transport neutron

8/14/2019 Transport Neutron

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last edited : august 6th 2007last edited : august 6th 2007

Komputasi dan SimulasiKomputasi dan Simulasi

Transport NeutronTransport Neutron

Coaching Neutronik 2008Coaching Neutronik 2008Computational DivisionComputational Division

PPIN BATANPPIN BATAN


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PendahuluanPendahuluan

Masalah utama dalam fisika reaktorMasalah utama dalam fisika reaktornuklir adalah penentuan distribusinuklir adalah penentuan distribusineutron dalam teras reaktor.neutron dalam teras reaktor.

Distribusi neutron menentukan lajuDistribusi neutron menentukan lajuterjadinya berbagai reaksi nuklirterjadinya berbagai reaksi nuklirdalam teras reaktor.dalam teras reaktor.

Dengan memahami keadaanDengan memahami keadaan

populasi neutron maka stabilitas daripopulasi neutron maka stabilitas darireaksi fisi berantai dapat diprediksireaksi fisi berantai dapat diprediksidengan baik.dengan baik.


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Proses transport neutron..Proses transport neutron..

Untuk menentukan distribusi neutronUntuk menentukan distribusi neutrondalam teras reaktor kita harusdalam teras reaktor kita harusmemahami dengan baik prosesmemahami dengan baik prosestransport neutron.transport neutron.

Yaitu proses yang terjadi selamaYaitu proses yang terjadi selamaneutron bergerak dalam terasneutron bergerak dalam terasreaktor, yang melibatkan berbagaireaktor, yang melibatkan berbagaiinteraksi neutron dengan intiinteraksi neutron dengan inti

penyusun teras reaktor berupapenyusun teras reaktor berupatumbukan hingga akhirnya neutrontumbukan hingga akhirnya neutronhilang karena diserap atau keluarhilang karena diserap atau keluardari teras reaktor.dari teras reaktor.


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Proses difusi..Proses difusi..

Kebanyakan studi neutronik terasKebanyakan studi neutronik terasreaktor memperlakukan gerakreaktor memperlakukan gerakneutron sebagai proses difusi.neutron sebagai proses difusi.

Dimana diasumsikan bahwa neutronDimana diasumsikan bahwa neutroncendrung untuk berdifusi dari daerahcendrung untuk berdifusi dari daerahdengan densitas neutron tinggi kedengan densitas neutron tinggi kedaerah dengan densitas neutrondaerah dengan densitas neutronlebih rendah, seperti difusi panaslebih rendah, seperti difusi panasdari daerah bertemperatur tinggi kedari daerah bertemperatur tinggi ketemperatur rendah.temperatur rendah.


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Keterbatasan difusi..Keterbatasan difusi..

Namun,berbeda denganNamun,berbeda denganpenanganan difusi padapenanganan difusi padakonduksi panas dan gas yangkonduksi panas dan gas yangyang memberikan simulasi yangyang memberikan simulasi yangakurat, pendekatan difusiakurat, pendekatan difusiterhadap transport neutronterhadap transport neutron

memiliki validitas yang terbatas.memiliki validitas yang terbatas.


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Diffusions limitation..(contd)Diffusions limitation..(contd)

The reason for this failure is easilyThe reason for this failure is easilyunderstood when it is noted that inunderstood when it is noted that inmost diffusion process the diffusingmost diffusion process the diffusingparticles areparticles are characterized by verycharacterized by very

frequent collisionsfrequent collisions that give rise tothat give rise tovery irregular, almost random, zigzagvery irregular, almost random, zigzagtrajectories.trajectories.

However, the cross-section forHowever, the cross-section for

neutron-nuclear collisions is quiteneutron-nuclear collisions is quitesmall (about 10small (about 10-24-24 cmcm22). Hence). Henceneutron tend to stream relativelyneutron tend to stream relativelylarge distances between interactions.large distances between interactions.


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The mean free path (mfp)The mean free path (mfp)characterizing fast neutrons ischaracterizing fast neutrons istypically on the order of centimeters.typically on the order of centimeters.

And the dimensions characterizingAnd the dimensions characterizing

changes in reactor core compositionchanges in reactor core compositionare usually comparable to a neutronare usually comparable to a neutronmfp. (noted that a reactor fuel pin ismfp. (noted that a reactor fuel pin istypically about 1 cm in diameter).typically about 1 cm in diameter).

Hence, it is required a more accurateHence, it is required a more accuratedescription of neutron transport thatdescription of neutron transport thattakes into account the relatively longtakes into account the relatively longneutron mfp and neutron streaming.neutron mfp and neutron streaming.


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In practical problem, neutronIn practical problem, neutrondiffusion theory is invalid neardiffusion theory is invalid nearthe boundary of a reactor, orthe boundary of a reactor, ornear a highly absorbing materialnear a highly absorbing materialsuch as a fuel rod or controlsuch as a fuel rod or controlelement.element.


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More accurate ???More accurate ???

Such a description has beenSuch a description has beenborrowed from the kinetic theory ofborrowed from the kinetic theory ofrarefied gases (which are alsorarefied gases (which are also

characterized by long mfp).characterized by long mfp). The fundamental equation describingThe fundamental equation describing

dilute gases was first proposed moredilute gases was first proposed morethan one century ago by Boltzmann,than one century ago by Boltzmann,

and even today the Boltzmannand even today the Boltzmannequation remain the principal tool ofequation remain the principal tool ofthe gas dynamicist.the gas dynamicist.


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Neutron transport equationNeutron transport equation

Its counter part for the neutron gas called Its counter part for the neutron gas called neutronneutrontransport equationtransport equation..

It is far simpler than Boltzmann equation. Where itIt is far simpler than Boltzmann equation. Where itis linear equation while the boltzmann equation isis linear equation while the boltzmann equation isnon linear equation.non linear equation.

Neutron transport equation is much simpler toNeutron transport equation is much simpler toderive, requiring only thederive, requiring only the concept of neutronconcept of neutronconservation plus a bit of vector calculusconservation plus a bit of vector calculus, and, andeasier to understand than the neutron diffusioneasier to understand than the neutron diffusionequation.equation.

It is far more fundamental and exact description ofIt is far more fundamental and exact description ofthe neutron population in reactor, indeed,the neutron population in reactor, indeed, it is theit is thefundamental cornerstone on which all of the variousfundamental cornerstone on which all of the variousapproximate methods used in nuclear reactorapproximate methods used in nuclear reactoranalysis are basedanalysis are based..


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Transport problem..Transport problem..

But, neutron transport theory hasBut, neutron transport theory hascome to be associated with acome to be associated with ahideous plethora of impenetrablehideous plethora of impenetrable

mathematics, unwieldy formulas, andmathematics, unwieldy formulas, andthe expenditure of enourmosthe expenditure of enourmosamounts of money on computeramounts of money on computernumber-crunching.number-crunching.

It is usually very dificult to solve theIt is usually very dificult to solve thetransport equation for any buttransport equation for any butsimplest modeled problemssimplest modeled problems


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But..But..

However that is quite all right, since it is notHowever that is quite all right, since it is notthe intent to attack the transport equationthe intent to attack the transport equationhead on (for a while).head on (for a while).

Rather the job of the reactor analyst is toRather the job of the reactor analyst is todevelop suitable (calculationally feasibledevelop suitable (calculationally feasibleand accurate) approximation to it.and accurate) approximation to it.

Usually, only by comparing these variousUsually, only by comparing these variousapproximation theories to the transportapproximation theories to the transportequation from which they originated canequation from which they originated canone really assess their range of validity.one really assess their range of validity.

The effort in understanding the neutronThe effort in understanding the neutrontransport equation will provide one with atransport equation will provide one with amuch deeper and more thoroughmuch deeper and more thoroughunderstanding of the approximate methodsunderstanding of the approximate methods


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Some Introductory ConceptSome Introductory Concept


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Neutron Density and FluxNeutron Density and Flux

Start by defining the neutron densityStart by defining the neutron densityN(r,t) at any point in reactor core byN(r,t) at any point in reactor core by

N(r,t) dN(r,t) d33rr expected number ofexpected number ofneutrons in dneutrons in d33r about r at a time tr about r at a time t..

It is a statistical theory in which onlyIt is a statistical theory in which onlymean or average values aremean or average values arecalculated.calculated.

The neutron density N(r,t) is ofThe neutron density N(r,t) is of

interest because it allows us tointerest because it allows us tocalculate the rate at which nuclearcalculate the rate at which nuclearreactions are occuring at any point inreactions are occuring at any point inthe reactor.the reactor.


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Neutron Density and Flux (contd)Neutron Density and Flux (contd)

Let us suppose that all the neutronsLet us suppose that all the neutronsin the reactor have the same speedin the reactor have the same speed..

The frequency with which a neutronThe frequency with which a neutronwill experience a given neutron-will experience a given neutron-nuclear reaction in terms of thenuclear reaction in terms of themacrocospic cross sectionmacrocospic cross sectioncharacteizing that reactioncharacteizing that reaction and theand theneutron speed v isneutron speed v is

vv = interaction frequency= interaction frequency


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Hence, the reaction-rate densityHence, the reaction-rate densityF(r,t) at any point in the system isF(r,t) at any point in the system isdefined by multiplying the neutrondefined by multiplying the neutrondensity N(r,t) by the interactiondensity N(r,t) by the interactionfrequencyfrequency vv ::F(r,t) dF(r,t) d33rr vv N(r,t) dN(r,t) d33rr

expected rate atexpected rate at

which interactions arewhich interactions areoccuring in doccuring in d33r about r atr about r ata time t.a time t.



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Example :Example :

Neutron density of N=10Neutron density of N=1088 cmcm-3-3 in ain agraphite medium where its totalgraphite medium where its total

cross sectioncross section tt=0.385 cm=0.385 cm-1-1, neutron, neutronspeed 2.2x10speed 2.2x1055 cm/sec.cm/sec. We wouldWe wouldfind a reaction rate density offind a reaction rate density of8.47x108.47x101212 reactions/cmreactions/cm33/sec./sec. In thisIn thisparticular case, most of theseparticular case, most of thesereactions would consist of scatteringreactions would consist of scatteringcollisions.collisions.



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These concept can easily beThese concept can easily beextended to the case in which theextended to the case in which theneutron density is different forneutron density is different for

various neutron energies E byvarious neutron energies E bydefining :defining :

N(r,E,t) dN(r,E,t) d33r dE expected number ofr dE expected number ofneutrons in dneutrons in d33r about r, energies inr about r, energies in

dE about E, at time t.dE about E, at time t. Also the reaction rate densityAlso the reaction rate density

F(r,t) dF(r,t) d33r dE vr dE v(E) N(r,E,t) d(E) N(r,E,t) d33r dEr dE



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The product vN(r,t) occurs veryThe product vN(r,t) occurs veryfrequently in reactor theory, andfrequently in reactor theory, andtherefore it is given a special nametherefore it is given a special name

(r,t) vN(r,t)(r,t) vN(r,t) neutron flux neutron flux Its unit isIts unit is [cm[cm-2-2 secsec-1-1]] Noted thatNoted that neutron fux is scalarneutron fux is scalar

quantityquantity not as others definition ofnot as others definition offlux in electromagnetic or heatflux in electromagnetic or heatconduction.conduction.



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Angular Densities and CurrentsAngular Densities and Currents

So far, we already use three variableSo far, we already use three variableto characterize the state of individualto characterize the state of individualneutron; the neutron positionneutron; the neutron position (r)(r), its, its

energyenergy (E),(E), and the timeand the time (t)(t) at whichat whichthe neutron is observed.the neutron is observed. Yet, notice that to specify the state ofYet, notice that to specify the state of

the neutron, we must also give itsthe neutron, we must also give its

direction of motion characterized bydirection of motion characterized bythe unit vectorthe unit vector=v/|v|.=v/|v|.


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By introducing this new variable letsBy introducing this new variable letsgeneralize the concept of density bygeneralize the concept of density bydefining thedefining the angular neutron densityangular neutron density ::

n(r,E,,t) dn(r,E,,t) d33r dE d =r dE d = expectedexpected

number of neutrons in dnumber of neutrons in d33

r about r,r about r,energy dE about E, moving inenergy dE about E, moving indirection in solid angle d at timedirection in solid angle d at time

tt.. This is the most general neutron densityThis is the most general neutron density

function we need to define since it happensfunction we need to define since it happensthatthat one can derive an essential exactone can derive an essential exactequationequation, the neutron transport equation,, the neutron transport equation,for the angular neutron density n(e,E,,t).for the angular neutron density n(e,E,,t).

Angular Densities and CurrentsAngular Densities and Currents(contd)(contd)


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Angular neutron fluxAngular neutron flux(r,E,,t) v(r,E,,t) v n(e,E,,t)n(e,E,,t)

Angular current densityAngular current density

j(r,E,,t) vj(r,E,,t) vn(e,E,,t)n(e,E,,t) (r,E,,t)(r,E,,t)

Notice that sinceNotice that since is a unit vector, the is a unit vector, theangular flux is actually nothing moreangular flux is actually nothing morethan the magnitude of the angularthan the magnitude of the angularcurrent density.current density.

|j|=||j|=||| ==



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Angular Densities and CurrentsAngular Densities and Currents(contd)(contd) The angular current density has aThe angular current density has a

useful physical interpretation.useful physical interpretation.j(r,E,,t) dA dE d j(r,E,,t) dA dE d expectedexpected

number of neutrons passing throughnumber of neutrons passing throughan area dA per unit time with eergy Ean area dA per unit time with eergy Ein dE, direction in d at time t.in dE, direction in d at time t.

We can also define an angularWe can also define an angular

interaction rateinteraction ratef(r,E,,t) = vf(r,E,,t) = v (r,E)(r,E) n(e,E,,t)n(e,E,,t)== (r,E)(r,E) (r,E,,t)(r,E,,t)


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Angular Densities and CurrentsAngular Densities and Currents(contd)(contd) All of the angle-dependent quantitiesAll of the angle-dependent quantities

can be related to the earlier definitioncan be related to the earlier definitionby simply integrating over theby simply integrating over theangular variables.angular variables.

For neutron density :For neutron density :

N(r,E,t)N(r,E,t) == 44dd n(e,E,,t)n(e,E,,t)

furtherfurther

N(r,t)N(r,t) == 00 dE N(r,E,t)dE N(r,E,t)

= = 00 dEdE44dd n(e,E,,t)n(e,E,,t)


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For neutron fluxFor neutron flux

(r,E,t) =(r,E,t) = 44dd (e,E,,t)(e,E,,t)

andand(r,t)(r,t) == 00

dEdE (r,E,t)(r,E,t)

= = 00

dEdE44dd (e,E,,t)(e,E,,t)


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For neutron currentFor neutron current

J(r,E,t) =J(r,E,t) = 44d jd j(e,E,,t)(e,E,,t)

J(r,E,t) is called neutron currentJ(r,E,t) is called neutron currentdensitydensity. Also,. Also,

J(r,t)J(r,t) == 00 dEdE J(r,E,t)J(r,E,t)

= = 00 dEdE44ddjj(e,E,,t)(e,E,,t)


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More about J(r,t) andMore about J(r,t) and (r,t)(r,t)

Notice that J(r,t) is actually what would beNotice that J(r,t) is actually what would bereffered to as the flux in other fields ofreffered to as the flux in other fields ofphysics, since if we have a small area dAphysics, since if we have a small area dAat a position r, thenat a position r, thenJ(r,t)J(r,t)..dAdA = net rate at which neutrons pass= net rate at which neutrons pass

through a surface area dA.through a surface area dA. The unit of both J(r,t) andThe unit of both J(r,t) and (r,t)(r,t) areare

identicalidentical [cm[cm-2-2..secsec-1-1].]. However, J is aHowever, J is a vector quantityvectorquantity thatthat

characterize the net rate at which neutronscharacterize the net rate at which neutronspass through a surfacepass through a surface oriented in a givenoriented in a givendirectiondirection, whereas, whereas simply characterizesimply characterizethe totalrate at which neutron pass throughthe totalrate at which neutron pass througha unit area,a unit area, regardless of orientationregardless of orientation..


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Persamaan Transport NeutronPersamaan Transport Neutron


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PendahuluanPendahuluan

Persamaan yangPersamaan yangmenggambarkan kerapatanmenggambarkan kerapatanneutron angular pada sistemneutron angular pada sistem

nuklir akan diturunkan dengannuklir akan diturunkan denganmelakukanmelakukan akuntansiakuntansi terhadapterhadapproses-prosesproses-proses yang dapatyang dapatmemunculkanmemunculkan neutron danneutron dan

menghilangkanmenghilangkan neutron darineutron darisembarangsembarang volume vvolume v dalamdalamsistem.sistem.


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Mekanisme pada volume VMekanisme pada volume V

Mekanisme neutron muncul :Mekanisme neutron muncul :

1. sumber neutron dalam volume V1. sumber neutron dalam volume V2. neutron yang terhambur dengan variabel2. neutron yang terhambur dengan variabel

akhirakhir E,E, dari sembarang E,dari sembarang E, ..

((ruang energi dan arahruang energi dan arah

))

3. neutron3. neutron masuk volume Vmasuk volume V melaluimelaluipermukaan S.(permukaan S.(ruang spasialruang spasial))

Mekanisme neutron hilang :

4. neutron bocormelalui permukaan S.5. neutron dalam V (dengan variabel E, )

mengalami tumbukan sehinggavariabelnya menjadi E, .


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Mekanisme Neutron MunculMekanisme Neutron Muncul

1.1.Sumber neutron pada V,Sumber neutron pada V,dengan definisi sumber berikutdengan definisi sumber berikut

)

,

,,3

ddErdtErs

Maka suku sumber dinyatakan sbb:

,,,3 ddErdtErs

V


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Mekanisme Neutron MunculMekanisme Neutron Muncul

2.2. Neutron muncul karena tumbukanNeutron muncul karena tumbukandan terhambur ke ruang V.dan terhambur ke ruang V.Laju neutron terhambur dari suatu ruang (E,)ke (E, ) adalah

( ) ( ) ,,,','' 3 ddErdtErnEEV sKarena harus diperhitungkan neutron darisemua ruang lain maka

( ) ( )( )

,,,',''''

04

3 ddEtErnEEdEdrdV

s


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Mekanisme Neutron hilangMekanisme Neutron hilang

5.5. Neutron yang terhambur ke ruangNeutron yang terhambur ke ruanglain dari V. Laju neutron mengalamilain dari V. Laju neutron mengalamiinteraksi adalahinteraksi adalah

( ( ) ( tErnErtErf tt ,,,,,,, = Maka neutron yang terhambur ke ruang Vdinyatakan sebagai berikut

( ) ) ,,,,3 ddErdtErnEr

Vt


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Mekanisme hilang+munculMekanisme hilang+muncul

(3+4)(3+4) Bocor kedalam dan keluar volume VBocor kedalam dan keluar volume Vdigabung.digabung.Dengan konsep rapat arus angularj, maka lajupada E, akan bocor dari permukaan dS adalah

( ) ( ) dStErndStErj = ,,,,,, Untuk seluruh permukaan, total bocor keluar danmasuk, ( )tErndS

S,,,

Dari pers. Gauss berikut ( ) = VS rArdrAdS )()(3

Didapat,

= ,,,,,,3

ddEtErnrdddEtErndSVS

,,,3

ddEtErnrdV Atau


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Total semua mekanismeTotal semua mekanisme

Dengan mensubstitusi semuanya keDengan mensubstitusi semuanya kepers.pers. AwalAwal diperoleh :diperoleh :

( ) 0)','('''0 4

3 =

++

ddEsnEEvddEnn

t

nrd

V

st

Karena volume V sembarang maka integran diatas harus nol

Maka didapat hubungan kesetimbanganberikut :

( ) ),,,(),,,()','('''),,,(0

4tErstErnEEvddEtErnn

t

nst +=++


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FormulasiFormulasiPersamaan TransporPersamaan Transpor Dengan menggunakan notasi fluks angularDengan menggunakan notasi fluks angular

maka persamaan transport biasa ditulismaka persamaan transport biasa ditulissbb:sbb:

( ) ),

,,(),

,,()'

,'('

'),

,,(),(

1

04 tErstErEEddEtErErt st

+=++

Dimana :

-Syarat awal :

- Syarat batas :

),,,()0,,,( 0 tErEr =

0),,,( = tErs

Bila 0


39/69

Persamaan DiffusiPersamaan DiffusiSatu EnergiSatu Energi

Dengan akuntansi yang sama, untukDengan akuntansi yang sama, untukasumsi satu energi diperolehasumsi satu energi diperolehpersamaan berikutpersamaan berikut

=

++

V

a JS

t

rd 013

SehinggaSJ

ta +=

1

Dari pers. Diatas, untuk dapat diselesaikan lebih

lanjut diperlukan hubungan antara J dan . Inidiberikan oleh Hukum Ficks berikut

( ) ),()(, trrDtrJ Konstanta

diffusi


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Persamaan DiffusiPersamaan DiffusiSatu EnergiSatu Energi

Setelah disubstitusikan kembaliSetelah disubstitusikan kembalimaka diperolehmaka diperoleh

),(),()(),()(

1

trStrrtrrDt a=+

Untuk D yang homogen :),(),()(),(

1 2 trStrrtrD

t

a =+

Lebih jauh, untuk masalah statis :

)()()()(2 rSrrrD a =+

Persamaan Helmholtz

Pers.Difusi

Satu Grup


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Pers.Difusi : Kasus 1-DPers.Difusi : Kasus 1-D

Untuk satu dimensi (mis.X)Untuk satu dimensi (mis.X)makamaka

)()(2

2

rSx

dx

dD a =+

0)()0( == a

Diskritisasi ruang, operator diff.menjadi :( ) ...

2

22

11 +++= ++ii

iiidx

d

dx

dx

( ) ...2

22

11 += ii

iiidx

d

dx

dx

2

11

2

2 2

+

+ iii

idx

d


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Pers.Difusi : Kasus 1-DPers.Difusi : Kasus 1-D

Setelah substitusi diperoleh,Setelah substitusi diperoleh,

iiaiii SD =+

+ +

2

11 2

iiiaiS

DDD=

+

+

+ 122122

Dengan pengaturan variabel :

iiiiiiiiii Saaa =++ ++ 11,,11, Atau (untuk i=1,2,,N-1)

SA = A matriks (n-1)x(N-1),Svektor kolom (N-1)


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Bentuk lebih umum u/ 1-DBentuk lebih umum u/ 1-D

Pers.diff umum 1-D pada geometriPers.diff umum 1-D pada geometribidang datar :bidang datar :

)()()()( xSxx

dx

dxD

dx

da =+

Cara memecahkan persamaan initerbagi kedalam dua langkah :1. menurunkan persamaaan beda

(diskritisasi).2. menyelesaikan persamaan bedadengan algoritma tertentu.


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DiskritisasiDiskritisasi

Metoda umum untuk memperolehMetoda umum untuk memperolehpers.bedapers.beda (difference eq.)(difference eq.) adalah denganadalah denganmelakukanmelakukan integrasiintegrasi terhadap pers.diffterhadap pers.diffpada sembarangpada sembarang meshmesh interval.interval.

Integrasi dari tiap suku pers.diff dilakukanterhadap mesh interval berikut :

2

1+

+

i

ix2

i

ix

ix

1ix 1+ix


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Integrasi tiap sukuIntegrasi tiap sukusuku sumber dan penyerapansuku sumber dan penyerapan

Suku sumberSuku sumber

+

+

+

+

22)( 1

2

2

1

ii

i

x

x

SxSdx

i

i

i

i

Suku penyerapan

+

+

+

+

22

)()( 12

2

1

ii

iaa

x

x

i

i

i

i

i

xxdx


46/69

Integrasi tiap sukuIntegrasi tiap sukusuku bocorsuku bocor

Suku bocor :Suku bocor :

2

2

2

2

11

)()(

++

+

+

i

i

ii

ii

ii

x

x

x

x

dx

dxD

dx

dxD

dx

ddx

Suku ini memerlukan beberapa langkah detail berikut :

1

1

2

1

+

+

+

+

i

ii

x

i

i

dx

d

i

ii

x iidx

d

1

2

21++ i

ix

ix 1+ix

2

i

ix

1ix

ix


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Integrasi tiap sukuIntegrasi tiap sukusuku bocorsuku bocor

Untuk nilai D,Untuk nilai D,

[ ] 1,112

1

2++

+ +=

+

iiii

i

iDDDxD

[ ] 1,1212 += iiiii

i DDDxD

Sehingga total suku bocor,

1

1

1,1,

1

1,

1

1,2

2

)(

1

++

+

+

+

+

+

+

+

i

i

ii

i

i

ii

i

ii

i

i

ii

x

x

DDDD

dx

dxD

dx

ddx

ii

ii


48/69

Hasil integrasiHasil integrasi

Substitusi hasil integrasi terhadapSubstitusi hasil integrasi terhadappers.diffusi awal sbb:pers.diffusi awal sbb:

iiiiiiiiiiSaaa =++ ++ 11,,11,

Dimana koefisiennya adalah

1

1

1,

1

+

+

+

=iii

ii

ii

DDa

1

1

1

1,

1

+

+

+

+

+

++

+=iii

ii

i

iiaii

DDDDa

1

11,

1

+

++ +

+

=iii

ii

ii

DDa

Diperoleh N -1 pers.beda tiga titik (three-pointdifference equations) untuk N+1 variabel tak

diketahui yaitu 0,1,, N.


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Syarat batasSyarat batas

Syarat batas umum dapat diberikanSyarat batas umum dapat diberikansbb:sbb:

011,000,0 Saa =+

NNNNNNN Saa =+ ,11,


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Solusi pers.differensial 3-titikSolusi pers.differensial 3-titik

Persamaan terakhir yang kitaPersamaan terakhir yang kitadapatkan adalahdapatkan adalah

SA =

Lebih eksplisitnya

=

1

3

2

1

1

3

2

1

1111

111111

1111

000

00

0

00

NN S

S

S

S

aa

aaa

aa


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Matrik tridiagonal dapat langsungMatrik tridiagonal dapat langsungdipecahkan dengan eliminasi Gaussian.dipecahkan dengan eliminasi Gaussian.Sehingga diperoleh matriks berikut :Sehingga diperoleh matriks berikut :

=

1

3

2

1

1

3

2

1

3

2

1

1000

0

100

010

001

NN

A

A

A

dimana

11,,

1,

+

+=

nnnnn

nn

nAaa

aA1,1

2,1

1a

aA =

11,,

11,

=

nnnnn

nnnn

nAaa

aS

1,1

11

a

s=


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Maka, nilai fluks diperoleh denganMaka, nilai fluks diperoleh dengansubstitusi kembali, dan diperoleh :substitusi kembali, dan diperoleh :

11 = nN 2122122 +=+= NNNNNNN AA

dst,


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Dekomposisi LUDekomposisi LU

Secara formal yang telah dilakukan adalahSecara formal yang telah dilakukan adalahdekomposisi LU berikutdekomposisi LU berikut

=

1000

0100

010

001

000)(0

00)(

000

3

2

1

2323332

1212221

11

A

A

A

Aaaa

Aaaa

a

A

Sehingga penyelesaiannya sebagai berikut

SULA ==

== SLU 1

== 111 USLUA

Forwardelimination

Back

substitution


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Perhitungan KritikalitasPerhitungan Kritikalitas

Sekarang kita beralih kepadaSekarang kita beralih kepadaperhitungan yang sangatperhitungan yang sangatpenting, yaitu tingkat kritikalitaspenting, yaitu tingkat kritikalitas

suatu sistem nuklir dengansuatu sistem nuklir denganmengetahuimengetahui komposisi bahankomposisi bahandandan geometrigeometrinya.nya.


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Mencari KesetimbanganMencari Kesetimbangan

Untuk menentukan komposisi agarUntuk menentukan komposisi agardiperoleh kesetimbangan makadiperoleh kesetimbangan makadiberikanlah koefisien k berikutdiberikanlah koefisien k berikut

( )rkrrrD fa =+

1

)()()(

2

Cara lain dengan menganggap vvariabel, dimana keadaan kritis dicapaipada nilai v tertentu yaitu vC

( )rrrrD fCa =+ )()()(2 Hubungannya

C

k

=


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Perhitungan KritikalitasPerhitungan Kritikalitas

Secara sederhana persamaan yangSecara sederhana persamaan yangakan dipecahkan berbentukakan dipecahkan berbentuk

= Fk

M1

+ )(2 rDM a

= )(rF f

dengan

Operator destruksi

Operator sumber


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Metoda iterasiMetoda iterasi

Solusi dilakukan dengan metoda iterasiSolusi dilakukan dengan metoda iterasiberikut, diawali dengan memberi sumberberikut, diawali dengan memberi sumberawal dan k tebakan.awal dan k tebakan.

)()( )0( rSFrS

Lalu tentukan flux (1) sbb :

)0(kk

)0(

)1(

)1()1(2)1( 1)( Sk

rDM a =+

Dengan hasil diatas dapat kita hitung sumberdan k baru sbb

)1()1()1( fFS ==


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Bagan AlgoritmaBagan AlgoritmaInput geometri dan

komposisi bahan

Tebak sumberawal (S(0)) dan k(0)

)1()(

)1( 1 ++ =n

nn F

kM

)1()1( ++ = nn FS

++

)(1

)(

)(3

)(

)1(3

)1(

rrSdk

rrSdk

n

n

n

n

1)(

)1()(


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Tim Java [Arya dan SintaTim Java [Arya dan SintaAW,Aniq)AW,Aniq)

Tim Visual Basic [Elfrida,Utaja]Tim Visual Basic [Elfrida,Utaja] Tim Fortran [Marsodi, Sangadji]Tim Fortran [Marsodi, Sangadji] Tim MATLABTim MATLAB

[Mike,Entin,Wahyu,Dinan][Mike,Entin,Wahyu,Dinan]


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InputInput

K[i]Array Bil.riilK-awal

Fl[n,i]Array 2-D bil.riilFluks awalPerhitungan

D[n]Array 1-D bil.riilNeutron per fisi

pla[n], plf[n],

D[n]

Array 1-D Bil.riilPen.lintang

absorpsi,fisi,

konstanta difusi

Material

HBil.riilLebar partisi

LBil.riilPanjang bahanGeometri

INPUT


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SolverSolver

Iterasi dalamIterasi dalaminloop(inloop(

Iterasi luarIterasi luar


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Untuk memantapkanUntuk memantapkanpemahaman kita, mari kitapemahaman kita, mari kitasimak penjelasan untuk halsimak penjelasan untuk halyang sama dari pengembangyang sama dari pengembangMCNP F.Brown dari Los AlamosMCNP F.Brown dari Los Alamos

National Laboratory.National Laboratory.


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transport neutron

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