskii V A Galaktionov S P Kurdyumov and A PMikhailov Blow-Up in Quasilinear Parabolic Equations Walterde Gruyter Berlin Germany 1995

[56] S P KurdyumovBlow-upModes Evolution of the IdeaTheLawsof Co-Evolution of Complex Systems Nauka Moscow Russia1999


[57] S Kurdyumov Blow-Up Modes Fizmatlit Moscow Russia2006

[58] E Knyazeva and S Kurdyumov Synergetics Nonlinearity ofTime and Landscape of Co Evolution KomKniga MoscowRussia 2007

[59] V Rusov V Tarasov and S Chernegenko ldquoBlow-up modes inuranium-plutonium fissile medium in technical nuclear reac-tors and georeactorrdquoProblems of Atomic Science andTechnologyvol 97 pp 123ndash131 2011 (Russian)

[60] D Skorov Y Bychkov and A Dashkovskii Reactor MaterialScience Atomizdat Moscow Russia 1979 (Russian)

[61] B Nadykto Ed Plutonium Fundamental problems RFNC-AREPRI Sarov Russia 2003 (Russian)

[62] Los Alamos National Laboratory ENSDFB-VI 1998[63] N Fedorov Short Reference Book for Engineer-Physicist Nuclear

Physics and Atomic Physics State Publishing Company forAtomic Science and Technology Literature Moscow Russia1961 (Russian)

[64] V Vladimirov Practical Problems on Nuclear Reactors Opera-tion Energoatomizdat Moscow Russia 1986 (Russian)

[65] A Samarskii and A GulinNumerical Methods inMathematicalPhysics Nauchnyi mir Moscow Russia 2003 (Russian)

[66] G H Kinchin and R S Pease ldquoThe displacement of atoms insolids by radiationrdquo Reports on Progress in Physics vol 18 no 1article 301 pp 590ndash615 1955

[67] B V Ershler and F S Lapteva ldquoThe evaporation of metals byfission fragmentsrdquo Journal of Nuclear Energy (1954) vol 4 no4 pp 471ndash474 1957

[68] IM LifshitsM I Kaganov and L V Tanatarov ldquoOn the theoryof the changes produced in metals by radiationrdquo The SovietJournal of Atomic Energy vol 6 no 4 pp 261ndash270 1960

[69] V Rusov V Tarasov M Eingorn S Chernezhenko andA Kakaev ldquoUltraslow wave nuclear burning of uranium-plutonium fissile medium on epithermal neutronsrdquo In prepa-ration httpxxxtauacilabs14097343

[70] V D Rusov V N Pavlovich V N Vaschenko et al ldquoGeoan-tineutrino spectrum and slow nuclear burning on the boundaryof the liquid and solid phases of the Earthrsquos corerdquo Journal ofGeophysical Research B Solid Earth vol 112 no 9 Article IDB09203 2007

[71] V F Anisichkin A P Ershov A Bezborodov et al ldquoThepossible modes of chain nuclear reactions in the Earthrsquos corerdquoin VII Zababa Khinrsquos Sientific Lectures 2003

[72] V Anisichkin A Bezborodov and I Suslov ldquoChain fissionreactions of nuclides in the earths core during billions yearsrdquoAtomic Energy vol 98 pp 370ndash379 2005

[73] V F Anisichkin ldquoDo the planets exploderdquo Burning andExplosion Physics vol 33 no 1 pp 138ndash142 1997 (Russian)

[74] V Rusov V Tarasov and D Litvinov Reactor AntineutrinoPhysics URSS Moscow Russia 2008 (Russian)

[75] V Zharkov The Inner Structure of Earth and Planets NaukaMoscow Russia 1983

[76] A G Lanin and I I Fedik ldquoSelecting and using materials fora nuclear rocket engine reactorrdquo Physics-Uspekhi vol 54 no 3pp 305ndash318 2011

[77] V D Rusov T N Zelentsova V A Tarasov and D A LitvinovldquoInverse problem of remote neutrino diagnostics of intrareactorprocessesrdquo Journal of Applied Physics vol 96 no 3 pp 1734ndash1739 2004

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High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


or neutrons respectively is born and destroyed at the sametime during the nuclear cycle) 119899crit [3] These conditions arevery important though they are almost always practicallyimplementable in case when the nuclear transformationschain of Feoktistovrsquos uranium-plutonium cycle type [3] issignificant among other reactions in the reactor

238U (119899 120574) 997888rarr239U

120573minus

997888rarr239

119873119901

120573minus

997888rarr239Pu (119899 fission) (1)

The same is also true for the Teller-Ishikawa-Woodthorium-uranium cycle type [4]

232Th (119899 120574) 997888rarr233Th

120573minus

997888rarr233Pa

120573minus

997888rarr233U (119899 fission) (2)

In these cases the fissionable isotopes form (239Pu in (1) or233U in (2)) which are the active components of the nuclearfuel The characteristic time of such reaction depends onthe time of the corresponding 120573-decays and is approximatelyequal to 120591

120573= 23 ln 2 asymp 33 days in case (1) and 120591

120573asymp 395

days in case (2) which is many orders of magnitude higherthan the corresponding time for the delayed neutrons

The effect of the nuclear burning process self-regulation isprovided by the fact that the system being left by itself cannotsurpass the critical state and enter the uncontrolled reactorrunaway mode because the critical concentration is limitedfrom above by a finite value of the active fuel componentequilibrium concentration (plutonium in (1) or uranium in(2)) 119899fis gt 119899crit (the Feoktistovrsquos wave existence condition [3])

Phenomenologically the process of the nuclear burningself-regulation is as follows Any increase in neutron flowleads to a quick burnout of the active fuel component(plutonium in (1) or uranium in (2)) that is to a reductionof their concentration and neutron flow meanwhile theformation of the new nuclei by the corresponding active fuelcomponent proceeds with the prior rate during the time 120591

120573

On the other hand if the neutron flow drops due to someexternal impact the burnout speed reduces and the activecomponent nuclei generation rate increases followed by theincrease of a number of neutrons generated in the reactorduring approximately the same time 120591

120573

The system of kinetic equations for nuclei (the compo-nents of nuclear fuel) and neutrons (in diffuse approxima-tion) in such chains is rather simple They differ only by thedepth of description of all the possible active fuel componentsand nonburnable poison (here by poisonwemean the oxygennuclei or other elements chemically bound to heavy nuclidesconstruction materials coolant and the poison itself thatis the nuclei added to the initial reactor composition inorder to control the neutron balance) Figure 1 shows thecharacteristic solutions for such problem (equations (3)ndash(9)in [5]) in a form of the soliton-like waves of the nuclearfuel components and neutrons concentrations for uranium-plutonium cycle in a cylindrical geometry case Within thetheory of the soliton-like fast reactors it is easy to show thatin general case the phase speed 119906 of soliton-like neutron wave

of nuclear burning is defined by the following approximateequality [5]

119906120591120573

2119871

≃ (

8

3radic120587

)

6

exp (minus 64

9120587

1198862)

1198862=

1205872

4

sdot


(3)

where 119899fis and 119899crit are the equilibrium and critical concen-tration of the active (fissile) isotope 119871 is the mean neutrondiffusion length and 120591

120573is the delay time associated with the

birth of the active (fissile) isotope and equal to an effective120573-decay period of the intermediate nuclei in Feoktistovrsquosuranium-plutonium cycle (1) or in Teller-Ishikawa-Woodthorium-uranium cycle (2)

Let us note that expression (3) automatically incorporatesa condition of nuclear burning process self-regulation sincethe fact of a wave existence is obviously predetermined by theinequality 119899fis gt 119899crit In other words the expression (3) is anecessary physical condition of the soliton-like neutron waveexistence Let us note for comparison that the maximal valueof the nuclear burning wave phase velocity as follows from(3) is characterized for both uranium and thorium cyclesby the equal average diffusion length (119871 sim 5 cm) of the fastneutrons (1MeV) and is equal to 370 cmday for uranium-plutonium cycle (4) and 031 cmday for thorium-uraniumcycle (2)

Generalizing the results of a wide range of numericalexperiments [5ndash23] we can positively affirm that the princi-pal possibility of themain stationarywave parameters controlwas reliably established within the theory of a self-regulatingfast reactor in traveling wave mode or in other words thetraveling wave reactor (TWR) It is possible both to increasethe speed the thermal power and the final fluence and todecrease them Obviously according to (3) it is achieved byvarying the equilibrium and critical concentrations of theactive fuel component that is by the purposeful change ofthe initial nuclear fuel composition

The technological problems of TWR are actively dis-cussed in science nowadays The essence of these problemsusually comes to a principal impossibility of such projectrealization and is defined by the following insurmountableflaws

(i) high degree of nuclear fuel burnup (over 20 in aver-age) leading to the following adverse consequences

(a) high damaging dose of fast neutrons acting atat the constructional materials (sim500 dpa) (forcomparison the claimed parameters for theRussian FN-800 reactor are 93 dpa at the sametime it is known that one of the main tasks ofthe construction materials radioactive stabilityinvestigations conducted at the BochvarHi-techInstitute for nonorganic materials (Moscow) isto achieve 133 to 164 dpa by 2020)

(b) high gas release which requires an increased gascavity length on top of a long fuel rod as it is


4353

252

151

050

140120

10080

6040

200 0

200 400600

8001000

1200

n

r (cm)z (cm)

times109

4

5

35

45

3252

151

050

140120

10080

6040

200 0

200400

600800 1000

1200

r (cm)z (cm)

times1022

238U

25

2

15

1

05

0140

120100

8060

4020

0 0200

400600 800

10001200

r (cm)z (cm)

times1021

239U

14

12

10

8

6

4

2

0140

120100

8060

4020

0 0200

400600

8001000

1200

r (cm)z (cm)

times1020

239Pu





through the reactor





119898 phase 120572 and












119863

1205972120593

1205971199092+ (119896infin

minus 1 + 120574120593) Σ119886120593 =

1

V120597120593

120597119905

(4)


119886




120593 (119909 119905) = 120593 (119909 minus 119906119905) equiv 120593 (120585) (5)


1

V120597120593

120597119905

= minus

119906

V120597120593

120597119905

(6)


1198712 1205972120593

1205971205852+ [119896infin(120595) minus 1 + 120574120593] 120593 = 0 (7)



120595 (119909 119905) = int

119905

0

120593 (119909 1199051015840) 1198891199051015840 (8)





119896infin

= 119896max + (1198960minus 119896max) (

120595

120595maxminus 1)

2

(9)



1198712120593120585120585+ 120588max120593 + 120574

01205932minus 120575[

[

int

infin

120585120593119889120585

119906120595119898

minus 1]

]

2

120593 = 0 (10)


0equiv 120574


1198712 1198892120593

1198891205852+(120588max + 120574

0120593 minus 120575(

int

+infin

120585120593119889120585

119906120595119898

minus 1)

2

)120593 = 0

(11)


120594 (120585) = int

+infin

120585

120593119889120585 997904rArr 120593 (120585) = minus

119889120594 (120585)

119889120585

(12)




11987121198893120594 (120585)

1198891205853

+ (120588max minus 1205740

119889120594 (120585)

119889120585

minus 120575(

120594 (120585)

119906120595119898

minus 1)

2

)

sdot

119889120594 (120585)

119889120585

= 0

(13)


120588max minus 1205740

119889120594 (120585)

119889120585

minus 120575(

120594(120585)

119906120595119898

minus 1)

2

= 119891 (120594 (120585)) (14)


11987121198893120594 (120585)

1198891205853

+ 119891 (120594 (120585))

119889120594 (120585)

119889120585

= 0 (15)


11987121198892120594 (120585)

1198891205852

+ 1198651(120594 (120585)) = 119862 (16)



1198712

2

(

119889120594(120585)

119889120585

)

2

+ 1198652(120594 (120585)) minus 119862120594 (120585) = 119861 (17)





119889120585 = plusmn

119889120594

radic(21198712) (119861 minus 119865

2(120594 (120585)) + 119862120594 (120585))

(18)


119889120585 = 119889120594(

120588max1205740

minus

120575

1205740

(

120594(120585)

119906120595119898

minus 1)

2

minus

1

1205740

119891 (120594(120585)))

minus1

(19)


plusmn radic2

1198712(119861 minus 119865

2(120594 (120585)) + 119862120594 (120585))

=

120588max1205740

minus

120575

1205740

(

120594(120585)

119906120595119898

minus 1)

2

minus

1

1205740

119891 (120594 (120585))

(20)




119891 (120594) = 11990421205942+ 1199041120594 + 1199040 (21)

where 1199040 1199041 and 119904



1198652(120594) =

1199042

12

1205944+

1199041

6

1205943+

1199040

2

1205942+ 1198881120594 + 1198882 (22)



(

120588max1205740

minus

120575

1205740

(

120594

119906120595119898

minus 1)

2

minus

1

1205740

(11990421205942+ 1199041120594 + 1199040))

2

=

2

1198712(119861 minus (

1199042

12

1205944+

1199041

6

1205943+

1199040

2

1205942+ 1198881120594 + 1198882) + 119862120594)

(23)


(

120575

120574011990621205952

119898

+

1199042

1205740

)

2

= minus

1199042

61198712 (24)

minus2(

120575

120574011990621205952

119898

+

1199042

1205740

)(

2120575

1205740119906120595119898

minus

1199041

1205740

) = minus

1199041

31198712 (25)

(

2120575

1205740119906120595119898

minus

1199041

1205740

)

2

minus 2(

120575

120574011990621205952

119898

+

1199042

1205740

)

sdot (

120588max1205740

minus

120575

1205740

minus

1199040

1205740

) = minus

1199040

1198712

(26)

2 (

2120575

1205740119906120595119898

minus

1199041

1205740

)(

120588max1205740

minus

120575

1205740

minus

1199040

1205740

)

= minus(

21198881

1198712+

2119862

1198712)

(27)

(

120588max1205740

minus

120575

1205740

minus

1199040

1205740

)

2

=

2

1198712119861 minus

21198882

1198712 (28)


0 1199041 and 119904



1198882as follows

1199040= 120588max minus 120575

1199041=

2120575

119906120595119898

minus

1205740

119871


1199042=

1205751205740

3119871119906120595119898radic120575 minus 120588max

minus

1205742

0

61198712minus

120575

11990621205952

119898

(29)





119889120585 = 119889120594(minus(

120575

120574011990621205952

119898

+

1199042

1205740

)1205942+ (

2120575

1205740119906120595119898

+

1199041

1205740

)120594

+ (

120588max1205740

minus

120575

1205740

minus

1199040

1205740

))

minus1

(30)


119889120594

(120594 minus 119870)2

minus1198722

= minus119873119889120585 (31)

where

119870 =

1205751205740119906120595119898minus 119904121205740

120575120574011990621205952

119898+ 11990421205740

1198722= (

1205751205740119906120595119898minus 119904121205740

120575120574011990621205952

119898+ 11990421205740

) +

(120588max1205740 minus 1205751205740minus 11990401205740)

(120575120574011990621205952

119898+ 11990421205740)

119873 =

120575

120574011990621205952

119898

+

1199042

1205740

(32)


the following

120594 minus 119870 = 1198721205941 119889120594 = 119872119889120594

1 (33)


1198891205941

(1205941)2

minus 1

= minus119872119873119889120585 (34)

Hence

1205941= minus tanh (119872119873120585 minus 119863) (35)


1

120594 = 119870 minus119872 tanh (119872119873120585 minus 119863) (36)


120593 (120585) = 1198722119873sech2 (119872119873120585 minus 119863) (37)


lim120585rarrinfin

120594 (120585) = 0 (38)


lim120585rarrinfin

120594 (120585) = lim120585rarrinfin

[119870 minus119872 tanh (119872119873120585 minus 119863)] = 0 (39)

leads to

119870 = 119872 (40)



120572 = 119872119873 =

radic120575 minus 120588max2119871

(41)

120593119898= 1198722119873 =

120575 minus 3120588max21205740

=

3120588max minus 120575

210038161003816100381610038161205740

1003816100381610038161003816

(42)

119906 =

120593119898

120572120595119898

(43)



0gt 0 (44)







0gt 0

(45)




115

111

107

103

099

09500 05 10 15 20

kinfin





120593 = 120593119898sech2 (120572120585) = 120593

119898sech2 [120572 (119909 minus 119906119905)] (46)



Δ12


[cm] (47)


119860area =2120593119898

120572

[cm2] (48)


119878 = minus119896119861int

infin

minusinfin

119901 (119909) ln119901 (119909) 119889119909 (49)




119878 = minus2119896119861int

infin

0

sech2 (120572120585) ln [sech2 (120572120585)] 119889( 119909

119906120591120573

)

= 4119896119861int

infin

0

ln cosh (120573119910)cosh2 (120573119910)

119889119910

(50)





119878 =

4 (1 minus ln 2)120573

= 4 (1 minus ln 2)119896119861

120572119906120591120573

=

4119896119861(1 minus ln 2)


2119871

119906120591120573

(51)


119878 sim 119896119861

2119871

119906120591120573

= const (52)




1=

1120572 and 1198972










≫ 1198972 then



119906 prop (1198972

11198972







120595119898sim

Δ12

119906

120593119898= 120591120593sdot 120593119898 (53)


005

1 0

02040608

0

1

005

01

01502120601

minus1

minus05

r(m

)

z (m)05m

(a)

02

46

810 0

05115

200025005007501

120601

r(m

)

z (m)

22m

(b)


0= 102 119896max = 106 120601



0= 099955 120601

119898= 3 sdot 10


jex

r

Z













119886 =

120587

2

radic


(54)



alowast

25

2

15

1

05

0

0 05 1 15 2 25 3




lowast) = 119906120591

1205732119871 on the parameter 119886 [5]











119899dpa = 119905 sdot

119873

sum

119894

⟨120590119894

dpa⟩int119864119894

119864119894

minus1

Φ(119864119894) 119889119864119894=

119873

sum

119894=1

⟨120590119894

dpa⟩120593119894119905 (55)

where

⟨120590119894

dpa⟩ (119864119894) = ⟨120590119889(119864119894 119885 119860)⟩ sdot 119889 (119864

119889(119885 119860)) (56)


)

10

9

8

7

6

5

4

3

2

1

060 70 80 90 100 110 120 130 140 150 160 170 180 190 200

Dose (dpa)

Average316Ti


Embrittlement limit









119903is not


119864dam = 119864119903120585 (57)


119889 (119864119889(119885 119860)) = 120578

119864dam2 ⟨119864119889⟩

(58)




119899dpa ≃ ⟨120590dpa⟩ sdot 120593 sdot

2Δ12

119906

⩽ 200 [dpa] (59)


⟨120590dpa⟩ sdot 120593 sdot

Δ12

119906

⩽ 100 [dpa] (60)


1000

750

500

250

01 1 10 100 1000


Ope

ratin

g te

mpe

ratu

re (∘

C)

HTR materialsFusion


materials

Thermal

materials


ODS1000

950

900

850 100 150 200 250

reactor



103

102

10

1

10minus1

10minus2



120590D(E)

(bar

n)







Δ12











Be 9 021 185 11 139 153C 12 0158 160 15 356 534H2O 18 0924 10 25 167 416

H2O + B 25 100 250

He 4 0425 018 541 112 607



1205951000

=

Δ12

119906

120593 ≃ 1023

[cmminus2] (61)








119877 mod ≃

1

Σ119878+ Σ119886

sdot

1

120585


(62)

where Σ119878

asymp ⟨120590119878⟩119873mod and Σ

119886asymp ⟨120590

119886⟩119873mod are the


119878⟩ and ⟨120590



2 ln[(119860 minus 1)119860 + 1]2119860





120595100

=

Δ12

119906

120593 ≃ 1024

[cmminus2] (63)

where 120595100













120588 ( 119903 119879 119905) sdot 119888 ( 119903 119879 119905) sdot

120597119879 ( 119903 119905)

120597119905


+ 119902119891

119879( 119903 119879 119905)

(64)


120588 ( 119903 119879 119905) = sum

119894

119873119894( 119903 119879 119905) sdot 120588

119894 (65)







119888 ( 119903 119879 119905) = sum

119894

119888119894(119879)119873

119894( 119903 119879 119905) (66)

alefsym ( 119903 119879 119905) = sum

119894

alefsym119894(119879)119873

119894( 119903 119879 119905) (67)



time









0] (68)



0is the



119879( 119903 Φ 119879 119905) that character-


119902119891

119879( 119903 Φ 119879 119905)

= Φ ( 119903 119879 119905)sum

119894

119876119891

119894120590119894

119891( 119903 119879 119905)119873

119894( 119903 119879 119905) [Wcm3]

(69)

where

Φ ( 119903 119879 119905) = int

119864max119899

0

Φ ( 119903 119864 119879 119905) 119889119864 (70)




120590119894

119891( 119903 119879 119905) = int

119864max119899

0

120590119894

119891(119864 119879) 120588 ( 119903 119864 119879 119905) 119889119864 (71)


120588 ( 119903 119864 119879 119905) =

Φ ( 119903 119864 119879 119905)

Φ ( 119903 119879 119905)

(72)




35

3

25

2

15

1

05

0

0 1 2 3 4 5 6

0

E (eV)

3000

2000

1000 T(K

)

(times100

barn

)



300K to 3000K


119891( 119903 119879 119905) averaged over



92U 23892U 23992U and 239





⟨120590 (119864lim 119879)⟩ =

int

119864lim

011986412

119890minus119864119896119879

120590 (119864 119879) 119889119864

int

119864lim

011986412

119890minus119864119896119879

119889119864

(73)



⟨120590 (119864lim 119879)⟩ =

int

infin

119864lim120590 (119864 119879) 119864

minus1119889119864

int

infin

119864lim119864minus1119889119864

(74)


119872(119864119899)




max119899


119899119896119879 where 119864



119879119899= 1198790[1 + 120578 sdot

Σ119886(1198961198790)

⟨120585⟩ Σ119878

] (75)


119886(1198961198790) is




ΦFermi (119864 119864119865) =119878

⟨120585⟩ Σ119905119864

exp[minusint

119864119891

119864lim

Σ119886(1198641015840) 1198891198641015840

⟨120585⟩ Σ119905(1198641015840) 1198641015840]

(76)


sum119894(120585119894Σ119894




119878+ Σ119894



119878is the total


119865is the







119878 ( 119903 119879 119905)

= int

119864max119899

119864119865

( 119903 119864 119879 119905)

sdot [sum

119894

]119894(119864) sdot Φ ( 119903 119864 119879 119905) sdot 120590

119894

119891(119864 119879) sdot 119873

119894( 119903 119879 119905)] 119889119864

(78)

where 119864max119899


119899asymp 10MeV)119864



119865asymp





119899for


119899119887sim 1radic119864

119899



2000

1500

1000

500

00 1000 2000 3000 4000 5000 6000

T (K)

120590f

(bar

n)

3kT35kT4kT

5kT6kT

(a)

2000

1500

1000

500

00 1000 2000 3000 4000 5000 6000

T (K)

3kT35kT4kT

5kT6kT

120590c

(bar

n)(b)


200

180

160

140

120

100

80

60

400 1000 2000 3000 4000 5000 6000

T (K)

120590f

(bar

n)

3kT35 kT4kT

5kT6kT

(a)

140

120

100

80

60

0

20

40

0 1000 2000 3000 4000 5000 6000

T (K)

3kT35kT4kT

5kT6kT

120590c

(bar

n)

(b)




119879( 119903 Φ 119879 119905) (69) on temperature



036

034

032

03

028

026

0240 1000 2000 3000 4000 5000 6000

T (K)

120590f

(bar

n)

3kT35 kT4kT

5kT6kT

(a)

800

700

600

500

400

300

200

120590c

(bar

n)

0 1000 2000 3000 4000 5000 6000

T (K)

3kT35kT4kT

5kT6kT

(b)



25

20

15

10

05

0600 800 1000 1200 1400

T (K)

qf T(rarrrΦTN

it)

(times10

20eV

)

1

2

3


119879(119903 Φ 119879119873

119894


1013 n(cm2sdots)






119899(119909 119905)






119891

119879( 119903 Φ 119879 119905) (69) temperature







119902119879(119879) = const sdot 119879(1+120575) (79)







2+ 00000126119879

3


minus101198795


1198796

(80)

25

20

15

10

5

00 200 400 600 800 1000 1200 1400

T (K)

c p(J

molmiddotK

)

alefsym(W

mmiddotK

)

40

375

35

325

30

275

25

225



alefsym (119879) asymp 21575 + 00152661119879 (81)








T (K)x (120583m)

t (s)

0

1

2

3 0

025

05

075

10

0

1025

05

075

103

(d)

0

1

2

3 0

025

05

075

10

1025

05

075

(b)

0

1

2

3 0

025

05

075

10

0

1

05

075

(c)

0

1

2

3 0

025

05

075

10

0

1025

05

075

(a)

4 middot 104

2 middot 103

6 middot 104








119879119899asymp [1 + 18

80 sdot 1198702

⟨120585⟩ sdot 45

] (82)




= 45


1198702=

119873238

11987312

= 05 (83)






sdot 119879 (84)







0

50

100

150

200

250

300

350

400

600

700

800

900

1000

1100

1200

1300

1400

0 1000 2000 3000 4000 5000 6000

0 1000 2000 3000 4000 5000 6000

T (K)

239Pu

fiss

ion

cros

s sec

tion

(bar

n)

235U

fiss

ion

cros

s sec

tion

(bar

n)

239Pu

235U








7 Conclusions



(d)

250

500

750

1000

0250

500750

6000

250

500

750

250500

750

(b)

0

250

500

750

1000

0

250500

7501000

01000 01000

250

500

750

0

250500

750

6000

(c)

0250

500

7500

250

500

750

6000

250

500

750

1000

(a)

0

250

500

750

1000

60000

250

500

750

10

0

y (cm)

x (cm)

t (s)t (s)

t (s) t (s)

T (K)

x(cm

)

x(cm

)

x(cm

)

x(cm

)


3 middot 1011

2 middot 1011

4 middot 1012

2 middot 1012

4 middot 1012

2 middot 1012


1 middot 1010

2 middot 10minus7

4 middot 10minus7

6 middot 10minus7

8 middot 10minus7

1 middot 10minus6














References






















































































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of


















4353

252

151

050

140120

10080

6040

200 0

200 400600

8001000

1200

n

r (cm)z (cm)

times109

4

5

35

45

3252

151

050

140120

10080

6040

200 0

200400

600800 1000

1200

r (cm)z (cm)

times1022

238U

25

2

15

1

05

0140

120100

8060

4020

0 0200

400600 800

10001200

r (cm)z (cm)

times1021

239U

14

12

10

8

6

4

2

0140

120100

8060

4020

0 0200

400600

8001000

1200

r (cm)z (cm)

times1020

239Pu





through the reactor





119898 phase 120572 and












119863

1205972120593

1205971199092+ (119896infin

minus 1 + 120574120593) Σ119886120593 =

1

V120597120593

120597119905

(4)


119886




120593 (119909 119905) = 120593 (119909 minus 119906119905) equiv 120593 (120585) (5)


1

V120597120593

120597119905

= minus

119906

V120597120593

120597119905

(6)


1198712 1205972120593

1205971205852+ [119896infin(120595) minus 1 + 120574120593] 120593 = 0 (7)



120595 (119909 119905) = int

119905

0

120593 (119909 1199051015840) 1198891199051015840 (8)





119896infin

= 119896max + (1198960minus 119896max) (

120595

120595maxminus 1)

2

(9)



1198712120593120585120585+ 120588max120593 + 120574

01205932minus 120575[

[

int

infin

120585120593119889120585

119906120595119898

minus 1]

]

2

120593 = 0 (10)


0equiv 120574


1198712 1198892120593

1198891205852+(120588max + 120574

0120593 minus 120575(

int

+infin

120585120593119889120585

119906120595119898

minus 1)

2

)120593 = 0

(11)


120594 (120585) = int

+infin

120585

120593119889120585 997904rArr 120593 (120585) = minus

119889120594 (120585)

119889120585

(12)




11987121198893120594 (120585)

1198891205853

+ (120588max minus 1205740

119889120594 (120585)

119889120585

minus 120575(

120594 (120585)

119906120595119898

minus 1)

2

)

sdot

119889120594 (120585)

119889120585

= 0

(13)


120588max minus 1205740

119889120594 (120585)

119889120585

minus 120575(

120594(120585)

119906120595119898

minus 1)

2

= 119891 (120594 (120585)) (14)


11987121198893120594 (120585)

1198891205853

+ 119891 (120594 (120585))

119889120594 (120585)

119889120585

= 0 (15)


11987121198892120594 (120585)

1198891205852

+ 1198651(120594 (120585)) = 119862 (16)



1198712

2

(

119889120594(120585)

119889120585

)

2

+ 1198652(120594 (120585)) minus 119862120594 (120585) = 119861 (17)





119889120585 = plusmn

119889120594

radic(21198712) (119861 minus 119865

2(120594 (120585)) + 119862120594 (120585))

(18)


119889120585 = 119889120594(

120588max1205740

minus

120575

1205740

(

120594(120585)

119906120595119898

minus 1)

2

minus

1

1205740

119891 (120594(120585)))

minus1

(19)


plusmn radic2

1198712(119861 minus 119865

2(120594 (120585)) + 119862120594 (120585))

=

120588max1205740

minus

120575

1205740

(

120594(120585)

119906120595119898

minus 1)

2

minus

1

1205740

119891 (120594 (120585))

(20)




119891 (120594) = 11990421205942+ 1199041120594 + 1199040 (21)

where 1199040 1199041 and 119904



1198652(120594) =

1199042

12

1205944+

1199041

6

1205943+

1199040

2

1205942+ 1198881120594 + 1198882 (22)



(

120588max1205740

minus

120575

1205740

(

120594

119906120595119898

minus 1)

2

minus

1

1205740

(11990421205942+ 1199041120594 + 1199040))

2

=

2

1198712(119861 minus (

1199042

12

1205944+

1199041

6

1205943+

1199040

2

1205942+ 1198881120594 + 1198882) + 119862120594)

(23)


(

120575

120574011990621205952

119898

+

1199042

1205740

)

2

= minus

1199042

61198712 (24)

minus2(

120575

120574011990621205952

119898

+

1199042

1205740

)(

2120575

1205740119906120595119898

minus

1199041

1205740

) = minus

1199041

31198712 (25)

(

2120575

1205740119906120595119898

minus

1199041

1205740

)

2

minus 2(

120575

120574011990621205952

119898

+

1199042

1205740

)

sdot (

120588max1205740

minus

120575

1205740

minus

1199040

1205740

) = minus

1199040

1198712

(26)

2 (

2120575

1205740119906120595119898

minus

1199041

1205740

)(

120588max1205740

minus

120575

1205740

minus

1199040

1205740

)

= minus(

21198881

1198712+

2119862

1198712)

(27)

(

120588max1205740

minus

120575

1205740

minus

1199040

1205740

)

2

=

2

1198712119861 minus

21198882

1198712 (28)


0 1199041 and 119904



1198882as follows

1199040= 120588max minus 120575

1199041=

2120575

119906120595119898

minus

1205740

119871


1199042=

1205751205740

3119871119906120595119898radic120575 minus 120588max

minus

1205742

0

61198712minus

120575

11990621205952

119898

(29)





119889120585 = 119889120594(minus(

120575

120574011990621205952

119898

+

1199042

1205740

)1205942+ (

2120575

1205740119906120595119898

+

1199041

1205740

)120594

+ (

120588max1205740

minus

120575

1205740

minus

1199040

1205740

))

minus1

(30)


119889120594

(120594 minus 119870)2

minus1198722

= minus119873119889120585 (31)

where

119870 =

1205751205740119906120595119898minus 119904121205740

120575120574011990621205952

119898+ 11990421205740

1198722= (

1205751205740119906120595119898minus 119904121205740

120575120574011990621205952

119898+ 11990421205740

) +

(120588max1205740 minus 1205751205740minus 11990401205740)

(120575120574011990621205952

119898+ 11990421205740)

119873 =

120575

120574011990621205952

119898

+

1199042

1205740

(32)


the following

120594 minus 119870 = 1198721205941 119889120594 = 119872119889120594

1 (33)


1198891205941

(1205941)2

minus 1

= minus119872119873119889120585 (34)

Hence

1205941= minus tanh (119872119873120585 minus 119863) (35)


1

120594 = 119870 minus119872 tanh (119872119873120585 minus 119863) (36)


120593 (120585) = 1198722119873sech2 (119872119873120585 minus 119863) (37)


lim120585rarrinfin

120594 (120585) = 0 (38)


lim120585rarrinfin

120594 (120585) = lim120585rarrinfin

[119870 minus119872 tanh (119872119873120585 minus 119863)] = 0 (39)

leads to

119870 = 119872 (40)



120572 = 119872119873 =

radic120575 minus 120588max2119871

(41)

120593119898= 1198722119873 =

120575 minus 3120588max21205740

=

3120588max minus 120575

210038161003816100381610038161205740

1003816100381610038161003816

(42)

119906 =

120593119898

120572120595119898

(43)



0gt 0 (44)







0gt 0

(45)




115

111

107

103

099

09500 05 10 15 20

kinfin





120593 = 120593119898sech2 (120572120585) = 120593

119898sech2 [120572 (119909 minus 119906119905)] (46)



Δ12


[cm] (47)


119860area =2120593119898

120572

[cm2] (48)


119878 = minus119896119861int

infin

minusinfin

119901 (119909) ln119901 (119909) 119889119909 (49)




119878 = minus2119896119861int

infin

0

sech2 (120572120585) ln [sech2 (120572120585)] 119889( 119909

119906120591120573

)

= 4119896119861int

infin

0

ln cosh (120573119910)cosh2 (120573119910)

119889119910

(50)





119878 =

4 (1 minus ln 2)120573

= 4 (1 minus ln 2)119896119861

120572119906120591120573

=

4119896119861(1 minus ln 2)


2119871

119906120591120573

(51)


119878 sim 119896119861

2119871

119906120591120573

= const (52)




1=

1120572 and 1198972










≫ 1198972 then



119906 prop (1198972

11198972







120595119898sim

Δ12

119906

120593119898= 120591120593sdot 120593119898 (53)


005

1 0

02040608

0

1

005

01

01502120601

minus1

minus05

r(m

)

z (m)05m

(a)

02

46

810 0

05115

200025005007501

120601

r(m

)

z (m)

22m

(b)


0= 102 119896max = 106 120601



0= 099955 120601

119898= 3 sdot 10


jex

r

Z













119886 =

120587

2

radic


(54)



alowast

25

2

15

1

05

0

0 05 1 15 2 25 3




lowast) = 119906120591

1205732119871 on the parameter 119886 [5]











119899dpa = 119905 sdot

119873

sum

119894

⟨120590119894

dpa⟩int119864119894

119864119894

minus1

Φ(119864119894) 119889119864119894=

119873

sum

119894=1

⟨120590119894

dpa⟩120593119894119905 (55)

where

⟨120590119894

dpa⟩ (119864119894) = ⟨120590119889(119864119894 119885 119860)⟩ sdot 119889 (119864

119889(119885 119860)) (56)


)

10

9

8

7

6

5

4

3

2

1

060 70 80 90 100 110 120 130 140 150 160 170 180 190 200

Dose (dpa)

Average316Ti


Embrittlement limit









119903is not


119864dam = 119864119903120585 (57)


119889 (119864119889(119885 119860)) = 120578

119864dam2 ⟨119864119889⟩

(58)




119899dpa ≃ ⟨120590dpa⟩ sdot 120593 sdot

2Δ12

119906

⩽ 200 [dpa] (59)


⟨120590dpa⟩ sdot 120593 sdot

Δ12

119906

⩽ 100 [dpa] (60)


1000

750

500

250

01 1 10 100 1000


Ope

ratin

g te

mpe

ratu

re (∘

C)

HTR materialsFusion


materials

Thermal

materials


ODS1000

950

900

850 100 150 200 250

reactor



103

102

10

1

10minus1

10minus2



120590D(E)

(bar

n)







Δ12











Be 9 021 185 11 139 153C 12 0158 160 15 356 534H2O 18 0924 10 25 167 416

H2O + B 25 100 250

He 4 0425 018 541 112 607



1205951000

=

Δ12

119906

120593 ≃ 1023

[cmminus2] (61)








119877 mod ≃

1

Σ119878+ Σ119886

sdot

1

120585


(62)

where Σ119878

asymp ⟨120590119878⟩119873mod and Σ

119886asymp ⟨120590

119886⟩119873mod are the


119878⟩ and ⟨120590



2 ln[(119860 minus 1)119860 + 1]2119860





120595100

=

Δ12

119906

120593 ≃ 1024

[cmminus2] (63)

where 120595100













120588 ( 119903 119879 119905) sdot 119888 ( 119903 119879 119905) sdot

120597119879 ( 119903 119905)

120597119905


+ 119902119891

119879( 119903 119879 119905)

(64)


120588 ( 119903 119879 119905) = sum

119894

119873119894( 119903 119879 119905) sdot 120588

119894 (65)







119888 ( 119903 119879 119905) = sum

119894

119888119894(119879)119873

119894( 119903 119879 119905) (66)

alefsym ( 119903 119879 119905) = sum

119894

alefsym119894(119879)119873

119894( 119903 119879 119905) (67)



time









0] (68)



0is the



119879( 119903 Φ 119879 119905) that character-


119902119891

119879( 119903 Φ 119879 119905)

= Φ ( 119903 119879 119905)sum

119894

119876119891

119894120590119894

119891( 119903 119879 119905)119873

119894( 119903 119879 119905) [Wcm3]

(69)

where

Φ ( 119903 119879 119905) = int

119864max119899

0

Φ ( 119903 119864 119879 119905) 119889119864 (70)




120590119894

119891( 119903 119879 119905) = int

119864max119899

0

120590119894

119891(119864 119879) 120588 ( 119903 119864 119879 119905) 119889119864 (71)


120588 ( 119903 119864 119879 119905) =

Φ ( 119903 119864 119879 119905)

Φ ( 119903 119879 119905)

(72)




35

3

25

2

15

1

05

0

0 1 2 3 4 5 6

0

E (eV)

3000

2000

1000 T(K

)

(times100

barn

)



300K to 3000K


119891( 119903 119879 119905) averaged over



92U 23892U 23992U and 239





⟨120590 (119864lim 119879)⟩ =

int

119864lim

011986412

119890minus119864119896119879

120590 (119864 119879) 119889119864

int

119864lim

011986412

119890minus119864119896119879

119889119864

(73)



⟨120590 (119864lim 119879)⟩ =

int

infin

119864lim120590 (119864 119879) 119864

minus1119889119864

int

infin

119864lim119864minus1119889119864

(74)


119872(119864119899)




max119899


119899119896119879 where 119864



119879119899= 1198790[1 + 120578 sdot

Σ119886(1198961198790)

⟨120585⟩ Σ119878

] (75)


119886(1198961198790) is




ΦFermi (119864 119864119865) =119878

⟨120585⟩ Σ119905119864

exp[minusint

119864119891

119864lim

Σ119886(1198641015840) 1198891198641015840

⟨120585⟩ Σ119905(1198641015840) 1198641015840]

(76)


sum119894(120585119894Σ119894




119878+ Σ119894



119878is the total


119865is the







119878 ( 119903 119879 119905)

= int

119864max119899

119864119865

( 119903 119864 119879 119905)

sdot [sum

119894

]119894(119864) sdot Φ ( 119903 119864 119879 119905) sdot 120590

119894

119891(119864 119879) sdot 119873

119894( 119903 119879 119905)] 119889119864

(78)

where 119864max119899


119899asymp 10MeV)119864



119865asymp





119899for


119899119887sim 1radic119864

119899



2000

1500

1000

500

00 1000 2000 3000 4000 5000 6000

T (K)

120590f

(bar

n)

3kT35kT4kT

5kT6kT

(a)

2000

1500

1000

500

00 1000 2000 3000 4000 5000 6000

T (K)

3kT35kT4kT

5kT6kT

120590c

(bar

n)(b)


200

180

160

140

120

100

80

60

400 1000 2000 3000 4000 5000 6000

T (K)

120590f

(bar

n)

3kT35 kT4kT

5kT6kT

(a)

140

120

100

80

60

0

20

40

0 1000 2000 3000 4000 5000 6000

T (K)

3kT35kT4kT

5kT6kT

120590c

(bar

n)

(b)




119879( 119903 Φ 119879 119905) (69) on temperature



036

034

032

03

028

026

0240 1000 2000 3000 4000 5000 6000

T (K)

120590f

(bar

n)

3kT35 kT4kT

5kT6kT

(a)

800

700

600

500

400

300

200

120590c

(bar

n)

0 1000 2000 3000 4000 5000 6000

T (K)

3kT35kT4kT

5kT6kT

(b)



25

20

15

10

05

0600 800 1000 1200 1400

T (K)

qf T(rarrrΦTN

it)

(times10

20eV

)

1

2

3


119879(119903 Φ 119879119873

119894


1013 n(cm2sdots)






119899(119909 119905)






119891

119879( 119903 Φ 119879 119905) (69) temperature







119902119879(119879) = const sdot 119879(1+120575) (79)







2+ 00000126119879

3


minus101198795


1198796

(80)

25

20

15

10

5

00 200 400 600 800 1000 1200 1400

T (K)

c p(J

molmiddotK

)

alefsym(W

mmiddotK

)

40

375

35

325

30

275

25

225



alefsym (119879) asymp 21575 + 00152661119879 (81)








T (K)x (120583m)

t (s)

0

1

2

3 0

025

05

075

10

0

1025

05

075

103

(d)

0

1

2

3 0

025

05

075

10

1025

05

075

(b)

0

1

2

3 0

025

05

075

10

0

1

05

075

(c)

0

1

2

3 0

025

05

075

10

0

1025

05

075

(a)

4 middot 104

2 middot 103

6 middot 104








119879119899asymp [1 + 18

80 sdot 1198702

⟨120585⟩ sdot 45

] (82)




= 45


1198702=

119873238

11987312

= 05 (83)






sdot 119879 (84)







0

50

100

150

200

250

300

350

400

600

700

800

900

1000

1100

1200

1300

1400

0 1000 2000 3000 4000 5000 6000

0 1000 2000 3000 4000 5000 6000

T (K)

239Pu

fiss

ion

cros

s sec

tion

(bar

n)

235U

fiss

ion

cros

s sec

tion

(bar

n)

239Pu

235U








7 Conclusions



(d)

250

500

750

1000

0250

500750

6000

250

500

750

250500

750

(b)

0

250

500

750

1000

0

250500

7501000

01000 01000

250

500

750

0

250500

750

6000

(c)

0250

500

7500

250

500

750

6000

250

500

750

1000

(a)

0

250

500

750

1000

60000

250

500

750

10

0

y (cm)

x (cm)

t (s)t (s)

t (s) t (s)

T (K)

x(cm

)

x(cm

)

x(cm

)

x(cm

)


3 middot 1011

2 middot 1011

4 middot 1012

2 middot 1012

4 middot 1012

2 middot 1012


1 middot 1010

2 middot 10minus7

4 middot 10minus7

6 middot 10minus7

8 middot 10minus7

1 middot 10minus6














References






















































































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of



























119863

1205972120593

1205971199092+ (119896infin

minus 1 + 120574120593) Σ119886120593 =

1

V120597120593

120597119905

(4)


119886




120593 (119909 119905) = 120593 (119909 minus 119906119905) equiv 120593 (120585) (5)


1

V120597120593

120597119905

= minus

119906

V120597120593

120597119905

(6)


1198712 1205972120593

1205971205852+ [119896infin(120595) minus 1 + 120574120593] 120593 = 0 (7)



120595 (119909 119905) = int

119905

0

120593 (119909 1199051015840) 1198891199051015840 (8)





119896infin

= 119896max + (1198960minus 119896max) (

120595

120595maxminus 1)

2

(9)



1198712120593120585120585+ 120588max120593 + 120574

01205932minus 120575[

[

int

infin

120585120593119889120585

119906120595119898

minus 1]

]

2

120593 = 0 (10)


0equiv 120574


1198712 1198892120593

1198891205852+(120588max + 120574

0120593 minus 120575(

int

+infin

120585120593119889120585

119906120595119898

minus 1)

2

)120593 = 0

(11)


120594 (120585) = int

+infin

120585

120593119889120585 997904rArr 120593 (120585) = minus

119889120594 (120585)

119889120585

(12)




11987121198893120594 (120585)

1198891205853

+ (120588max minus 1205740

119889120594 (120585)

119889120585

minus 120575(

120594 (120585)

119906120595119898

minus 1)

2

)

sdot

119889120594 (120585)

119889120585

= 0

(13)


120588max minus 1205740

119889120594 (120585)

119889120585

minus 120575(

120594(120585)

119906120595119898

minus 1)

2

= 119891 (120594 (120585)) (14)


11987121198893120594 (120585)

1198891205853

+ 119891 (120594 (120585))

119889120594 (120585)

119889120585

= 0 (15)


11987121198892120594 (120585)

1198891205852

+ 1198651(120594 (120585)) = 119862 (16)



1198712

2

(

119889120594(120585)

119889120585

)

2

+ 1198652(120594 (120585)) minus 119862120594 (120585) = 119861 (17)





119889120585 = plusmn

119889120594

radic(21198712) (119861 minus 119865

2(120594 (120585)) + 119862120594 (120585))

(18)


119889120585 = 119889120594(

120588max1205740

minus

120575

1205740

(

120594(120585)

119906120595119898

minus 1)

2

minus

1

1205740

119891 (120594(120585)))

minus1

(19)


plusmn radic2

1198712(119861 minus 119865

2(120594 (120585)) + 119862120594 (120585))

=

120588max1205740

minus

120575

1205740

(

120594(120585)

119906120595119898

minus 1)

2

minus

1

1205740

119891 (120594 (120585))

(20)




119891 (120594) = 11990421205942+ 1199041120594 + 1199040 (21)

where 1199040 1199041 and 119904



1198652(120594) =

1199042

12

1205944+

1199041

6

1205943+

1199040

2

1205942+ 1198881120594 + 1198882 (22)



(

120588max1205740

minus

120575

1205740

(

120594

119906120595119898

minus 1)

2

minus

1

1205740

(11990421205942+ 1199041120594 + 1199040))

2

=

2

1198712(119861 minus (

1199042

12

1205944+

1199041

6

1205943+

1199040

2

1205942+ 1198881120594 + 1198882) + 119862120594)

(23)


(

120575

120574011990621205952

119898

+

1199042

1205740

)

2

= minus

1199042

61198712 (24)

minus2(

120575

120574011990621205952

119898

+

1199042

1205740

)(

2120575

1205740119906120595119898

minus

1199041

1205740

) = minus

1199041

31198712 (25)

(

2120575

1205740119906120595119898

minus

1199041

1205740

)

2

minus 2(

120575

120574011990621205952

119898

+

1199042

1205740

)

sdot (

120588max1205740

minus

120575

1205740

minus

1199040

1205740

) = minus

1199040

1198712

(26)

2 (

2120575

1205740119906120595119898

minus

1199041

1205740

)(

120588max1205740

minus

120575

1205740

minus

1199040

1205740

)

= minus(

21198881

1198712+

2119862

1198712)

(27)

(

120588max1205740

minus

120575

1205740

minus

1199040

1205740

)

2

=

2

1198712119861 minus

21198882

1198712 (28)


0 1199041 and 119904



1198882as follows

1199040= 120588max minus 120575

1199041=

2120575

119906120595119898

minus

1205740

119871


1199042=

1205751205740

3119871119906120595119898radic120575 minus 120588max

minus

1205742

0

61198712minus

120575

11990621205952

119898

(29)





119889120585 = 119889120594(minus(

120575

120574011990621205952

119898

+

1199042

1205740

)1205942+ (

2120575

1205740119906120595119898

+

1199041

1205740

)120594

+ (

120588max1205740

minus

120575

1205740

minus

1199040

1205740

))

minus1

(30)


119889120594

(120594 minus 119870)2

minus1198722

= minus119873119889120585 (31)

where

119870 =

1205751205740119906120595119898minus 119904121205740

120575120574011990621205952

119898+ 11990421205740

1198722= (

1205751205740119906120595119898minus 119904121205740

120575120574011990621205952

119898+ 11990421205740

) +

(120588max1205740 minus 1205751205740minus 11990401205740)

(120575120574011990621205952

119898+ 11990421205740)

119873 =

120575

120574011990621205952

119898

+

1199042

1205740

(32)


the following

120594 minus 119870 = 1198721205941 119889120594 = 119872119889120594

1 (33)


1198891205941

(1205941)2

minus 1

= minus119872119873119889120585 (34)

Hence

1205941= minus tanh (119872119873120585 minus 119863) (35)


1

120594 = 119870 minus119872 tanh (119872119873120585 minus 119863) (36)


120593 (120585) = 1198722119873sech2 (119872119873120585 minus 119863) (37)


lim120585rarrinfin

120594 (120585) = 0 (38)


lim120585rarrinfin

120594 (120585) = lim120585rarrinfin

[119870 minus119872 tanh (119872119873120585 minus 119863)] = 0 (39)

leads to

119870 = 119872 (40)



120572 = 119872119873 =

radic120575 minus 120588max2119871

(41)

120593119898= 1198722119873 =

120575 minus 3120588max21205740

=

3120588max minus 120575

210038161003816100381610038161205740

1003816100381610038161003816

(42)

119906 =

120593119898

120572120595119898

(43)



0gt 0 (44)







0gt 0

(45)




115

111

107

103

099

09500 05 10 15 20

kinfin





120593 = 120593119898sech2 (120572120585) = 120593

119898sech2 [120572 (119909 minus 119906119905)] (46)



Δ12


[cm] (47)


119860area =2120593119898

120572

[cm2] (48)


119878 = minus119896119861int

infin

minusinfin

119901 (119909) ln119901 (119909) 119889119909 (49)




119878 = minus2119896119861int

infin

0

sech2 (120572120585) ln [sech2 (120572120585)] 119889( 119909

119906120591120573

)

= 4119896119861int

infin

0

ln cosh (120573119910)cosh2 (120573119910)

119889119910

(50)





119878 =

4 (1 minus ln 2)120573

= 4 (1 minus ln 2)119896119861

120572119906120591120573

=

4119896119861(1 minus ln 2)


2119871

119906120591120573

(51)


119878 sim 119896119861

2119871

119906120591120573

= const (52)




1=

1120572 and 1198972










≫ 1198972 then



119906 prop (1198972

11198972







120595119898sim

Δ12

119906

120593119898= 120591120593sdot 120593119898 (53)


005

1 0

02040608

0

1

005

01

01502120601

minus1

minus05

r(m

)

z (m)05m

(a)

02

46

810 0

05115

200025005007501

120601

r(m

)

z (m)

22m

(b)


0= 102 119896max = 106 120601



0= 099955 120601

119898= 3 sdot 10


jex

r

Z













119886 =

120587

2

radic


(54)



alowast

25

2

15

1

05

0

0 05 1 15 2 25 3




lowast) = 119906120591

1205732119871 on the parameter 119886 [5]











119899dpa = 119905 sdot

119873

sum

119894

⟨120590119894

dpa⟩int119864119894

119864119894

minus1

Φ(119864119894) 119889119864119894=

119873

sum

119894=1

⟨120590119894

dpa⟩120593119894119905 (55)

where

⟨120590119894

dpa⟩ (119864119894) = ⟨120590119889(119864119894 119885 119860)⟩ sdot 119889 (119864

119889(119885 119860)) (56)


)

10

9

8

7

6

5

4

3

2

1

060 70 80 90 100 110 120 130 140 150 160 170 180 190 200

Dose (dpa)

Average316Ti


Embrittlement limit









119903is not


119864dam = 119864119903120585 (57)


119889 (119864119889(119885 119860)) = 120578

119864dam2 ⟨119864119889⟩

(58)




119899dpa ≃ ⟨120590dpa⟩ sdot 120593 sdot

2Δ12

119906

⩽ 200 [dpa] (59)


⟨120590dpa⟩ sdot 120593 sdot

Δ12

119906

⩽ 100 [dpa] (60)


1000

750

500

250

01 1 10 100 1000


Ope

ratin

g te

mpe

ratu

re (∘

C)

HTR materialsFusion


materials

Thermal

materials


ODS1000

950

900

850 100 150 200 250

reactor



103

102

10

1

10minus1

10minus2



120590D(E)

(bar

n)







Δ12











Be 9 021 185 11 139 153C 12 0158 160 15 356 534H2O 18 0924 10 25 167 416

H2O + B 25 100 250

He 4 0425 018 541 112 607



1205951000

=

Δ12

119906

120593 ≃ 1023

[cmminus2] (61)








119877 mod ≃

1

Σ119878+ Σ119886

sdot

1

120585


(62)

where Σ119878

asymp ⟨120590119878⟩119873mod and Σ

119886asymp ⟨120590

119886⟩119873mod are the


119878⟩ and ⟨120590



2 ln[(119860 minus 1)119860 + 1]2119860





120595100

=

Δ12

119906

120593 ≃ 1024

[cmminus2] (63)

where 120595100













120588 ( 119903 119879 119905) sdot 119888 ( 119903 119879 119905) sdot

120597119879 ( 119903 119905)

120597119905


+ 119902119891

119879( 119903 119879 119905)

(64)


120588 ( 119903 119879 119905) = sum

119894

119873119894( 119903 119879 119905) sdot 120588

119894 (65)







119888 ( 119903 119879 119905) = sum

119894

119888119894(119879)119873

119894( 119903 119879 119905) (66)

alefsym ( 119903 119879 119905) = sum

119894

alefsym119894(119879)119873

119894( 119903 119879 119905) (67)



time









0] (68)



0is the



119879( 119903 Φ 119879 119905) that character-


119902119891

119879( 119903 Φ 119879 119905)

= Φ ( 119903 119879 119905)sum

119894

119876119891

119894120590119894

119891( 119903 119879 119905)119873

119894( 119903 119879 119905) [Wcm3]

(69)

where

Φ ( 119903 119879 119905) = int

119864max119899

0

Φ ( 119903 119864 119879 119905) 119889119864 (70)




120590119894

119891( 119903 119879 119905) = int

119864max119899

0

120590119894

119891(119864 119879) 120588 ( 119903 119864 119879 119905) 119889119864 (71)


120588 ( 119903 119864 119879 119905) =

Φ ( 119903 119864 119879 119905)

Φ ( 119903 119879 119905)

(72)




35

3

25

2

15

1

05

0

0 1 2 3 4 5 6

0

E (eV)

3000

2000

1000 T(K

)

(times100

barn

)



300K to 3000K


119891( 119903 119879 119905) averaged over



92U 23892U 23992U and 239





⟨120590 (119864lim 119879)⟩ =

int

119864lim

011986412

119890minus119864119896119879

120590 (119864 119879) 119889119864

int

119864lim

011986412

119890minus119864119896119879

119889119864

(73)



⟨120590 (119864lim 119879)⟩ =

int

infin

119864lim120590 (119864 119879) 119864

minus1119889119864

int

infin

119864lim119864minus1119889119864

(74)


119872(119864119899)




max119899


119899119896119879 where 119864



119879119899= 1198790[1 + 120578 sdot

Σ119886(1198961198790)

⟨120585⟩ Σ119878

] (75)


119886(1198961198790) is




ΦFermi (119864 119864119865) =119878

⟨120585⟩ Σ119905119864

exp[minusint

119864119891

119864lim

Σ119886(1198641015840) 1198891198641015840

⟨120585⟩ Σ119905(1198641015840) 1198641015840]

(76)


sum119894(120585119894Σ119894




119878+ Σ119894



119878is the total


119865is the







119878 ( 119903 119879 119905)

= int

119864max119899

119864119865

( 119903 119864 119879 119905)

sdot [sum

119894

]119894(119864) sdot Φ ( 119903 119864 119879 119905) sdot 120590

119894

119891(119864 119879) sdot 119873

119894( 119903 119879 119905)] 119889119864

(78)

where 119864max119899


119899asymp 10MeV)119864



119865asymp





119899for


119899119887sim 1radic119864

119899



2000

1500

1000

500

00 1000 2000 3000 4000 5000 6000

T (K)

120590f

(bar

n)

3kT35kT4kT

5kT6kT

(a)

2000

1500

1000

500

00 1000 2000 3000 4000 5000 6000

T (K)

3kT35kT4kT

5kT6kT

120590c

(bar

n)(b)


200

180

160

140

120

100

80

60

400 1000 2000 3000 4000 5000 6000

T (K)

120590f

(bar

n)

3kT35 kT4kT

5kT6kT

(a)

140

120

100

80

60

0

20

40

0 1000 2000 3000 4000 5000 6000

T (K)

3kT35kT4kT

5kT6kT

120590c

(bar

n)

(b)




119879( 119903 Φ 119879 119905) (69) on temperature



036

034

032

03

028

026

0240 1000 2000 3000 4000 5000 6000

T (K)

120590f

(bar

n)

3kT35 kT4kT

5kT6kT

(a)

800

700

600

500

400

300

200

120590c

(bar

n)

0 1000 2000 3000 4000 5000 6000

T (K)

3kT35kT4kT

5kT6kT

(b)



25

20

15

10

05

0600 800 1000 1200 1400

T (K)

qf T(rarrrΦTN

it)

(times10

20eV

)

1

2

3


119879(119903 Φ 119879119873

119894


1013 n(cm2sdots)






119899(119909 119905)






119891

119879( 119903 Φ 119879 119905) (69) temperature







119902119879(119879) = const sdot 119879(1+120575) (79)







2+ 00000126119879

3


minus101198795


1198796

(80)

25

20

15

10

5

00 200 400 600 800 1000 1200 1400

T (K)

c p(J

molmiddotK

)

alefsym(W

mmiddotK

)

40

375

35

325

30

275

25

225



alefsym (119879) asymp 21575 + 00152661119879 (81)








T (K)x (120583m)

t (s)

0

1

2

3 0

025

05

075

10

0

1025

05

075

103

(d)

0

1

2

3 0

025

05

075

10

1025

05

075

(b)

0

1

2

3 0

025

05

075

10

0

1

05

075

(c)

0

1

2

3 0

025

05

075

10

0

1025

05

075

(a)

4 middot 104

2 middot 103

6 middot 104








119879119899asymp [1 + 18

80 sdot 1198702

⟨120585⟩ sdot 45

] (82)




= 45


1198702=

119873238

11987312

= 05 (83)






sdot 119879 (84)







0

50

100

150

200

250

300

350

400

600

700

800

900

1000

1100

1200

1300

1400

0 1000 2000 3000 4000 5000 6000

0 1000 2000 3000 4000 5000 6000

T (K)

239Pu

fiss

ion

cros

s sec

tion

(bar

n)

235U

fiss

ion

cros

s sec

tion

(bar

n)

239Pu

235U








7 Conclusions



(d)

250

500

750

1000

0250

500750

6000

250

500

750

250500

750

(b)

0

250

500

750

1000

0

250500

7501000

01000 01000

250

500

750

0

250500

750

6000

(c)

0250

500

7500

250

500

750

6000

250

500

750

1000

(a)

0

250

500

750

1000

60000

250

500

750

10

0

y (cm)

x (cm)

t (s)t (s)

t (s) t (s)

T (K)

x(cm

)

x(cm

)

x(cm

)

x(cm

)


3 middot 1011

2 middot 1011

4 middot 1012

2 middot 1012

4 middot 1012

2 middot 1012


1 middot 1010

2 middot 10minus7

4 middot 10minus7

6 middot 10minus7

8 middot 10minus7

1 middot 10minus6














References






















































































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of




















11987121198893120594 (120585)

1198891205853

+ (120588max minus 1205740

119889120594 (120585)

119889120585

minus 120575(

120594 (120585)

119906120595119898

minus 1)

2

)

sdot

119889120594 (120585)

119889120585

= 0

(13)


120588max minus 1205740

119889120594 (120585)

119889120585

minus 120575(

120594(120585)

119906120595119898

minus 1)

2

= 119891 (120594 (120585)) (14)


11987121198893120594 (120585)

1198891205853

+ 119891 (120594 (120585))

119889120594 (120585)

119889120585

= 0 (15)


11987121198892120594 (120585)

1198891205852

+ 1198651(120594 (120585)) = 119862 (16)



1198712

2

(

119889120594(120585)

119889120585

)

2

+ 1198652(120594 (120585)) minus 119862120594 (120585) = 119861 (17)





119889120585 = plusmn

119889120594

radic(21198712) (119861 minus 119865

2(120594 (120585)) + 119862120594 (120585))

(18)


119889120585 = 119889120594(

120588max1205740

minus

120575

1205740

(

120594(120585)

119906120595119898

minus 1)

2

minus

1

1205740

119891 (120594(120585)))

minus1

(19)


plusmn radic2

1198712(119861 minus 119865

2(120594 (120585)) + 119862120594 (120585))

=

120588max1205740

minus

120575

1205740

(

120594(120585)

119906120595119898

minus 1)

2

minus

1

1205740

119891 (120594 (120585))

(20)




119891 (120594) = 11990421205942+ 1199041120594 + 1199040 (21)

where 1199040 1199041 and 119904



1198652(120594) =

1199042

12

1205944+

1199041

6

1205943+

1199040

2

1205942+ 1198881120594 + 1198882 (22)



(

120588max1205740

minus

120575

1205740

(

120594

119906120595119898

minus 1)

2

minus

1

1205740

(11990421205942+ 1199041120594 + 1199040))

2

=

2

1198712(119861 minus (

1199042

12

1205944+

1199041

6

1205943+

1199040

2

1205942+ 1198881120594 + 1198882) + 119862120594)

(23)


(

120575

120574011990621205952

119898

+

1199042

1205740

)

2

= minus

1199042

61198712 (24)

minus2(

120575

120574011990621205952

119898

+

1199042

1205740

)(

2120575

1205740119906120595119898

minus

1199041

1205740

) = minus

1199041

31198712 (25)

(

2120575

1205740119906120595119898

minus

1199041

1205740

)

2

minus 2(

120575

120574011990621205952

119898

+

1199042

1205740

)

sdot (

120588max1205740

minus

120575

1205740

minus

1199040

1205740

) = minus

1199040

1198712

(26)

2 (

2120575

1205740119906120595119898

minus

1199041

1205740

)(

120588max1205740

minus

120575

1205740

minus

1199040

1205740

)

= minus(

21198881

1198712+

2119862

1198712)

(27)

(

120588max1205740

minus

120575

1205740

minus

1199040

1205740

)

2

=

2

1198712119861 minus

21198882

1198712 (28)


0 1199041 and 119904



1198882as follows

1199040= 120588max minus 120575

1199041=

2120575

119906120595119898

minus

1205740

119871


1199042=

1205751205740

3119871119906120595119898radic120575 minus 120588max

minus

1205742

0

61198712minus

120575

11990621205952

119898

(29)





119889120585 = 119889120594(minus(

120575

120574011990621205952

119898

+

1199042

1205740

)1205942+ (

2120575

1205740119906120595119898

+

1199041

1205740

)120594

+ (

120588max1205740

minus

120575

1205740

minus

1199040

1205740

))

minus1

(30)


119889120594

(120594 minus 119870)2

minus1198722

= minus119873119889120585 (31)

where

119870 =

1205751205740119906120595119898minus 119904121205740

120575120574011990621205952

119898+ 11990421205740

1198722= (

1205751205740119906120595119898minus 119904121205740

120575120574011990621205952

119898+ 11990421205740

) +

(120588max1205740 minus 1205751205740minus 11990401205740)

(120575120574011990621205952

119898+ 11990421205740)

119873 =

120575

120574011990621205952

119898

+

1199042

1205740

(32)


the following

120594 minus 119870 = 1198721205941 119889120594 = 119872119889120594

1 (33)


1198891205941

(1205941)2

minus 1

= minus119872119873119889120585 (34)

Hence

1205941= minus tanh (119872119873120585 minus 119863) (35)


1

120594 = 119870 minus119872 tanh (119872119873120585 minus 119863) (36)


120593 (120585) = 1198722119873sech2 (119872119873120585 minus 119863) (37)


lim120585rarrinfin

120594 (120585) = 0 (38)


lim120585rarrinfin

120594 (120585) = lim120585rarrinfin

[119870 minus119872 tanh (119872119873120585 minus 119863)] = 0 (39)

leads to

119870 = 119872 (40)



120572 = 119872119873 =

radic120575 minus 120588max2119871

(41)

120593119898= 1198722119873 =

120575 minus 3120588max21205740

=

3120588max minus 120575

210038161003816100381610038161205740

1003816100381610038161003816

(42)

119906 =

120593119898

120572120595119898

(43)



0gt 0 (44)







0gt 0

(45)




115

111

107

103

099

09500 05 10 15 20

kinfin





120593 = 120593119898sech2 (120572120585) = 120593

119898sech2 [120572 (119909 minus 119906119905)] (46)



Δ12


[cm] (47)


119860area =2120593119898

120572

[cm2] (48)


119878 = minus119896119861int

infin

minusinfin

119901 (119909) ln119901 (119909) 119889119909 (49)




119878 = minus2119896119861int

infin

0

sech2 (120572120585) ln [sech2 (120572120585)] 119889( 119909

119906120591120573

)

= 4119896119861int

infin

0

ln cosh (120573119910)cosh2 (120573119910)

119889119910

(50)





119878 =

4 (1 minus ln 2)120573

= 4 (1 minus ln 2)119896119861

120572119906120591120573

=

4119896119861(1 minus ln 2)


2119871

119906120591120573

(51)


119878 sim 119896119861

2119871

119906120591120573

= const (52)




1=

1120572 and 1198972










≫ 1198972 then



119906 prop (1198972

11198972







120595119898sim

Δ12

119906

120593119898= 120591120593sdot 120593119898 (53)


005

1 0

02040608

0

1

005

01

01502120601

minus1

minus05

r(m

)

z (m)05m

(a)

02

46

810 0

05115

200025005007501

120601

r(m

)

z (m)

22m

(b)


0= 102 119896max = 106 120601



0= 099955 120601

119898= 3 sdot 10


jex

r

Z













119886 =

120587

2

radic


(54)



alowast

25

2

15

1

05

0

0 05 1 15 2 25 3




lowast) = 119906120591

1205732119871 on the parameter 119886 [5]











119899dpa = 119905 sdot

119873

sum

119894

⟨120590119894

dpa⟩int119864119894

119864119894

minus1

Φ(119864119894) 119889119864119894=

119873

sum

119894=1

⟨120590119894

dpa⟩120593119894119905 (55)

where

⟨120590119894

dpa⟩ (119864119894) = ⟨120590119889(119864119894 119885 119860)⟩ sdot 119889 (119864

119889(119885 119860)) (56)


)

10

9

8

7

6

5

4

3

2

1

060 70 80 90 100 110 120 130 140 150 160 170 180 190 200

Dose (dpa)

Average316Ti


Embrittlement limit









119903is not


119864dam = 119864119903120585 (57)


119889 (119864119889(119885 119860)) = 120578

119864dam2 ⟨119864119889⟩

(58)




119899dpa ≃ ⟨120590dpa⟩ sdot 120593 sdot

2Δ12

119906

⩽ 200 [dpa] (59)


⟨120590dpa⟩ sdot 120593 sdot

Δ12

119906

⩽ 100 [dpa] (60)


1000

750

500

250

01 1 10 100 1000


Ope

ratin

g te

mpe

ratu

re (∘

C)

HTR materialsFusion


materials

Thermal

materials


ODS1000

950

900

850 100 150 200 250

reactor



103

102

10

1

10minus1

10minus2



120590D(E)

(bar

n)







Δ12











Be 9 021 185 11 139 153C 12 0158 160 15 356 534H2O 18 0924 10 25 167 416

H2O + B 25 100 250

He 4 0425 018 541 112 607



1205951000

=

Δ12

119906

120593 ≃ 1023

[cmminus2] (61)








119877 mod ≃

1

Σ119878+ Σ119886

sdot

1

120585


(62)

where Σ119878

asymp ⟨120590119878⟩119873mod and Σ

119886asymp ⟨120590

119886⟩119873mod are the


119878⟩ and ⟨120590



2 ln[(119860 minus 1)119860 + 1]2119860





120595100

=

Δ12

119906

120593 ≃ 1024

[cmminus2] (63)

where 120595100













120588 ( 119903 119879 119905) sdot 119888 ( 119903 119879 119905) sdot

120597119879 ( 119903 119905)

120597119905


+ 119902119891

119879( 119903 119879 119905)

(64)


120588 ( 119903 119879 119905) = sum

119894

119873119894( 119903 119879 119905) sdot 120588

119894 (65)







119888 ( 119903 119879 119905) = sum

119894

119888119894(119879)119873

119894( 119903 119879 119905) (66)

alefsym ( 119903 119879 119905) = sum

119894

alefsym119894(119879)119873

119894( 119903 119879 119905) (67)



time









0] (68)



0is the



119879( 119903 Φ 119879 119905) that character-


119902119891

119879( 119903 Φ 119879 119905)

= Φ ( 119903 119879 119905)sum

119894

119876119891

119894120590119894

119891( 119903 119879 119905)119873

119894( 119903 119879 119905) [Wcm3]

(69)

where

Φ ( 119903 119879 119905) = int

119864max119899

0

Φ ( 119903 119864 119879 119905) 119889119864 (70)




120590119894

119891( 119903 119879 119905) = int

119864max119899

0

120590119894

119891(119864 119879) 120588 ( 119903 119864 119879 119905) 119889119864 (71)


120588 ( 119903 119864 119879 119905) =

Φ ( 119903 119864 119879 119905)

Φ ( 119903 119879 119905)

(72)




35

3

25

2

15

1

05

0

0 1 2 3 4 5 6

0

E (eV)

3000

2000

1000 T(K

)

(times100

barn

)



300K to 3000K


119891( 119903 119879 119905) averaged over



92U 23892U 23992U and 239





⟨120590 (119864lim 119879)⟩ =

int

119864lim

011986412

119890minus119864119896119879

120590 (119864 119879) 119889119864

int

119864lim

011986412

119890minus119864119896119879

119889119864

(73)



⟨120590 (119864lim 119879)⟩ =

int

infin

119864lim120590 (119864 119879) 119864

minus1119889119864

int

infin

119864lim119864minus1119889119864

(74)


119872(119864119899)




max119899


119899119896119879 where 119864



119879119899= 1198790[1 + 120578 sdot

Σ119886(1198961198790)

⟨120585⟩ Σ119878

] (75)


119886(1198961198790) is




ΦFermi (119864 119864119865) =119878

⟨120585⟩ Σ119905119864

exp[minusint

119864119891

119864lim

Σ119886(1198641015840) 1198891198641015840

⟨120585⟩ Σ119905(1198641015840) 1198641015840]

(76)


sum119894(120585119894Σ119894




119878+ Σ119894



119878is the total


119865is the







119878 ( 119903 119879 119905)

= int

119864max119899

119864119865

( 119903 119864 119879 119905)

sdot [sum

119894

]119894(119864) sdot Φ ( 119903 119864 119879 119905) sdot 120590

119894

119891(119864 119879) sdot 119873

119894( 119903 119879 119905)] 119889119864

(78)

where 119864max119899


119899asymp 10MeV)119864



119865asymp





119899for


119899119887sim 1radic119864

119899



2000

1500

1000

500

00 1000 2000 3000 4000 5000 6000

T (K)

120590f

(bar

n)

3kT35kT4kT

5kT6kT

(a)

2000

1500

1000

500

00 1000 2000 3000 4000 5000 6000

T (K)

3kT35kT4kT

5kT6kT

120590c

(bar

n)(b)


200

180

160

140

120

100

80

60

400 1000 2000 3000 4000 5000 6000

T (K)

120590f

(bar

n)

3kT35 kT4kT

5kT6kT

(a)

140

120

100

80

60

0

20

40

0 1000 2000 3000 4000 5000 6000

T (K)

3kT35kT4kT

5kT6kT

120590c

(bar

n)

(b)




119879( 119903 Φ 119879 119905) (69) on temperature



036

034

032

03

028

026

0240 1000 2000 3000 4000 5000 6000

T (K)

120590f

(bar

n)

3kT35 kT4kT

5kT6kT

(a)

800

700

600

500

400

300

200

120590c

(bar

n)

0 1000 2000 3000 4000 5000 6000

T (K)

3kT35kT4kT

5kT6kT

(b)



25

20

15

10

05

0600 800 1000 1200 1400

T (K)

qf T(rarrrΦTN

it)

(times10

20eV

)

1

2

3


119879(119903 Φ 119879119873

119894


1013 n(cm2sdots)






119899(119909 119905)






119891

119879( 119903 Φ 119879 119905) (69) temperature







119902119879(119879) = const sdot 119879(1+120575) (79)







2+ 00000126119879

3


minus101198795


1198796

(80)

25

20

15

10

5

00 200 400 600 800 1000 1200 1400

T (K)

c p(J

molmiddotK

)

alefsym(W

mmiddotK

)

40

375

35

325

30

275

25

225



alefsym (119879) asymp 21575 + 00152661119879 (81)








T (K)x (120583m)

t (s)

0

1

2

3 0

025

05

075

10

0

1025

05

075

103

(d)

0

1

2

3 0

025

05

075

10

1025

05

075

(b)

0

1

2

3 0

025

05

075

10

0

1

05

075

(c)

0

1

2

3 0

025

05

075

10

0

1025

05

075

(a)

4 middot 104

2 middot 103

6 middot 104








119879119899asymp [1 + 18

80 sdot 1198702

⟨120585⟩ sdot 45

] (82)




= 45


1198702=

119873238

11987312

= 05 (83)






sdot 119879 (84)







0

50

100

150

200

250

300

350

400

600

700

800

900

1000

1100

1200

1300

1400

0 1000 2000 3000 4000 5000 6000

0 1000 2000 3000 4000 5000 6000

T (K)

239Pu

fiss

ion

cros

s sec

tion

(bar

n)

235U

fiss

ion

cros

s sec

tion

(bar

n)

239Pu

235U








7 Conclusions



(d)

250

500

750

1000

0250

500750

6000

250

500

750

250500

750

(b)

0

250

500

750

1000

0

250500

7501000

01000 01000

250

500

750

0

250500

750

6000

(c)

0250

500

7500

250

500

750

6000

250

500

750

1000

(a)

0

250

500

750

1000

60000

250

500

750

10

0

y (cm)

x (cm)

t (s)t (s)

t (s) t (s)

T (K)

x(cm

)

x(cm

)

x(cm

)

x(cm

)


3 middot 1011

2 middot 1011

4 middot 1012

2 middot 1012

4 middot 1012

2 middot 1012


1 middot 1010

2 middot 10minus7

4 middot 10minus7

6 middot 10minus7

8 middot 10minus7

1 middot 10minus6














References






















































































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of





















119889120585 = 119889120594(minus(

120575

120574011990621205952

119898

+

1199042

1205740

)1205942+ (

2120575

1205740119906120595119898

+

1199041

1205740

)120594

+ (

120588max1205740

minus

120575

1205740

minus

1199040

1205740

))

minus1

(30)


119889120594

(120594 minus 119870)2

minus1198722

= minus119873119889120585 (31)

where

119870 =

1205751205740119906120595119898minus 119904121205740

120575120574011990621205952

119898+ 11990421205740

1198722= (

1205751205740119906120595119898minus 119904121205740

120575120574011990621205952

119898+ 11990421205740

) +

(120588max1205740 minus 1205751205740minus 11990401205740)

(120575120574011990621205952

119898+ 11990421205740)

119873 =

120575

120574011990621205952

119898

+

1199042

1205740

(32)


the following

120594 minus 119870 = 1198721205941 119889120594 = 119872119889120594

1 (33)


1198891205941

(1205941)2

minus 1

= minus119872119873119889120585 (34)

Hence

1205941= minus tanh (119872119873120585 minus 119863) (35)


1

120594 = 119870 minus119872 tanh (119872119873120585 minus 119863) (36)


120593 (120585) = 1198722119873sech2 (119872119873120585 minus 119863) (37)


lim120585rarrinfin

120594 (120585) = 0 (38)


lim120585rarrinfin

120594 (120585) = lim120585rarrinfin

[119870 minus119872 tanh (119872119873120585 minus 119863)] = 0 (39)

leads to

119870 = 119872 (40)



120572 = 119872119873 =

radic120575 minus 120588max2119871

(41)

120593119898= 1198722119873 =

120575 minus 3120588max21205740

=

3120588max minus 120575

210038161003816100381610038161205740

1003816100381610038161003816

(42)

119906 =

120593119898

120572120595119898

(43)



0gt 0 (44)







0gt 0

(45)




115

111

107

103

099

09500 05 10 15 20

kinfin





120593 = 120593119898sech2 (120572120585) = 120593

119898sech2 [120572 (119909 minus 119906119905)] (46)



Δ12


[cm] (47)


119860area =2120593119898

120572

[cm2] (48)


119878 = minus119896119861int

infin

minusinfin

119901 (119909) ln119901 (119909) 119889119909 (49)




119878 = minus2119896119861int

infin

0

sech2 (120572120585) ln [sech2 (120572120585)] 119889( 119909

119906120591120573

)

= 4119896119861int

infin

0

ln cosh (120573119910)cosh2 (120573119910)

119889119910

(50)





119878 =

4 (1 minus ln 2)120573

= 4 (1 minus ln 2)119896119861

120572119906120591120573

=

4119896119861(1 minus ln 2)


2119871

119906120591120573

(51)


119878 sim 119896119861

2119871

119906120591120573

= const (52)




1=

1120572 and 1198972










≫ 1198972 then



119906 prop (1198972

11198972







120595119898sim

Δ12

119906

120593119898= 120591120593sdot 120593119898 (53)


005

1 0

02040608

0

1

005

01

01502120601

minus1

minus05

r(m

)

z (m)05m

(a)

02

46

810 0

05115

200025005007501

120601

r(m

)

z (m)

22m

(b)


0= 102 119896max = 106 120601



0= 099955 120601

119898= 3 sdot 10


jex

r

Z













119886 =

120587

2

radic


(54)



alowast

25

2

15

1

05

0

0 05 1 15 2 25 3




lowast) = 119906120591

1205732119871 on the parameter 119886 [5]











119899dpa = 119905 sdot

119873

sum

119894

⟨120590119894

dpa⟩int119864119894

119864119894

minus1

Φ(119864119894) 119889119864119894=

119873

sum

119894=1

⟨120590119894

dpa⟩120593119894119905 (55)

where

⟨120590119894

dpa⟩ (119864119894) = ⟨120590119889(119864119894 119885 119860)⟩ sdot 119889 (119864

119889(119885 119860)) (56)


)

10

9

8

7

6

5

4

3

2

1

060 70 80 90 100 110 120 130 140 150 160 170 180 190 200

Dose (dpa)

Average316Ti


Embrittlement limit









119903is not


119864dam = 119864119903120585 (57)


119889 (119864119889(119885 119860)) = 120578

119864dam2 ⟨119864119889⟩

(58)




119899dpa ≃ ⟨120590dpa⟩ sdot 120593 sdot

2Δ12

119906

⩽ 200 [dpa] (59)


⟨120590dpa⟩ sdot 120593 sdot

Δ12

119906

⩽ 100 [dpa] (60)


1000

750

500

250

01 1 10 100 1000


Ope

ratin

g te

mpe

ratu

re (∘

C)

HTR materialsFusion


materials

Thermal

materials


ODS1000

950

900

850 100 150 200 250

reactor



103

102

10

1

10minus1

10minus2



120590D(E)

(bar

n)







Δ12











Be 9 021 185 11 139 153C 12 0158 160 15 356 534H2O 18 0924 10 25 167 416

H2O + B 25 100 250

He 4 0425 018 541 112 607



1205951000

=

Δ12

119906

120593 ≃ 1023

[cmminus2] (61)








119877 mod ≃

1

Σ119878+ Σ119886

sdot

1

120585


(62)

where Σ119878

asymp ⟨120590119878⟩119873mod and Σ

119886asymp ⟨120590

119886⟩119873mod are the


119878⟩ and ⟨120590



2 ln[(119860 minus 1)119860 + 1]2119860





120595100

=

Δ12

119906

120593 ≃ 1024

[cmminus2] (63)

where 120595100













120588 ( 119903 119879 119905) sdot 119888 ( 119903 119879 119905) sdot

120597119879 ( 119903 119905)

120597119905


+ 119902119891

119879( 119903 119879 119905)

(64)


120588 ( 119903 119879 119905) = sum

119894

119873119894( 119903 119879 119905) sdot 120588

119894 (65)







119888 ( 119903 119879 119905) = sum

119894

119888119894(119879)119873

119894( 119903 119879 119905) (66)

alefsym ( 119903 119879 119905) = sum

119894

alefsym119894(119879)119873

119894( 119903 119879 119905) (67)



time









0] (68)



0is the



119879( 119903 Φ 119879 119905) that character-


119902119891

119879( 119903 Φ 119879 119905)

= Φ ( 119903 119879 119905)sum

119894

119876119891

119894120590119894

119891( 119903 119879 119905)119873

119894( 119903 119879 119905) [Wcm3]

(69)

where

Φ ( 119903 119879 119905) = int

119864max119899

0

Φ ( 119903 119864 119879 119905) 119889119864 (70)




120590119894

119891( 119903 119879 119905) = int

119864max119899

0

120590119894

119891(119864 119879) 120588 ( 119903 119864 119879 119905) 119889119864 (71)


120588 ( 119903 119864 119879 119905) =

Φ ( 119903 119864 119879 119905)

Φ ( 119903 119879 119905)

(72)




35

3

25

2

15

1

05

0

0 1 2 3 4 5 6

0

E (eV)

3000

2000

1000 T(K

)

(times100

barn

)



300K to 3000K


119891( 119903 119879 119905) averaged over



92U 23892U 23992U and 239





⟨120590 (119864lim 119879)⟩ =

int

119864lim

011986412

119890minus119864119896119879

120590 (119864 119879) 119889119864

int

119864lim

011986412

119890minus119864119896119879

119889119864

(73)



⟨120590 (119864lim 119879)⟩ =

int

infin

119864lim120590 (119864 119879) 119864

minus1119889119864

int

infin

119864lim119864minus1119889119864

(74)


119872(119864119899)




max119899


119899119896119879 where 119864



119879119899= 1198790[1 + 120578 sdot

Σ119886(1198961198790)

⟨120585⟩ Σ119878

] (75)


119886(1198961198790) is




ΦFermi (119864 119864119865) =119878

⟨120585⟩ Σ119905119864

exp[minusint

119864119891

119864lim

Σ119886(1198641015840) 1198891198641015840

⟨120585⟩ Σ119905(1198641015840) 1198641015840]

(76)


sum119894(120585119894Σ119894




119878+ Σ119894



119878is the total


119865is the







119878 ( 119903 119879 119905)

= int

119864max119899

119864119865

( 119903 119864 119879 119905)

sdot [sum

119894

]119894(119864) sdot Φ ( 119903 119864 119879 119905) sdot 120590

119894

119891(119864 119879) sdot 119873

119894( 119903 119879 119905)] 119889119864

(78)

where 119864max119899


119899asymp 10MeV)119864



119865asymp





119899for


119899119887sim 1radic119864

119899



2000

1500

1000

500

00 1000 2000 3000 4000 5000 6000

T (K)

120590f

(bar

n)

3kT35kT4kT

5kT6kT

(a)

2000

1500

1000

500

00 1000 2000 3000 4000 5000 6000

T (K)

3kT35kT4kT

5kT6kT

120590c

(bar

n)(b)


200

180

160

140

120

100

80

60

400 1000 2000 3000 4000 5000 6000

T (K)

120590f

(bar

n)

3kT35 kT4kT

5kT6kT

(a)

140

120

100

80

60

0

20

40

0 1000 2000 3000 4000 5000 6000

T (K)

3kT35kT4kT

5kT6kT

120590c

(bar

n)

(b)




119879( 119903 Φ 119879 119905) (69) on temperature



036

034

032

03

028

026

0240 1000 2000 3000 4000 5000 6000

T (K)

120590f

(bar

n)

3kT35 kT4kT

5kT6kT

(a)

800

700

600

500

400

300

200

120590c

(bar

n)

0 1000 2000 3000 4000 5000 6000

T (K)

3kT35kT4kT

5kT6kT

(b)



25

20

15

10

05

0600 800 1000 1200 1400

T (K)

qf T(rarrrΦTN

it)

(times10

20eV

)

1

2

3


119879(119903 Φ 119879119873

119894


1013 n(cm2sdots)






119899(119909 119905)






119891

119879( 119903 Φ 119879 119905) (69) temperature







119902119879(119879) = const sdot 119879(1+120575) (79)







2+ 00000126119879

3


minus101198795


1198796

(80)

25

20

15

10

5

00 200 400 600 800 1000 1200 1400

T (K)

c p(J

molmiddotK

)

alefsym(W

mmiddotK

)

40

375

35

325

30

275

25

225



alefsym (119879) asymp 21575 + 00152661119879 (81)








T (K)x (120583m)

t (s)

0

1

2

3 0

025

05

075

10

0

1025

05

075

103

(d)

0

1

2

3 0

025

05

075

10

1025

05

075

(b)

0

1

2

3 0

025

05

075

10

0

1

05

075

(c)

0

1

2

3 0

025

05

075

10

0

1025

05

075

(a)

4 middot 104

2 middot 103

6 middot 104








119879119899asymp [1 + 18

80 sdot 1198702

⟨120585⟩ sdot 45

] (82)




= 45


1198702=

119873238

11987312

= 05 (83)






sdot 119879 (84)







0

50

100

150

200

250

300

350

400

600

700

800

900

1000

1100

1200

1300

1400

0 1000 2000 3000 4000 5000 6000

0 1000 2000 3000 4000 5000 6000

T (K)

239Pu

fiss

ion

cros

s sec

tion

(bar

n)

235U

fiss

ion

cros

s sec

tion

(bar

n)

239Pu

235U








7 Conclusions



(d)

250

500

750

1000

0250

500750

6000

250

500

750

250500

750

(b)

0

250

500

750

1000

0

250500

7501000

01000 01000

250

500

750

0

250500

750

6000

(c)

0250

500

7500

250

500

750

6000

250

500

750

1000

(a)

0

250

500

750

1000

60000

250

500

750

10

0

y (cm)

x (cm)

t (s)t (s)

t (s) t (s)

T (K)

x(cm

)

x(cm

)

x(cm

)

x(cm

)


3 middot 1011

2 middot 1011

4 middot 1012

2 middot 1012

4 middot 1012

2 middot 1012


1 middot 1010

2 middot 10minus7

4 middot 10minus7

6 middot 10minus7

8 middot 10minus7

1 middot 10minus6














References






















































































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of


















115

111

107

103

099

09500 05 10 15 20

kinfin





120593 = 120593119898sech2 (120572120585) = 120593

119898sech2 [120572 (119909 minus 119906119905)] (46)



Δ12


[cm] (47)


119860area =2120593119898

120572

[cm2] (48)


119878 = minus119896119861int

infin

minusinfin

119901 (119909) ln119901 (119909) 119889119909 (49)




119878 = minus2119896119861int

infin

0

sech2 (120572120585) ln [sech2 (120572120585)] 119889( 119909

119906120591120573

)

= 4119896119861int

infin

0

ln cosh (120573119910)cosh2 (120573119910)

119889119910

(50)





119878 =

4 (1 minus ln 2)120573

= 4 (1 minus ln 2)119896119861

120572119906120591120573

=

4119896119861(1 minus ln 2)


2119871

119906120591120573

(51)


119878 sim 119896119861

2119871

119906120591120573

= const (52)




1=

1120572 and 1198972










≫ 1198972 then



119906 prop (1198972

11198972







120595119898sim

Δ12

119906

120593119898= 120591120593sdot 120593119898 (53)


005

1 0

02040608

0

1

005

01

01502120601

minus1

minus05

r(m

)

z (m)05m

(a)

02

46

810 0

05115

200025005007501

120601

r(m

)

z (m)

22m

(b)


0= 102 119896max = 106 120601



0= 099955 120601

119898= 3 sdot 10


jex

r

Z













119886 =

120587

2

radic


(54)



alowast

25

2

15

1

05

0

0 05 1 15 2 25 3




lowast) = 119906120591

1205732119871 on the parameter 119886 [5]











119899dpa = 119905 sdot

119873

sum

119894

⟨120590119894

dpa⟩int119864119894

119864119894

minus1

Φ(119864119894) 119889119864119894=

119873

sum

119894=1

⟨120590119894

dpa⟩120593119894119905 (55)

where

⟨120590119894

dpa⟩ (119864119894) = ⟨120590119889(119864119894 119885 119860)⟩ sdot 119889 (119864

119889(119885 119860)) (56)


)

10

9

8

7

6

5

4

3

2

1

060 70 80 90 100 110 120 130 140 150 160 170 180 190 200

Dose (dpa)

Average316Ti


Embrittlement limit









119903is not


119864dam = 119864119903120585 (57)


119889 (119864119889(119885 119860)) = 120578

119864dam2 ⟨119864119889⟩

(58)




119899dpa ≃ ⟨120590dpa⟩ sdot 120593 sdot

2Δ12

119906

⩽ 200 [dpa] (59)


⟨120590dpa⟩ sdot 120593 sdot

Δ12

119906

⩽ 100 [dpa] (60)


1000

750

500

250

01 1 10 100 1000


Ope

ratin

g te

mpe

ratu

re (∘

C)

HTR materialsFusion


materials

Thermal

materials


ODS1000

950

900

850 100 150 200 250

reactor



103

102

10

1

10minus1

10minus2



120590D(E)

(bar

n)







Δ12











Be 9 021 185 11 139 153C 12 0158 160 15 356 534H2O 18 0924 10 25 167 416

H2O + B 25 100 250

He 4 0425 018 541 112 607



1205951000

=

Δ12

119906

120593 ≃ 1023

[cmminus2] (61)








119877 mod ≃

1

Σ119878+ Σ119886

sdot

1

120585


(62)

where Σ119878

asymp ⟨120590119878⟩119873mod and Σ

119886asymp ⟨120590

119886⟩119873mod are the


119878⟩ and ⟨120590



2 ln[(119860 minus 1)119860 + 1]2119860





120595100

=

Δ12

119906

120593 ≃ 1024

[cmminus2] (63)

where 120595100













120588 ( 119903 119879 119905) sdot 119888 ( 119903 119879 119905) sdot

120597119879 ( 119903 119905)

120597119905


+ 119902119891

119879( 119903 119879 119905)

(64)


120588 ( 119903 119879 119905) = sum

119894

119873119894( 119903 119879 119905) sdot 120588

119894 (65)







119888 ( 119903 119879 119905) = sum

119894

119888119894(119879)119873

119894( 119903 119879 119905) (66)

alefsym ( 119903 119879 119905) = sum

119894

alefsym119894(119879)119873

119894( 119903 119879 119905) (67)



time









0] (68)



0is the



119879( 119903 Φ 119879 119905) that character-


119902119891

119879( 119903 Φ 119879 119905)

= Φ ( 119903 119879 119905)sum

119894

119876119891

119894120590119894

119891( 119903 119879 119905)119873

119894( 119903 119879 119905) [Wcm3]

(69)

where

Φ ( 119903 119879 119905) = int

119864max119899

0

Φ ( 119903 119864 119879 119905) 119889119864 (70)




120590119894

119891( 119903 119879 119905) = int

119864max119899

0

120590119894

119891(119864 119879) 120588 ( 119903 119864 119879 119905) 119889119864 (71)


120588 ( 119903 119864 119879 119905) =

Φ ( 119903 119864 119879 119905)

Φ ( 119903 119879 119905)

(72)




35

3

25

2

15

1

05

0

0 1 2 3 4 5 6

0

E (eV)

3000

2000

1000 T(K

)

(times100

barn

)



300K to 3000K


119891( 119903 119879 119905) averaged over



92U 23892U 23992U and 239





⟨120590 (119864lim 119879)⟩ =

int

119864lim

011986412

119890minus119864119896119879

120590 (119864 119879) 119889119864

int

119864lim

011986412

119890minus119864119896119879

119889119864

(73)



⟨120590 (119864lim 119879)⟩ =

int

infin

119864lim120590 (119864 119879) 119864

minus1119889119864

int

infin

119864lim119864minus1119889119864

(74)


119872(119864119899)




max119899


119899119896119879 where 119864



119879119899= 1198790[1 + 120578 sdot

Σ119886(1198961198790)

⟨120585⟩ Σ119878

] (75)


119886(1198961198790) is




ΦFermi (119864 119864119865) =119878

⟨120585⟩ Σ119905119864

exp[minusint

119864119891

119864lim

Σ119886(1198641015840) 1198891198641015840

⟨120585⟩ Σ119905(1198641015840) 1198641015840]

(76)


sum119894(120585119894Σ119894




119878+ Σ119894



119878is the total


119865is the







119878 ( 119903 119879 119905)

= int

119864max119899

119864119865

( 119903 119864 119879 119905)

sdot [sum

119894

]119894(119864) sdot Φ ( 119903 119864 119879 119905) sdot 120590

119894

119891(119864 119879) sdot 119873

119894( 119903 119879 119905)] 119889119864

(78)

where 119864max119899


119899asymp 10MeV)119864



119865asymp





119899for


119899119887sim 1radic119864

119899



2000

1500

1000

500

00 1000 2000 3000 4000 5000 6000

T (K)

120590f

(bar

n)

3kT35kT4kT

5kT6kT

(a)

2000

1500

1000

500

00 1000 2000 3000 4000 5000 6000

T (K)

3kT35kT4kT

5kT6kT

120590c

(bar

n)(b)


200

180

160

140

120

100

80

60

400 1000 2000 3000 4000 5000 6000

T (K)

120590f

(bar

n)

3kT35 kT4kT

5kT6kT

(a)

140

120

100

80

60

0

20

40

0 1000 2000 3000 4000 5000 6000

T (K)

3kT35kT4kT

5kT6kT

120590c

(bar

n)

(b)




119879( 119903 Φ 119879 119905) (69) on temperature



036

034

032

03

028

026

0240 1000 2000 3000 4000 5000 6000

T (K)

120590f

(bar

n)

3kT35 kT4kT

5kT6kT

(a)

800

700

600

500

400

300

200

120590c

(bar

n)

0 1000 2000 3000 4000 5000 6000

T (K)

3kT35kT4kT

5kT6kT

(b)



25

20

15

10

05

0600 800 1000 1200 1400

T (K)

qf T(rarrrΦTN

it)

(times10

20eV

)

1

2

3


119879(119903 Φ 119879119873

119894


1013 n(cm2sdots)






119899(119909 119905)






119891

119879( 119903 Φ 119879 119905) (69) temperature







119902119879(119879) = const sdot 119879(1+120575) (79)







2+ 00000126119879

3


minus101198795


1198796

(80)

25

20

15

10

5

00 200 400 600 800 1000 1200 1400

T (K)

c p(J

molmiddotK

)

alefsym(W

mmiddotK

)

40

375

35

325

30

275

25

225



alefsym (119879) asymp 21575 + 00152661119879 (81)








T (K)x (120583m)

t (s)

0

1

2

3 0

025

05

075

10

0

1025

05

075

103

(d)

0

1

2

3 0

025

05

075

10

1025

05

075

(b)

0

1

2

3 0

025

05

075

10

0

1

05

075

(c)

0

1

2

3 0

025

05

075

10

0

1025

05

075

(a)

4 middot 104

2 middot 103

6 middot 104








119879119899asymp [1 + 18

80 sdot 1198702

⟨120585⟩ sdot 45

] (82)




= 45


1198702=

119873238

11987312

= 05 (83)






sdot 119879 (84)







0

50

100

150

200

250

300

350

400

600

700

800

900

1000

1100

1200

1300

1400

0 1000 2000 3000 4000 5000 6000

0 1000 2000 3000 4000 5000 6000

T (K)

239Pu

fiss

ion

cros

s sec

tion

(bar

n)

235U

fiss

ion

cros

s sec

tion

(bar

n)

239Pu

235U








7 Conclusions



(d)

250

500

750

1000

0250

500750

6000

250

500

750

250500

750

(b)

0

250

500

750

1000

0

250500

7501000

01000 01000

250

500

750

0

250500

750

6000

(c)

0250

500

7500

250

500

750

6000

250

500

750

1000

(a)

0

250

500

750

1000

60000

250

500

750

10

0

y (cm)

x (cm)

t (s)t (s)

t (s) t (s)

T (K)

x(cm

)

x(cm

)

x(cm

)

x(cm

)


3 middot 1011

2 middot 1011

4 middot 1012

2 middot 1012

4 middot 1012

2 middot 1012


1 middot 1010

2 middot 10minus7

4 middot 10minus7

6 middot 10minus7

8 middot 10minus7

1 middot 10minus6














References






















































































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of


















005

1 0

02040608

0

1

005

01

01502120601

minus1

minus05

r(m

)

z (m)05m

(a)

02

46

810 0

05115

200025005007501

120601

r(m

)

z (m)

22m

(b)


0= 102 119896max = 106 120601



0= 099955 120601

119898= 3 sdot 10


jex

r

Z













119886 =

120587

2

radic


(54)



alowast

25

2

15

1

05

0

0 05 1 15 2 25 3




lowast) = 119906120591

1205732119871 on the parameter 119886 [5]











119899dpa = 119905 sdot

119873

sum

119894

⟨120590119894

dpa⟩int119864119894

119864119894

minus1

Φ(119864119894) 119889119864119894=

119873

sum

119894=1

⟨120590119894

dpa⟩120593119894119905 (55)

where

⟨120590119894

dpa⟩ (119864119894) = ⟨120590119889(119864119894 119885 119860)⟩ sdot 119889 (119864

119889(119885 119860)) (56)


)

10

9

8

7

6

5

4

3

2

1

060 70 80 90 100 110 120 130 140 150 160 170 180 190 200

Dose (dpa)

Average316Ti


Embrittlement limit









119903is not


119864dam = 119864119903120585 (57)


119889 (119864119889(119885 119860)) = 120578

119864dam2 ⟨119864119889⟩

(58)




119899dpa ≃ ⟨120590dpa⟩ sdot 120593 sdot

2Δ12

119906

⩽ 200 [dpa] (59)


⟨120590dpa⟩ sdot 120593 sdot

Δ12

119906

⩽ 100 [dpa] (60)


1000

750

500

250

01 1 10 100 1000


Ope

ratin

g te

mpe

ratu

re (∘

C)

HTR materialsFusion


materials

Thermal

materials


ODS1000

950

900

850 100 150 200 250

reactor



103

102

10

1

10minus1

10minus2



120590D(E)

(bar

n)







Δ12











Be 9 021 185 11 139 153C 12 0158 160 15 356 534H2O 18 0924 10 25 167 416

H2O + B 25 100 250

He 4 0425 018 541 112 607



1205951000

=

Δ12

119906

120593 ≃ 1023

[cmminus2] (61)








119877 mod ≃

1

Σ119878+ Σ119886

sdot

1

120585


(62)

where Σ119878

asymp ⟨120590119878⟩119873mod and Σ

119886asymp ⟨120590

119886⟩119873mod are the


119878⟩ and ⟨120590



2 ln[(119860 minus 1)119860 + 1]2119860





120595100

=

Δ12

119906

120593 ≃ 1024

[cmminus2] (63)

where 120595100













120588 ( 119903 119879 119905) sdot 119888 ( 119903 119879 119905) sdot

120597119879 ( 119903 119905)

120597119905


+ 119902119891

119879( 119903 119879 119905)

(64)


120588 ( 119903 119879 119905) = sum

119894

119873119894( 119903 119879 119905) sdot 120588

119894 (65)







119888 ( 119903 119879 119905) = sum

119894

119888119894(119879)119873

119894( 119903 119879 119905) (66)

alefsym ( 119903 119879 119905) = sum

119894

alefsym119894(119879)119873

119894( 119903 119879 119905) (67)



time









0] (68)



0is the



119879( 119903 Φ 119879 119905) that character-


119902119891

119879( 119903 Φ 119879 119905)

= Φ ( 119903 119879 119905)sum

119894

119876119891

119894120590119894

119891( 119903 119879 119905)119873

119894( 119903 119879 119905) [Wcm3]

(69)

where

Φ ( 119903 119879 119905) = int

119864max119899

0

Φ ( 119903 119864 119879 119905) 119889119864 (70)




120590119894

119891( 119903 119879 119905) = int

119864max119899

0

120590119894

119891(119864 119879) 120588 ( 119903 119864 119879 119905) 119889119864 (71)


120588 ( 119903 119864 119879 119905) =

Φ ( 119903 119864 119879 119905)

Φ ( 119903 119879 119905)

(72)




35

3

25

2

15

1

05

0

0 1 2 3 4 5 6

0

E (eV)

3000

2000

1000 T(K

)

(times100

barn

)



300K to 3000K


119891( 119903 119879 119905) averaged over



92U 23892U 23992U and 239





⟨120590 (119864lim 119879)⟩ =

int

119864lim

011986412

119890minus119864119896119879

120590 (119864 119879) 119889119864

int

119864lim

011986412

119890minus119864119896119879

119889119864

(73)



⟨120590 (119864lim 119879)⟩ =

int

infin

119864lim120590 (119864 119879) 119864

minus1119889119864

int

infin

119864lim119864minus1119889119864

(74)


119872(119864119899)




max119899


119899119896119879 where 119864



119879119899= 1198790[1 + 120578 sdot

Σ119886(1198961198790)

⟨120585⟩ Σ119878

] (75)


119886(1198961198790) is




ΦFermi (119864 119864119865) =119878

⟨120585⟩ Σ119905119864

exp[minusint

119864119891

119864lim

Σ119886(1198641015840) 1198891198641015840

⟨120585⟩ Σ119905(1198641015840) 1198641015840]

(76)


sum119894(120585119894Σ119894




119878+ Σ119894



119878is the total


119865is the







119878 ( 119903 119879 119905)

= int

119864max119899

119864119865

( 119903 119864 119879 119905)

sdot [sum

119894

]119894(119864) sdot Φ ( 119903 119864 119879 119905) sdot 120590

119894

119891(119864 119879) sdot 119873

119894( 119903 119879 119905)] 119889119864

(78)

where 119864max119899


119899asymp 10MeV)119864



119865asymp





119899for


119899119887sim 1radic119864

119899



2000

1500

1000

500

00 1000 2000 3000 4000 5000 6000

T (K)

120590f

(bar

n)

3kT35kT4kT

5kT6kT

(a)

2000

1500

1000

500

00 1000 2000 3000 4000 5000 6000

T (K)

3kT35kT4kT

5kT6kT

120590c

(bar

n)(b)


200

180

160

140

120

100

80

60

400 1000 2000 3000 4000 5000 6000

T (K)

120590f

(bar

n)

3kT35 kT4kT

5kT6kT

(a)

140

120

100

80

60

0

20

40

0 1000 2000 3000 4000 5000 6000

T (K)

3kT35kT4kT

5kT6kT

120590c

(bar

n)

(b)




119879( 119903 Φ 119879 119905) (69) on temperature



036

034

032

03

028

026

0240 1000 2000 3000 4000 5000 6000

T (K)

120590f

(bar

n)

3kT35 kT4kT

5kT6kT

(a)

800

700

600

500

400

300

200

120590c

(bar

n)

0 1000 2000 3000 4000 5000 6000

T (K)

3kT35kT4kT

5kT6kT

(b)



25

20

15

10

05

0600 800 1000 1200 1400

T (K)

qf T(rarrrΦTN

it)

(times10

20eV

)

1

2

3


119879(119903 Φ 119879119873

119894


1013 n(cm2sdots)






119899(119909 119905)






119891

119879( 119903 Φ 119879 119905) (69) temperature







119902119879(119879) = const sdot 119879(1+120575) (79)







2+ 00000126119879

3


minus101198795


1198796

(80)

25

20

15

10

5

00 200 400 600 800 1000 1200 1400

T (K)

c p(J

molmiddotK

)

alefsym(W

mmiddotK

)

40

375

35

325

30

275

25

225



alefsym (119879) asymp 21575 + 00152661119879 (81)








T (K)x (120583m)

t (s)

0

1

2

3 0

025

05

075

10

0

1025

05

075

103

(d)

0

1

2

3 0

025

05

075

10

1025

05

075

(b)

0

1

2

3 0

025

05

075

10

0

1

05

075

(c)

0

1

2

3 0

025

05

075

10

0

1025

05

075

(a)

4 middot 104

2 middot 103

6 middot 104








119879119899asymp [1 + 18

80 sdot 1198702

⟨120585⟩ sdot 45

] (82)




= 45


1198702=

119873238

11987312

= 05 (83)






sdot 119879 (84)







0

50

100

150

200

250

300

350

400

600

700

800

900

1000

1100

1200

1300

1400

0 1000 2000 3000 4000 5000 6000

0 1000 2000 3000 4000 5000 6000

T (K)

239Pu

fiss

ion

cros

s sec

tion

(bar

n)

235U

fiss

ion

cros

s sec

tion

(bar

n)

239Pu

235U








7 Conclusions



(d)

250

500

750

1000

0250

500750

6000

250

500

750

250500

750

(b)

0

250

500

750

1000

0

250500

7501000

01000 01000

250

500

750

0

250500

750

6000

(c)

0250

500

7500

250

500

750

6000

250

500

750

1000

(a)

0

250

500

750

1000

60000

250

500

750

10

0

y (cm)

x (cm)

t (s)t (s)

t (s) t (s)

T (K)

x(cm

)

x(cm

)

x(cm

)

x(cm

)


3 middot 1011

2 middot 1011

4 middot 1012

2 middot 1012

4 middot 1012

2 middot 1012


1 middot 1010

2 middot 10minus7

4 middot 10minus7

6 middot 10minus7

8 middot 10minus7

1 middot 10minus6














References






















































































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of


















alowast

25

2

15

1

05

0

0 05 1 15 2 25 3




lowast) = 119906120591

1205732119871 on the parameter 119886 [5]











119899dpa = 119905 sdot

119873

sum

119894

⟨120590119894

dpa⟩int119864119894

119864119894

minus1

Φ(119864119894) 119889119864119894=

119873

sum

119894=1

⟨120590119894

dpa⟩120593119894119905 (55)

where

⟨120590119894

dpa⟩ (119864119894) = ⟨120590119889(119864119894 119885 119860)⟩ sdot 119889 (119864

119889(119885 119860)) (56)


)

10

9

8

7

6

5

4

3

2

1

060 70 80 90 100 110 120 130 140 150 160 170 180 190 200

Dose (dpa)

Average316Ti


Embrittlement limit









119903is not


119864dam = 119864119903120585 (57)


119889 (119864119889(119885 119860)) = 120578

119864dam2 ⟨119864119889⟩

(58)




119899dpa ≃ ⟨120590dpa⟩ sdot 120593 sdot

2Δ12

119906

⩽ 200 [dpa] (59)


⟨120590dpa⟩ sdot 120593 sdot

Δ12

119906

⩽ 100 [dpa] (60)


1000

750

500

250

01 1 10 100 1000


Ope

ratin

g te

mpe

ratu

re (∘

C)

HTR materialsFusion


materials

Thermal

materials


ODS1000

950

900

850 100 150 200 250

reactor



103

102

10

1

10minus1

10minus2



120590D(E)

(bar

n)







Δ12











Be 9 021 185 11 139 153C 12 0158 160 15 356 534H2O 18 0924 10 25 167 416

H2O + B 25 100 250

He 4 0425 018 541 112 607



1205951000

=

Δ12

119906

120593 ≃ 1023

[cmminus2] (61)








119877 mod ≃

1

Σ119878+ Σ119886

sdot

1

120585


(62)

where Σ119878

asymp ⟨120590119878⟩119873mod and Σ

119886asymp ⟨120590

119886⟩119873mod are the


119878⟩ and ⟨120590



2 ln[(119860 minus 1)119860 + 1]2119860





120595100

=

Δ12

119906

120593 ≃ 1024

[cmminus2] (63)

where 120595100













120588 ( 119903 119879 119905) sdot 119888 ( 119903 119879 119905) sdot

120597119879 ( 119903 119905)

120597119905


+ 119902119891

119879( 119903 119879 119905)

(64)


120588 ( 119903 119879 119905) = sum

119894

119873119894( 119903 119879 119905) sdot 120588

119894 (65)







119888 ( 119903 119879 119905) = sum

119894

119888119894(119879)119873

119894( 119903 119879 119905) (66)

alefsym ( 119903 119879 119905) = sum

119894

alefsym119894(119879)119873

119894( 119903 119879 119905) (67)



time









0] (68)



0is the



119879( 119903 Φ 119879 119905) that character-


119902119891

119879( 119903 Φ 119879 119905)

= Φ ( 119903 119879 119905)sum

119894

119876119891

119894120590119894

119891( 119903 119879 119905)119873

119894( 119903 119879 119905) [Wcm3]

(69)

where

Φ ( 119903 119879 119905) = int

119864max119899

0

Φ ( 119903 119864 119879 119905) 119889119864 (70)




120590119894

119891( 119903 119879 119905) = int

119864max119899

0

120590119894

119891(119864 119879) 120588 ( 119903 119864 119879 119905) 119889119864 (71)


120588 ( 119903 119864 119879 119905) =

Φ ( 119903 119864 119879 119905)

Φ ( 119903 119879 119905)

(72)




35

3

25

2

15

1

05

0

0 1 2 3 4 5 6

0

E (eV)

3000

2000

1000 T(K

)

(times100

barn

)



300K to 3000K


119891( 119903 119879 119905) averaged over



92U 23892U 23992U and 239





⟨120590 (119864lim 119879)⟩ =

int

119864lim

011986412

119890minus119864119896119879

120590 (119864 119879) 119889119864

int

119864lim

011986412

119890minus119864119896119879

119889119864

(73)



⟨120590 (119864lim 119879)⟩ =

int

infin

119864lim120590 (119864 119879) 119864

minus1119889119864

int

infin

119864lim119864minus1119889119864

(74)


119872(119864119899)




max119899


119899119896119879 where 119864



119879119899= 1198790[1 + 120578 sdot

Σ119886(1198961198790)

⟨120585⟩ Σ119878

] (75)


119886(1198961198790) is




ΦFermi (119864 119864119865) =119878

⟨120585⟩ Σ119905119864

exp[minusint

119864119891

119864lim

Σ119886(1198641015840) 1198891198641015840

⟨120585⟩ Σ119905(1198641015840) 1198641015840]

(76)


sum119894(120585119894Σ119894




119878+ Σ119894



119878is the total


119865is the







119878 ( 119903 119879 119905)

= int

119864max119899

119864119865

( 119903 119864 119879 119905)

sdot [sum

119894

]119894(119864) sdot Φ ( 119903 119864 119879 119905) sdot 120590

119894

119891(119864 119879) sdot 119873

119894( 119903 119879 119905)] 119889119864

(78)

where 119864max119899


119899asymp 10MeV)119864



119865asymp





119899for


119899119887sim 1radic119864

119899



2000

1500

1000

500

00 1000 2000 3000 4000 5000 6000

T (K)

120590f

(bar

n)

3kT35kT4kT

5kT6kT

(a)

2000

1500

1000

500

00 1000 2000 3000 4000 5000 6000

T (K)

3kT35kT4kT

5kT6kT

120590c

(bar

n)(b)


200

180

160

140

120

100

80

60

400 1000 2000 3000 4000 5000 6000

T (K)

120590f

(bar

n)

3kT35 kT4kT

5kT6kT

(a)

140

120

100

80

60

0

20

40

0 1000 2000 3000 4000 5000 6000

T (K)

3kT35kT4kT

5kT6kT

120590c

(bar

n)

(b)




119879( 119903 Φ 119879 119905) (69) on temperature



036

034

032

03

028

026

0240 1000 2000 3000 4000 5000 6000

T (K)

120590f

(bar

n)

3kT35 kT4kT

5kT6kT

(a)

800

700

600

500

400

300

200

120590c

(bar

n)

0 1000 2000 3000 4000 5000 6000

T (K)

3kT35kT4kT

5kT6kT

(b)



25

20

15

10

05

0600 800 1000 1200 1400

T (K)

qf T(rarrrΦTN

it)

(times10

20eV

)

1

2

3


119879(119903 Φ 119879119873

119894


1013 n(cm2sdots)






119899(119909 119905)






119891

119879( 119903 Φ 119879 119905) (69) temperature







119902119879(119879) = const sdot 119879(1+120575) (79)







2+ 00000126119879

3


minus101198795


1198796

(80)

25

20

15

10

5

00 200 400 600 800 1000 1200 1400

T (K)

c p(J

molmiddotK

)

alefsym(W

mmiddotK

)

40

375

35

325

30

275

25

225



alefsym (119879) asymp 21575 + 00152661119879 (81)








T (K)x (120583m)

t (s)

0

1

2

3 0

025

05

075

10

0

1025

05

075

103

(d)

0

1

2

3 0

025

05

075

10

1025

05

075

(b)

0

1

2

3 0

025

05

075

10

0

1

05

075

(c)

0

1

2

3 0

025

05

075

10

0

1025

05

075

(a)

4 middot 104

2 middot 103

6 middot 104








119879119899asymp [1 + 18

80 sdot 1198702

⟨120585⟩ sdot 45

] (82)




= 45


1198702=

119873238

11987312

= 05 (83)






sdot 119879 (84)







0

50

100

150

200

250

300

350

400

600

700

800

900

1000

1100

1200

1300

1400

0 1000 2000 3000 4000 5000 6000

0 1000 2000 3000 4000 5000 6000

T (K)

239Pu

fiss

ion

cros

s sec

tion

(bar

n)

235U

fiss

ion

cros

s sec

tion

(bar

n)

239Pu

235U








7 Conclusions



(d)

250

500

750

1000

0250

500750

6000

250

500

750

250500

750

(b)

0

250

500

750

1000

0

250500

7501000

01000 01000

250

500

750

0

250500

750

6000

(c)

0250

500

7500

250

500

750

6000

250

500

750

1000

(a)

0

250

500

750

1000

60000

250

500

750

10

0

y (cm)

x (cm)

t (s)t (s)

t (s) t (s)

T (K)

x(cm

)

x(cm

)

x(cm

)

x(cm

)


3 middot 1011

2 middot 1011

4 middot 1012

2 middot 1012

4 middot 1012

2 middot 1012


1 middot 1010

2 middot 10minus7

4 middot 10minus7

6 middot 10minus7

8 middot 10minus7

1 middot 10minus6














References






















































































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of


















)

10

9

8

7

6

5

4

3

2

1

060 70 80 90 100 110 120 130 140 150 160 170 180 190 200

Dose (dpa)

Average316Ti


Embrittlement limit









119903is not


119864dam = 119864119903120585 (57)


119889 (119864119889(119885 119860)) = 120578

119864dam2 ⟨119864119889⟩

(58)




119899dpa ≃ ⟨120590dpa⟩ sdot 120593 sdot

2Δ12

119906

⩽ 200 [dpa] (59)


⟨120590dpa⟩ sdot 120593 sdot

Δ12

119906

⩽ 100 [dpa] (60)


1000

750

500

250

01 1 10 100 1000


Ope

ratin

g te

mpe

ratu

re (∘

C)

HTR materialsFusion


materials

Thermal

materials


ODS1000

950

900

850 100 150 200 250

reactor



103

102

10

1

10minus1

10minus2



120590D(E)

(bar

n)







Δ12











Be 9 021 185 11 139 153C 12 0158 160 15 356 534H2O 18 0924 10 25 167 416

H2O + B 25 100 250

He 4 0425 018 541 112 607



1205951000

=

Δ12

119906

120593 ≃ 1023

[cmminus2] (61)








119877 mod ≃

1

Σ119878+ Σ119886

sdot

1

120585


(62)

where Σ119878

asymp ⟨120590119878⟩119873mod and Σ

119886asymp ⟨120590

119886⟩119873mod are the


119878⟩ and ⟨120590



2 ln[(119860 minus 1)119860 + 1]2119860





120595100

=

Δ12

119906

120593 ≃ 1024

[cmminus2] (63)

where 120595100













120588 ( 119903 119879 119905) sdot 119888 ( 119903 119879 119905) sdot

120597119879 ( 119903 119905)

120597119905


+ 119902119891

119879( 119903 119879 119905)

(64)


120588 ( 119903 119879 119905) = sum

119894

119873119894( 119903 119879 119905) sdot 120588

119894 (65)







119888 ( 119903 119879 119905) = sum

119894

119888119894(119879)119873

119894( 119903 119879 119905) (66)

alefsym ( 119903 119879 119905) = sum

119894

alefsym119894(119879)119873

119894( 119903 119879 119905) (67)



time









0] (68)



0is the



119879( 119903 Φ 119879 119905) that character-


119902119891

119879( 119903 Φ 119879 119905)

= Φ ( 119903 119879 119905)sum

119894

119876119891

119894120590119894

119891( 119903 119879 119905)119873

119894( 119903 119879 119905) [Wcm3]

(69)

where

Φ ( 119903 119879 119905) = int

119864max119899

0

Φ ( 119903 119864 119879 119905) 119889119864 (70)




120590119894

119891( 119903 119879 119905) = int

119864max119899

0

120590119894

119891(119864 119879) 120588 ( 119903 119864 119879 119905) 119889119864 (71)


120588 ( 119903 119864 119879 119905) =

Φ ( 119903 119864 119879 119905)

Φ ( 119903 119879 119905)

(72)




35

3

25

2

15

1

05

0

0 1 2 3 4 5 6

0

E (eV)

3000

2000

1000 T(K

)

(times100

barn

)



300K to 3000K


119891( 119903 119879 119905) averaged over



92U 23892U 23992U and 239





⟨120590 (119864lim 119879)⟩ =

int

119864lim

011986412

119890minus119864119896119879

120590 (119864 119879) 119889119864

int

119864lim

011986412

119890minus119864119896119879

119889119864

(73)



⟨120590 (119864lim 119879)⟩ =

int

infin

119864lim120590 (119864 119879) 119864

minus1119889119864

int

infin

119864lim119864minus1119889119864

(74)


119872(119864119899)




max119899


119899119896119879 where 119864



119879119899= 1198790[1 + 120578 sdot

Σ119886(1198961198790)

⟨120585⟩ Σ119878

] (75)


119886(1198961198790) is




ΦFermi (119864 119864119865) =119878

⟨120585⟩ Σ119905119864

exp[minusint

119864119891

119864lim

Σ119886(1198641015840) 1198891198641015840

⟨120585⟩ Σ119905(1198641015840) 1198641015840]

(76)


sum119894(120585119894Σ119894




119878+ Σ119894



119878is the total


119865is the







119878 ( 119903 119879 119905)

= int

119864max119899

119864119865

( 119903 119864 119879 119905)

sdot [sum

119894

]119894(119864) sdot Φ ( 119903 119864 119879 119905) sdot 120590

119894

119891(119864 119879) sdot 119873

119894( 119903 119879 119905)] 119889119864

(78)

where 119864max119899


119899asymp 10MeV)119864



119865asymp





119899for


119899119887sim 1radic119864

119899



2000

1500

1000

500

00 1000 2000 3000 4000 5000 6000

T (K)

120590f

(bar

n)

3kT35kT4kT

5kT6kT

(a)

2000

1500

1000

500

00 1000 2000 3000 4000 5000 6000

T (K)

3kT35kT4kT

5kT6kT

120590c

(bar

n)(b)


200

180

160

140

120

100

80

60

400 1000 2000 3000 4000 5000 6000

T (K)

120590f

(bar

n)

3kT35 kT4kT

5kT6kT

(a)

140

120

100

80

60

0

20

40

0 1000 2000 3000 4000 5000 6000

T (K)

3kT35kT4kT

5kT6kT

120590c

(bar

n)

(b)




119879( 119903 Φ 119879 119905) (69) on temperature



036

034

032

03

028

026

0240 1000 2000 3000 4000 5000 6000

T (K)

120590f

(bar

n)

3kT35 kT4kT

5kT6kT

(a)

800

700

600

500

400

300

200

120590c

(bar

n)

0 1000 2000 3000 4000 5000 6000

T (K)

3kT35kT4kT

5kT6kT

(b)



25

20

15

10

05

0600 800 1000 1200 1400

T (K)

qf T(rarrrΦTN

it)

(times10

20eV

)

1

2

3


119879(119903 Φ 119879119873

119894


1013 n(cm2sdots)






119899(119909 119905)






119891

119879( 119903 Φ 119879 119905) (69) temperature







119902119879(119879) = const sdot 119879(1+120575) (79)







2+ 00000126119879

3


minus101198795


1198796

(80)

25

20

15

10

5

00 200 400 600 800 1000 1200 1400

T (K)

c p(J

molmiddotK

)

alefsym(W

mmiddotK

)

40

375

35

325

30

275

25

225



alefsym (119879) asymp 21575 + 00152661119879 (81)








T (K)x (120583m)

t (s)

0

1

2

3 0

025

05

075

10

0

1025

05

075

103

(d)

0

1

2

3 0

025

05

075

10

1025

05

075

(b)

0

1

2

3 0

025

05

075

10

0

1

05

075

(c)

0

1

2

3 0

025

05

075

10

0

1025

05

075

(a)

4 middot 104

2 middot 103

6 middot 104








119879119899asymp [1 + 18

80 sdot 1198702

⟨120585⟩ sdot 45

] (82)




= 45


1198702=

119873238

11987312

= 05 (83)






sdot 119879 (84)







0

50

100

150

200

250

300

350

400

600

700

800

900

1000

1100

1200

1300

1400

0 1000 2000 3000 4000 5000 6000

0 1000 2000 3000 4000 5000 6000

T (K)

239Pu

fiss

ion

cros

s sec

tion

(bar

n)

235U

fiss

ion

cros

s sec

tion

(bar

n)

239Pu

235U








7 Conclusions



(d)

250

500

750

1000

0250

500750

6000

250

500

750

250500

750

(b)

0

250

500

750

1000

0

250500

7501000

01000 01000

250

500

750

0

250500

750

6000

(c)

0250

500

7500

250

500

750

6000

250

500

750

1000

(a)

0

250

500

750

1000

60000

250

500

750

10

0

y (cm)

x (cm)

t (s)t (s)

t (s) t (s)

T (K)

x(cm

)

x(cm

)

x(cm

)

x(cm

)


3 middot 1011

2 middot 1011

4 middot 1012

2 middot 1012

4 middot 1012

2 middot 1012


1 middot 1010

2 middot 10minus7

4 middot 10minus7

6 middot 10minus7

8 middot 10minus7

1 middot 10minus6














References






















































































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of



















Δ12











Be 9 021 185 11 139 153C 12 0158 160 15 356 534H2O 18 0924 10 25 167 416

H2O + B 25 100 250

He 4 0425 018 541 112 607



1205951000

=

Δ12

119906

120593 ≃ 1023

[cmminus2] (61)








119877 mod ≃

1

Σ119878+ Σ119886

sdot

1

120585


(62)

where Σ119878

asymp ⟨120590119878⟩119873mod and Σ

119886asymp ⟨120590

119886⟩119873mod are the


119878⟩ and ⟨120590



2 ln[(119860 minus 1)119860 + 1]2119860





120595100

=

Δ12

119906

120593 ≃ 1024

[cmminus2] (63)

where 120595100













120588 ( 119903 119879 119905) sdot 119888 ( 119903 119879 119905) sdot

120597119879 ( 119903 119905)

120597119905


+ 119902119891

119879( 119903 119879 119905)

(64)


120588 ( 119903 119879 119905) = sum

119894

119873119894( 119903 119879 119905) sdot 120588

119894 (65)







119888 ( 119903 119879 119905) = sum

119894

119888119894(119879)119873

119894( 119903 119879 119905) (66)

alefsym ( 119903 119879 119905) = sum

119894

alefsym119894(119879)119873

119894( 119903 119879 119905) (67)



time









0] (68)



0is the



119879( 119903 Φ 119879 119905) that character-


119902119891

119879( 119903 Φ 119879 119905)

= Φ ( 119903 119879 119905)sum

119894

119876119891

119894120590119894

119891( 119903 119879 119905)119873

119894( 119903 119879 119905) [Wcm3]

(69)

where

Φ ( 119903 119879 119905) = int

119864max119899

0

Φ ( 119903 119864 119879 119905) 119889119864 (70)




120590119894

119891( 119903 119879 119905) = int

119864max119899

0

120590119894

119891(119864 119879) 120588 ( 119903 119864 119879 119905) 119889119864 (71)


120588 ( 119903 119864 119879 119905) =

Φ ( 119903 119864 119879 119905)

Φ ( 119903 119879 119905)

(72)




35

3

25

2

15

1

05

0

0 1 2 3 4 5 6

0

E (eV)

3000

2000

1000 T(K

)

(times100

barn

)



300K to 3000K


119891( 119903 119879 119905) averaged over



92U 23892U 23992U and 239





⟨120590 (119864lim 119879)⟩ =

int

119864lim

011986412

119890minus119864119896119879

120590 (119864 119879) 119889119864

int

119864lim

011986412

119890minus119864119896119879

119889119864

(73)



⟨120590 (119864lim 119879)⟩ =

int

infin

119864lim120590 (119864 119879) 119864

minus1119889119864

int

infin

119864lim119864minus1119889119864

(74)


119872(119864119899)




max119899


119899119896119879 where 119864



119879119899= 1198790[1 + 120578 sdot

Σ119886(1198961198790)

⟨120585⟩ Σ119878

] (75)


119886(1198961198790) is




ΦFermi (119864 119864119865) =119878

⟨120585⟩ Σ119905119864

exp[minusint

119864119891

119864lim

Σ119886(1198641015840) 1198891198641015840

⟨120585⟩ Σ119905(1198641015840) 1198641015840]

(76)


sum119894(120585119894Σ119894




119878+ Σ119894



119878is the total


119865is the







119878 ( 119903 119879 119905)

= int

119864max119899

119864119865

( 119903 119864 119879 119905)

sdot [sum

119894

]119894(119864) sdot Φ ( 119903 119864 119879 119905) sdot 120590

119894

119891(119864 119879) sdot 119873

119894( 119903 119879 119905)] 119889119864

(78)

where 119864max119899


119899asymp 10MeV)119864



119865asymp





119899for


119899119887sim 1radic119864

119899



2000

1500

1000

500

00 1000 2000 3000 4000 5000 6000

T (K)

120590f

(bar

n)

3kT35kT4kT

5kT6kT

(a)

2000

1500

1000

500

00 1000 2000 3000 4000 5000 6000

T (K)

3kT35kT4kT

5kT6kT

120590c

(bar

n)(b)


200

180

160

140

120

100

80

60

400 1000 2000 3000 4000 5000 6000

T (K)

120590f

(bar

n)

3kT35 kT4kT

5kT6kT

(a)

140

120

100

80

60

0

20

40

0 1000 2000 3000 4000 5000 6000

T (K)

3kT35kT4kT

5kT6kT

120590c

(bar

n)

(b)




119879( 119903 Φ 119879 119905) (69) on temperature



036

034

032

03

028

026

0240 1000 2000 3000 4000 5000 6000

T (K)

120590f

(bar

n)

3kT35 kT4kT

5kT6kT

(a)

800

700

600

500

400

300

200

120590c

(bar

n)

0 1000 2000 3000 4000 5000 6000

T (K)

3kT35kT4kT

5kT6kT

(b)



25

20

15

10

05

0600 800 1000 1200 1400

T (K)

qf T(rarrrΦTN

it)

(times10

20eV

)

1

2

3


119879(119903 Φ 119879119873

119894


1013 n(cm2sdots)






119899(119909 119905)






119891

119879( 119903 Φ 119879 119905) (69) temperature







119902119879(119879) = const sdot 119879(1+120575) (79)







2+ 00000126119879

3


minus101198795


1198796

(80)

25

20

15

10

5

00 200 400 600 800 1000 1200 1400

T (K)

c p(J

molmiddotK

)

alefsym(W

mmiddotK

)

40

375

35

325

30

275

25

225



alefsym (119879) asymp 21575 + 00152661119879 (81)








T (K)x (120583m)

t (s)

0

1

2

3 0

025

05

075

10

0

1025

05

075

103

(d)

0

1

2

3 0

025

05

075

10

1025

05

075

(b)

0

1

2

3 0

025

05

075

10

0

1

05

075

(c)

0

1

2

3 0

025

05

075

10

0

1025

05

075

(a)

4 middot 104

2 middot 103

6 middot 104








119879119899asymp [1 + 18

80 sdot 1198702

⟨120585⟩ sdot 45

] (82)




= 45


1198702=

119873238

11987312

= 05 (83)






sdot 119879 (84)







0

50

100

150

200

250

300

350

400

600

700

800

900

1000

1100

1200

1300

1400

0 1000 2000 3000 4000 5000 6000

0 1000 2000 3000 4000 5000 6000

T (K)

239Pu

fiss

ion

cros

s sec

tion

(bar

n)

235U

fiss

ion

cros

s sec

tion

(bar

n)

239Pu

235U








7 Conclusions



(d)

250

500

750

1000

0250

500750

6000

250

500

750

250500

750

(b)

0

250

500

750

1000

0

250500

7501000

01000 01000

250

500

750

0

250500

750

6000

(c)

0250

500

7500

250

500

750

6000

250

500

750

1000

(a)

0

250

500

750

1000

60000

250

500

750

10

0

y (cm)

x (cm)

t (s)t (s)

t (s) t (s)

T (K)

x(cm

)

x(cm

)

x(cm

)

x(cm

)


3 middot 1011

2 middot 1011

4 middot 1012

2 middot 1012

4 middot 1012

2 middot 1012


1 middot 1010

2 middot 10minus7

4 middot 10minus7

6 middot 10minus7

8 middot 10minus7

1 middot 10minus6














References






















































































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of



















120595100

=

Δ12

119906

120593 ≃ 1024

[cmminus2] (63)

where 120595100













120588 ( 119903 119879 119905) sdot 119888 ( 119903 119879 119905) sdot

120597119879 ( 119903 119905)

120597119905


+ 119902119891

119879( 119903 119879 119905)

(64)


120588 ( 119903 119879 119905) = sum

119894

119873119894( 119903 119879 119905) sdot 120588

119894 (65)







119888 ( 119903 119879 119905) = sum

119894

119888119894(119879)119873

119894( 119903 119879 119905) (66)

alefsym ( 119903 119879 119905) = sum

119894

alefsym119894(119879)119873

119894( 119903 119879 119905) (67)



time









0] (68)



0is the



119879( 119903 Φ 119879 119905) that character-


119902119891

119879( 119903 Φ 119879 119905)

= Φ ( 119903 119879 119905)sum

119894

119876119891

119894120590119894

119891( 119903 119879 119905)119873

119894( 119903 119879 119905) [Wcm3]

(69)

where

Φ ( 119903 119879 119905) = int

119864max119899

0

Φ ( 119903 119864 119879 119905) 119889119864 (70)




120590119894

119891( 119903 119879 119905) = int

119864max119899

0

120590119894

119891(119864 119879) 120588 ( 119903 119864 119879 119905) 119889119864 (71)


120588 ( 119903 119864 119879 119905) =

Φ ( 119903 119864 119879 119905)

Φ ( 119903 119879 119905)

(72)




35

3

25

2

15

1

05

0

0 1 2 3 4 5 6

0

E (eV)

3000

2000

1000 T(K

)

(times100

barn

)



300K to 3000K


119891( 119903 119879 119905) averaged over



92U 23892U 23992U and 239





⟨120590 (119864lim 119879)⟩ =

int

119864lim

011986412

119890minus119864119896119879

120590 (119864 119879) 119889119864

int

119864lim

011986412

119890minus119864119896119879

119889119864

(73)



⟨120590 (119864lim 119879)⟩ =

int

infin

119864lim120590 (119864 119879) 119864

minus1119889119864

int

infin

119864lim119864minus1119889119864

(74)


119872(119864119899)




max119899


119899119896119879 where 119864



119879119899= 1198790[1 + 120578 sdot

Σ119886(1198961198790)

⟨120585⟩ Σ119878

] (75)


119886(1198961198790) is




ΦFermi (119864 119864119865) =119878

⟨120585⟩ Σ119905119864

exp[minusint

119864119891

119864lim

Σ119886(1198641015840) 1198891198641015840

⟨120585⟩ Σ119905(1198641015840) 1198641015840]

(76)


sum119894(120585119894Σ119894




119878+ Σ119894



119878is the total


119865is the







119878 ( 119903 119879 119905)

= int

119864max119899

119864119865

( 119903 119864 119879 119905)

sdot [sum

119894

]119894(119864) sdot Φ ( 119903 119864 119879 119905) sdot 120590

119894

119891(119864 119879) sdot 119873

119894( 119903 119879 119905)] 119889119864

(78)

where 119864max119899


119899asymp 10MeV)119864



119865asymp





119899for


119899119887sim 1radic119864

119899



2000

1500

1000

500

00 1000 2000 3000 4000 5000 6000

T (K)

120590f

(bar

n)

3kT35kT4kT

5kT6kT

(a)

2000

1500

1000

500

00 1000 2000 3000 4000 5000 6000

T (K)

3kT35kT4kT

5kT6kT

120590c

(bar

n)(b)


200

180

160

140

120

100

80

60

400 1000 2000 3000 4000 5000 6000

T (K)

120590f

(bar

n)

3kT35 kT4kT

5kT6kT

(a)

140

120

100

80

60

0

20

40

0 1000 2000 3000 4000 5000 6000

T (K)

3kT35kT4kT

5kT6kT

120590c

(bar

n)

(b)




119879( 119903 Φ 119879 119905) (69) on temperature



036

034

032

03

028

026

0240 1000 2000 3000 4000 5000 6000

T (K)

120590f

(bar

n)

3kT35 kT4kT

5kT6kT

(a)

800

700

600

500

400

300

200

120590c

(bar

n)

0 1000 2000 3000 4000 5000 6000

T (K)

3kT35kT4kT

5kT6kT

(b)



25

20

15

10

05

0600 800 1000 1200 1400

T (K)

qf T(rarrrΦTN

it)

(times10

20eV

)

1

2

3


119879(119903 Φ 119879119873

119894


1013 n(cm2sdots)






119899(119909 119905)






119891

119879( 119903 Φ 119879 119905) (69) temperature







119902119879(119879) = const sdot 119879(1+120575) (79)







2+ 00000126119879

3


minus101198795


1198796

(80)

25

20

15

10

5

00 200 400 600 800 1000 1200 1400

T (K)

c p(J

molmiddotK

)

alefsym(W

mmiddotK

)

40

375

35

325

30

275

25

225



alefsym (119879) asymp 21575 + 00152661119879 (81)








T (K)x (120583m)

t (s)

0

1

2

3 0

025

05

075

10

0

1025

05

075

103

(d)

0

1

2

3 0

025

05

075

10

1025

05

075

(b)

0

1

2

3 0

025

05

075

10

0

1

05

075

(c)

0

1

2

3 0

025

05

075

10

0

1025

05

075

(a)

4 middot 104

2 middot 103

6 middot 104








119879119899asymp [1 + 18

80 sdot 1198702

⟨120585⟩ sdot 45

] (82)




= 45


1198702=

119873238

11987312

= 05 (83)






sdot 119879 (84)







0

50

100

150

200

250

300

350

400

600

700

800

900

1000

1100

1200

1300

1400

0 1000 2000 3000 4000 5000 6000

0 1000 2000 3000 4000 5000 6000

T (K)

239Pu

fiss

ion

cros

s sec

tion

(bar

n)

235U

fiss

ion

cros

s sec

tion

(bar

n)

239Pu

235U








7 Conclusions



(d)

250

500

750

1000

0250

500750

6000

250

500

750

250500

750

(b)

0

250

500

750

1000

0

250500

7501000

01000 01000

250

500

750

0

250500

750

6000

(c)

0250

500

7500

250

500

750

6000

250

500

750

1000

(a)

0

250

500

750

1000

60000

250

500

750

10

0

y (cm)

x (cm)

t (s)t (s)

t (s) t (s)

T (K)

x(cm

)

x(cm

)

x(cm

)

x(cm

)


3 middot 1011

2 middot 1011

4 middot 1012

2 middot 1012

4 middot 1012

2 middot 1012


1 middot 1010

2 middot 10minus7

4 middot 10minus7

6 middot 10minus7

8 middot 10minus7

1 middot 10minus6














References






















































































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of

























0] (68)



0is the



119879( 119903 Φ 119879 119905) that character-


119902119891

119879( 119903 Φ 119879 119905)

= Φ ( 119903 119879 119905)sum

119894

119876119891

119894120590119894

119891( 119903 119879 119905)119873

119894( 119903 119879 119905) [Wcm3]

(69)

where

Φ ( 119903 119879 119905) = int

119864max119899

0

Φ ( 119903 119864 119879 119905) 119889119864 (70)




120590119894

119891( 119903 119879 119905) = int

119864max119899

0

120590119894

119891(119864 119879) 120588 ( 119903 119864 119879 119905) 119889119864 (71)


120588 ( 119903 119864 119879 119905) =

Φ ( 119903 119864 119879 119905)

Φ ( 119903 119879 119905)

(72)




35

3

25

2

15

1

05

0

0 1 2 3 4 5 6

0

E (eV)

3000

2000

1000 T(K

)

(times100

barn

)



300K to 3000K


119891( 119903 119879 119905) averaged over



92U 23892U 23992U and 239





⟨120590 (119864lim 119879)⟩ =

int

119864lim

011986412

119890minus119864119896119879

120590 (119864 119879) 119889119864

int

119864lim

011986412

119890minus119864119896119879

119889119864

(73)



⟨120590 (119864lim 119879)⟩ =

int

infin

119864lim120590 (119864 119879) 119864

minus1119889119864

int

infin

119864lim119864minus1119889119864

(74)


119872(119864119899)




max119899


119899119896119879 where 119864



119879119899= 1198790[1 + 120578 sdot

Σ119886(1198961198790)

⟨120585⟩ Σ119878

] (75)


119886(1198961198790) is




ΦFermi (119864 119864119865) =119878

⟨120585⟩ Σ119905119864

exp[minusint

119864119891

119864lim

Σ119886(1198641015840) 1198891198641015840

⟨120585⟩ Σ119905(1198641015840) 1198641015840]

(76)


sum119894(120585119894Σ119894




119878+ Σ119894



119878is the total


119865is the







119878 ( 119903 119879 119905)

= int

119864max119899

119864119865

( 119903 119864 119879 119905)

sdot [sum

119894

]119894(119864) sdot Φ ( 119903 119864 119879 119905) sdot 120590

119894

119891(119864 119879) sdot 119873

119894( 119903 119879 119905)] 119889119864

(78)

where 119864max119899


119899asymp 10MeV)119864



119865asymp





119899for


119899119887sim 1radic119864

119899



2000

1500

1000

500

00 1000 2000 3000 4000 5000 6000

T (K)

120590f

(bar

n)

3kT35kT4kT

5kT6kT

(a)

2000

1500

1000

500

00 1000 2000 3000 4000 5000 6000

T (K)

3kT35kT4kT

5kT6kT

120590c

(bar

n)(b)


200

180

160

140

120

100

80

60

400 1000 2000 3000 4000 5000 6000

T (K)

120590f

(bar

n)

3kT35 kT4kT

5kT6kT

(a)

140

120

100

80

60

0

20

40

0 1000 2000 3000 4000 5000 6000

T (K)

3kT35kT4kT

5kT6kT

120590c

(bar

n)

(b)




119879( 119903 Φ 119879 119905) (69) on temperature



036

034

032

03

028

026

0240 1000 2000 3000 4000 5000 6000

T (K)

120590f

(bar

n)

3kT35 kT4kT

5kT6kT

(a)

800

700

600

500

400

300

200

120590c

(bar

n)

0 1000 2000 3000 4000 5000 6000

T (K)

3kT35kT4kT

5kT6kT

(b)



25

20

15

10

05

0600 800 1000 1200 1400

T (K)

qf T(rarrrΦTN

it)

(times10

20eV

)

1

2

3


119879(119903 Φ 119879119873

119894


1013 n(cm2sdots)






119899(119909 119905)






119891

119879( 119903 Φ 119879 119905) (69) temperature







119902119879(119879) = const sdot 119879(1+120575) (79)







2+ 00000126119879

3


minus101198795


1198796

(80)

25

20

15

10

5

00 200 400 600 800 1000 1200 1400

T (K)

c p(J

molmiddotK

)

alefsym(W

mmiddotK

)

40

375

35

325

30

275

25

225



alefsym (119879) asymp 21575 + 00152661119879 (81)








T (K)x (120583m)

t (s)

0

1

2

3 0

025

05

075

10

0

1025

05

075

103

(d)

0

1

2

3 0

025

05

075

10

1025

05

075

(b)

0

1

2

3 0

025

05

075

10

0

1

05

075

(c)

0

1

2

3 0

025

05

075

10

0

1025

05

075

(a)

4 middot 104

2 middot 103

6 middot 104








119879119899asymp [1 + 18

80 sdot 1198702

⟨120585⟩ sdot 45

] (82)




= 45


1198702=

119873238

11987312

= 05 (83)






sdot 119879 (84)







0

50

100

150

200

250

300

350

400

600

700

800

900

1000

1100

1200

1300

1400

0 1000 2000 3000 4000 5000 6000

0 1000 2000 3000 4000 5000 6000

T (K)

239Pu

fiss

ion

cros

s sec

tion

(bar

n)

235U

fiss

ion

cros

s sec

tion

(bar

n)

239Pu

235U








7 Conclusions



(d)

250

500

750

1000

0250

500750

6000

250

500

750

250500

750

(b)

0

250

500

750

1000

0

250500

7501000

01000 01000

250

500

750

0

250500

750

6000

(c)

0250

500

7500

250

500

750

6000

250

500

750

1000

(a)

0

250

500

750

1000

60000

250

500

750

10

0

y (cm)

x (cm)

t (s)t (s)

t (s) t (s)

T (K)

x(cm

)

x(cm

)

x(cm

)

x(cm

)


3 middot 1011

2 middot 1011

4 middot 1012

2 middot 1012

4 middot 1012

2 middot 1012


1 middot 1010

2 middot 10minus7

4 middot 10minus7

6 middot 10minus7

8 middot 10minus7

1 middot 10minus6














References






















































































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of



















⟨120590 (119864lim 119879)⟩ =

int

infin

119864lim120590 (119864 119879) 119864

minus1119889119864

int

infin

119864lim119864minus1119889119864

(74)


119872(119864119899)




max119899


119899119896119879 where 119864



119879119899= 1198790[1 + 120578 sdot

Σ119886(1198961198790)

⟨120585⟩ Σ119878

] (75)


119886(1198961198790) is




ΦFermi (119864 119864119865) =119878

⟨120585⟩ Σ119905119864

exp[minusint

119864119891

119864lim

Σ119886(1198641015840) 1198891198641015840

⟨120585⟩ Σ119905(1198641015840) 1198641015840]

(76)


sum119894(120585119894Σ119894




119878+ Σ119894



119878is the total


119865is the







119878 ( 119903 119879 119905)

= int

119864max119899

119864119865

( 119903 119864 119879 119905)

sdot [sum

119894

]119894(119864) sdot Φ ( 119903 119864 119879 119905) sdot 120590

119894

119891(119864 119879) sdot 119873

119894( 119903 119879 119905)] 119889119864

(78)

where 119864max119899


119899asymp 10MeV)119864



119865asymp





119899for


119899119887sim 1radic119864

119899



2000

1500

1000

500

00 1000 2000 3000 4000 5000 6000

T (K)

120590f

(bar

n)

3kT35kT4kT

5kT6kT

(a)

2000

1500

1000

500

00 1000 2000 3000 4000 5000 6000

T (K)

3kT35kT4kT

5kT6kT

120590c

(bar

n)(b)


200

180

160

140

120

100

80

60

400 1000 2000 3000 4000 5000 6000

T (K)

120590f

(bar

n)

3kT35 kT4kT

5kT6kT

(a)

140

120

100

80

60

0

20

40

0 1000 2000 3000 4000 5000 6000

T (K)

3kT35kT4kT

5kT6kT

120590c

(bar

n)

(b)




119879( 119903 Φ 119879 119905) (69) on temperature



036

034

032

03

028

026

0240 1000 2000 3000 4000 5000 6000

T (K)

120590f

(bar

n)

3kT35 kT4kT

5kT6kT

(a)

800

700

600

500

400

300

200

120590c

(bar

n)

0 1000 2000 3000 4000 5000 6000

T (K)

3kT35kT4kT

5kT6kT

(b)



25

20

15

10

05

0600 800 1000 1200 1400

T (K)

qf T(rarrrΦTN

it)

(times10

20eV

)

1

2

3


119879(119903 Φ 119879119873

119894


1013 n(cm2sdots)






119899(119909 119905)






119891

119879( 119903 Φ 119879 119905) (69) temperature







119902119879(119879) = const sdot 119879(1+120575) (79)







2+ 00000126119879

3


minus101198795


1198796

(80)

25

20

15

10

5

00 200 400 600 800 1000 1200 1400

T (K)

c p(J

molmiddotK

)

alefsym(W

mmiddotK

)

40

375

35

325

30

275

25

225



alefsym (119879) asymp 21575 + 00152661119879 (81)








T (K)x (120583m)

t (s)

0

1

2

3 0

025

05

075

10

0

1025

05

075

103

(d)

0

1

2

3 0

025

05

075

10

1025

05

075

(b)

0

1

2

3 0

025

05

075

10

0

1

05

075

(c)

0

1

2

3 0

025

05

075

10

0

1025

05

075

(a)

4 middot 104

2 middot 103

6 middot 104








119879119899asymp [1 + 18

80 sdot 1198702

⟨120585⟩ sdot 45

] (82)




= 45


1198702=

119873238

11987312

= 05 (83)






sdot 119879 (84)







0

50

100

150

200

250

300

350

400

600

700

800

900

1000

1100

1200

1300

1400

0 1000 2000 3000 4000 5000 6000

0 1000 2000 3000 4000 5000 6000

T (K)

239Pu

fiss

ion

cros

s sec

tion

(bar

n)

235U

fiss

ion

cros

s sec

tion

(bar

n)

239Pu

235U








7 Conclusions



(d)

250

500

750

1000

0250

500750

6000

250

500

750

250500

750

(b)

0

250

500

750

1000

0

250500

7501000

01000 01000

250

500

750

0

250500

750

6000

(c)

0250

500

7500

250

500

750

6000

250

500

750

1000

(a)

0

250

500

750

1000

60000

250

500

750

10

0

y (cm)

x (cm)

t (s)t (s)

t (s) t (s)

T (K)

x(cm

)

x(cm

)

x(cm

)

x(cm

)


3 middot 1011

2 middot 1011

4 middot 1012

2 middot 1012

4 middot 1012

2 middot 1012


1 middot 1010

2 middot 10minus7

4 middot 10minus7

6 middot 10minus7

8 middot 10minus7

1 middot 10minus6














References






















































































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of


















2000

1500

1000

500

00 1000 2000 3000 4000 5000 6000

T (K)

120590f

(bar

n)

3kT35kT4kT

5kT6kT

(a)

2000

1500

1000

500

00 1000 2000 3000 4000 5000 6000

T (K)

3kT35kT4kT

5kT6kT

120590c

(bar

n)(b)


200

180

160

140

120

100

80

60

400 1000 2000 3000 4000 5000 6000

T (K)

120590f

(bar

n)

3kT35 kT4kT

5kT6kT

(a)

140

120

100

80

60

0

20

40

0 1000 2000 3000 4000 5000 6000

T (K)

3kT35kT4kT

5kT6kT

120590c

(bar

n)

(b)




119879( 119903 Φ 119879 119905) (69) on temperature



036

034

032

03

028

026

0240 1000 2000 3000 4000 5000 6000

T (K)

120590f

(bar

n)

3kT35 kT4kT

5kT6kT

(a)

800

700

600

500

400

300

200

120590c

(bar

n)

0 1000 2000 3000 4000 5000 6000

T (K)

3kT35kT4kT

5kT6kT

(b)



25

20

15

10

05

0600 800 1000 1200 1400

T (K)

qf T(rarrrΦTN

it)

(times10

20eV

)

1

2

3


119879(119903 Φ 119879119873

119894


1013 n(cm2sdots)






119899(119909 119905)






119891

119879( 119903 Φ 119879 119905) (69) temperature







119902119879(119879) = const sdot 119879(1+120575) (79)







2+ 00000126119879

3


minus101198795


1198796

(80)

25

20

15

10

5

00 200 400 600 800 1000 1200 1400

T (K)

c p(J

molmiddotK

)

alefsym(W

mmiddotK

)

40

375

35

325

30

275

25

225



alefsym (119879) asymp 21575 + 00152661119879 (81)








T (K)x (120583m)

t (s)

0

1

2

3 0

025

05

075

10

0

1025

05

075

103

(d)

0

1

2

3 0

025

05

075

10

1025

05

075

(b)

0

1

2

3 0

025

05

075

10

0

1

05

075

(c)

0

1

2

3 0

025

05

075

10

0

1025

05

075

(a)

4 middot 104

2 middot 103

6 middot 104








119879119899asymp [1 + 18

80 sdot 1198702

⟨120585⟩ sdot 45

] (82)




= 45


1198702=

119873238

11987312

= 05 (83)






sdot 119879 (84)







0

50

100

150

200

250

300

350

400

600

700

800

900

1000

1100

1200

1300

1400

0 1000 2000 3000 4000 5000 6000

0 1000 2000 3000 4000 5000 6000

T (K)

239Pu

fiss

ion

cros

s sec

tion

(bar

n)

235U

fiss

ion

cros

s sec

tion

(bar

n)

239Pu

235U








7 Conclusions



(d)

250

500

750

1000

0250

500750

6000

250

500

750

250500

750

(b)

0

250

500

750

1000

0

250500

7501000

01000 01000

250

500

750

0

250500

750

6000

(c)

0250

500

7500

250

500

750

6000

250

500

750

1000

(a)

0

250

500

750

1000

60000

250

500

750

10

0

y (cm)

x (cm)

t (s)t (s)

t (s) t (s)

T (K)

x(cm

)

x(cm

)

x(cm

)

x(cm

)


3 middot 1011

2 middot 1011

4 middot 1012

2 middot 1012

4 middot 1012

2 middot 1012


1 middot 1010

2 middot 10minus7

4 middot 10minus7

6 middot 10minus7

8 middot 10minus7

1 middot 10minus6














References






















































































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of


















036

034

032

03

028

026

0240 1000 2000 3000 4000 5000 6000

T (K)

120590f

(bar

n)

3kT35 kT4kT

5kT6kT

(a)

800

700

600

500

400

300

200

120590c

(bar

n)

0 1000 2000 3000 4000 5000 6000

T (K)

3kT35kT4kT

5kT6kT

(b)



25

20

15

10

05

0600 800 1000 1200 1400

T (K)

qf T(rarrrΦTN

it)

(times10

20eV

)

1

2

3


119879(119903 Φ 119879119873

119894


1013 n(cm2sdots)






119899(119909 119905)






119891

119879( 119903 Φ 119879 119905) (69) temperature







119902119879(119879) = const sdot 119879(1+120575) (79)







2+ 00000126119879

3


minus101198795


1198796

(80)

25

20

15

10

5

00 200 400 600 800 1000 1200 1400

T (K)

c p(J

molmiddotK

)

alefsym(W

mmiddotK

)

40

375

35

325

30

275

25

225



alefsym (119879) asymp 21575 + 00152661119879 (81)








T (K)x (120583m)

t (s)

0

1

2

3 0

025

05

075

10

0

1025

05

075

103

(d)

0

1

2

3 0

025

05

075

10

1025

05

075

(b)

0

1

2

3 0

025

05

075

10

0

1

05

075

(c)

0

1

2

3 0

025

05

075

10

0

1025

05

075

(a)

4 middot 104

2 middot 103

6 middot 104








119879119899asymp [1 + 18

80 sdot 1198702

⟨120585⟩ sdot 45

] (82)




= 45


1198702=

119873238

11987312

= 05 (83)






sdot 119879 (84)







0

50

100

150

200

250

300

350

400

600

700

800

900

1000

1100

1200

1300

1400

0 1000 2000 3000 4000 5000 6000

0 1000 2000 3000 4000 5000 6000

T (K)

239Pu

fiss

ion

cros

s sec

tion

(bar

n)

235U

fiss

ion

cros

s sec

tion

(bar

n)

239Pu

235U








7 Conclusions



(d)

250

500

750

1000

0250

500750

6000

250

500

750

250500

750

(b)

0

250

500

750

1000

0

250500

7501000

01000 01000

250

500

750

0

250500

750

6000

(c)

0250

500

7500

250

500

750

6000

250

500

750

1000

(a)

0

250

500

750

1000

60000

250

500

750

10

0

y (cm)

x (cm)

t (s)t (s)

t (s) t (s)

T (K)

x(cm

)

x(cm

)

x(cm

)

x(cm

)


3 middot 1011

2 middot 1011

4 middot 1012

2 middot 1012

4 middot 1012

2 middot 1012


1 middot 1010

2 middot 10minus7

4 middot 10minus7

6 middot 10minus7

8 middot 10minus7

1 middot 10minus6














References






















































































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of






















119902119879(119879) = const sdot 119879(1+120575) (79)







2+ 00000126119879

3


minus101198795


1198796

(80)

25

20

15

10

5

00 200 400 600 800 1000 1200 1400

T (K)

c p(J

molmiddotK

)

alefsym(W

mmiddotK

)

40

375

35

325

30

275

25

225



alefsym (119879) asymp 21575 + 00152661119879 (81)








T (K)x (120583m)

t (s)

0

1

2

3 0

025

05

075

10

0

1025

05

075

103

(d)

0

1

2

3 0

025

05

075

10

1025

05

075

(b)

0

1

2

3 0

025

05

075

10

0

1

05

075

(c)

0

1

2

3 0

025

05

075

10

0

1025

05

075

(a)

4 middot 104

2 middot 103

6 middot 104








119879119899asymp [1 + 18

80 sdot 1198702

⟨120585⟩ sdot 45

] (82)




= 45


1198702=

119873238

11987312

= 05 (83)






sdot 119879 (84)







0

50

100

150

200

250

300

350

400

600

700

800

900

1000

1100

1200

1300

1400

0 1000 2000 3000 4000 5000 6000

0 1000 2000 3000 4000 5000 6000

T (K)

239Pu

fiss

ion

cros

s sec

tion

(bar

n)

235U

fiss

ion

cros

s sec

tion

(bar

n)

239Pu

235U








7 Conclusions



(d)

250

500

750

1000

0250

500750

6000

250

500

750

250500

750

(b)

0

250

500

750

1000

0

250500

7501000

01000 01000

250

500

750

0

250500

750

6000

(c)

0250

500

7500

250

500

750

6000

250

500

750

1000

(a)

0

250

500

750

1000

60000

250

500

750

10

0

y (cm)

x (cm)

t (s)t (s)

t (s) t (s)

T (K)

x(cm

)

x(cm

)

x(cm

)

x(cm

)


3 middot 1011

2 middot 1011

4 middot 1012

2 middot 1012

4 middot 1012

2 middot 1012


1 middot 1010

2 middot 10minus7

4 middot 10minus7

6 middot 10minus7

8 middot 10minus7

1 middot 10minus6














References






















































































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of


















T (K)x (120583m)

t (s)

0

1

2

3 0

025

05

075

10

0

1025

05

075

103

(d)

0

1

2

3 0

025

05

075

10

1025

05

075

(b)

0

1

2

3 0

025

05

075

10

0

1

05

075

(c)

0

1

2

3 0

025

05

075

10

0

1025

05

075

(a)

4 middot 104

2 middot 103

6 middot 104








119879119899asymp [1 + 18

80 sdot 1198702

⟨120585⟩ sdot 45

] (82)




= 45


1198702=

119873238

11987312

= 05 (83)






sdot 119879 (84)







0

50

100

150

200

250

300

350

400

600

700

800

900

1000

1100

1200

1300

1400

0 1000 2000 3000 4000 5000 6000

0 1000 2000 3000 4000 5000 6000

T (K)

239Pu

fiss

ion

cros

s sec

tion

(bar

n)

235U

fiss

ion

cros

s sec

tion

(bar

n)

239Pu

235U








7 Conclusions



(d)

250

500

750

1000

0250

500750

6000

250

500

750

250500

750

(b)

0

250

500

750

1000

0

250500

7501000

01000 01000

250

500

750

0

250500

750

6000

(c)

0250

500

7500

250

500

750

6000

250

500

750

1000

(a)

0

250

500

750

1000

60000

250

500

750

10

0

y (cm)

x (cm)

t (s)t (s)

t (s) t (s)

T (K)

x(cm

)

x(cm

)

x(cm

)

x(cm

)


3 middot 1011

2 middot 1011

4 middot 1012

2 middot 1012

4 middot 1012

2 middot 1012


1 middot 1010

2 middot 10minus7

4 middot 10minus7

6 middot 10minus7

8 middot 10minus7

1 middot 10minus6














References






















































































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of



















119879119899asymp [1 + 18

80 sdot 1198702

⟨120585⟩ sdot 45

] (82)




= 45


1198702=

119873238

11987312

= 05 (83)






sdot 119879 (84)







0

50

100

150

200

250

300

350

400

600

700

800

900

1000

1100

1200

1300

1400

0 1000 2000 3000 4000 5000 6000

0 1000 2000 3000 4000 5000 6000

T (K)

239Pu

fiss

ion

cros

s sec

tion

(bar

n)

235U

fiss

ion

cros

s sec

tion

(bar

n)

239Pu

235U








7 Conclusions



(d)

250

500

750

1000

0250

500750

6000

250

500

750

250500

750

(b)

0

250

500

750

1000

0

250500

7501000

01000 01000

250

500

750

0

250500

750

6000

(c)

0250

500

7500

250

500

750

6000

250

500

750

1000

(a)

0

250

500

750

1000

60000

250

500

750

10

0

y (cm)

x (cm)

t (s)t (s)

t (s) t (s)

T (K)

x(cm

)

x(cm

)

x(cm

)

x(cm

)


3 middot 1011

2 middot 1011

4 middot 1012

2 middot 1012

4 middot 1012

2 middot 1012


1 middot 1010

2 middot 10minus7

4 middot 10minus7

6 middot 10minus7

8 middot 10minus7

1 middot 10minus6














References






















































































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of


















(d)

250

500

750

1000

0250

500750

6000

250

500

750

250500

750

(b)

0

250

500

750

1000

0

250500

7501000

01000 01000

250

500

750

0

250500

750

6000

(c)

0250

500

7500

250

500

750

6000

250

500

750

1000

(a)

0

250

500

750

1000

60000

250

500

750

10

0

y (cm)

x (cm)

t (s)t (s)

t (s) t (s)

T (K)

x(cm

)

x(cm

)

x(cm

)

x(cm

)


3 middot 1011

2 middot 1011

4 middot 1012

2 middot 1012

4 middot 1012

2 middot 1012


1 middot 1010

2 middot 10minus7

4 middot 10minus7

6 middot 10minus7

8 middot 10minus7

1 middot 10minus6














References






















































































FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of
























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





















FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of

















Documents

Research Article On Some Fundamental Peculiarities of the ...downloads.hindawi.com/journals/stni/2015/703069.pdf · Science and Technology of Nuclear Installations 4 3.5 3 2.5 2 1.5