SO LU TION OF THE RE AC TOR POINT KI NET ICSEQUA TIONS BY MATLAB COM PUT ING

by

Sudhansu S. SINGH and Dinakrushna MOHAPATRA*

Safety Re search In sti tute, Atomic En ergy Reg u la tory Board, IGCAR Cam pus, Kalpakkam, Tamil Nadu, In dia

Sci en tific pa perDOI: 10.2298/NTRP1501011S

The nu mer i cal so lu tion of the point ki net ics equa tions in the pres ence of New to nian tem per a -ture feed back has been a chal leng ing is sue for an a lyz ing the re ac tor tran sients. Re ac tor pointki net ics equa tions are a sys tem of stiff or di nary dif fer en tial equa tions which need spe cial nu -mer i cal treat ments. Al though a pleth ora of nu mer i cal in tri ca cies have been in tro duced tosolve the point ki net ics equa tions over the years, some of the sim ple and straight for wardmeth ods still work very ef fi ciently with ex traor di nary ac cu racy. As an ex am ple, it has beenshown re cently that the fun da men tal back ward Eu ler fi nite dif fer ence al go rithm with its sim -plic ity has proven to be one of the most ef fec tive leg acy meth ods. Com ple ment ing the back -ward Eu ler fi nite dif fer ence scheme, the pres ent work dem on strates the ap pli ca tion of or di -nary dif fer en tial equa tion suite avail able in the MATLAB soft ware pack age to solve the stiffre ac tor point ki net ics equa tions with New to nian tem per a ture feed back ef fects very ef fec tively by an a lyz ing var i ous clas sic bench mark cases. Fair ac cu racy of the re sults im plies the ef fi cientap pli ca tion of MATLAB or di nary dif fer en tial equa tion suite for solv ing the re ac tor point ki -net ics equa tions as an al ter nate method for fu ture ap pli ca tions.

Key words: point ki net ics equa tion, back ward Eu ler fi nite dif fer ence, MATLAB, ODE suite, NDF

IN TRO DUC TION

The nu mer i cal so lu tion of the point ki net icsequa tions (PKE) in the pres ence of New to nian feed -back has been a chal leng ing is sue for an a lyz ing re ac -tor tran sients. Re cently a very in ter est ing pa per high -light ing the var i ous old and mod ern al go rithms for theso lu tion of the re ac tor point ki net ics equa tions hasbeen pub lished by B. D. Ganapol [1]. In his sem i nalwork Ganapol has brought out the no ta ble his toric at -tempts made to pro vide the orig i nal and ef fi cient nu -mer i cal schemes for the so lu tion of the point ki net icsequa tions. In ad di tion, he has also cited the im provedso lu tion tech niques, which has evolved over time.

Dated back in 1958, Akcasu [2] had given a gen -eral so lu tion of the re ac tor ki net ics equa tions with outfeed back by trans fer func tion ap proach. In 1960,Keepin and Cox [3] had given a gen eral so lu tion of there ac tor ki net ics equa tions by re duc ing them to an in te -gral form con ve nient for ex plicit nu mer i cal so lu tion.Sub se quently in the 1960's there were sev eral at tempts to de velop nu mer i cal meth ods based on Runge-Kuttameth ods [4, 5], Eu ler in te gra tion schemes [6, 7], fi nitedif fer ence meth ods [8] and meth ods based on in te gralequa tion for mu la tions with the slowly vary ing fac tor

in each integrand rep re sented by an as sumed func -tional form [9-13]. A method based on an a lyticcon tin u a tion was pro posed by Vigil [14], which waswell suited for fast dig i tal ap pli ca tion dur ing that time.In or der to avoid smaller time steps and long com put -ing times by Runge-Kutta meth ods, Izumi and Noda[15] de vel oped a new ex trap o lated im plicit method.This method not only as sured sta bil ity and ac cu racybut also es ti mated the trun ca tion er ror at a par tic u lartime and could con trol the time in ter val by ad just ingthe mag ni tude of the es ti mated trun ca tion error.

There were sev eral at tempts made by ap ply ingPade and Chebyshev types of ap prox i ma tions,Hermite poly no mi als and power se ries so lu tion meth -ods for solv ing the point ki net ics equa tions for var i ousre ac tiv ity in ser tions [16-18]. Ac cu rate so lu tion of there ac tor ki net ics equa tions was ob tained by Basken and Lewins [19] by ap ply ing straight for ward power se riesre cur rence re la tion in a lumped model with time vary -ing re ac tiv ity. Dur ing the past one-de cade many novelnu mer i cal meth ods, to name a few, the an a lyt i cal in -ver sion method [20], the an a lyt i cal ex po nen tialmethod [21, 22], fourth or der Rosenbrock method [23] with an au to matic step size con trol, an in te gral methodbased on better ba sis func tion [24], have been adoptedto find the ac cu rate, ef fi cient and sta ble so lu tions ofthe point ki net ics equa tions. More re cently in an in ter -

S. S. Singh, et al.: So lu tion of the Re ac tor Point Ki net ics Equa tions by ....Nu clear Tech nol ogy & Ra di a tion Pro tec tion: Year 2015, Vol. 30, No. 1, pp. 11-17 11

* Cor re spond ing au thor; e-mail: dina@igcar.gov.in

est ing work Nahla [25] has in tro duced an ef fi cienttech nique for solv ing the non-lin ear point ki net icsequa tions in pres ence of the adi a batic tem per a turefeed back ef fects. In this work, the pre dicted val ues ofre ac tiv ity are de ter mined us ing back ward Eu ler incon junc tion with the Crank Nichol son ap prox i ma tionscheme. The pre dicted value of re ac tiv ity is re quiredlater to change the non lin ear dif fer en tial sys tem to lin -ear sys tem.

At this point our at ten tion is drawn to the re -marks made by Ganapol [1] that al though so many nu -mer i cal in tri ca cies have been in tro duced to solve thepoint ki net ics equa tions over years, ar gu ably the back -ward Eu ler fi nite dif fer ence (BEFD) al go rithm with its sim plic ity has proven to be one of the most ef fec tiveleg acy meth ods. Com ple ment ing the prop o si tion ofGanapol, in the pres ent work or di nary dif fer en tialequa tion suite (ODE suite) avail able in the MATLABsoft ware pack age has been ap plied to solve the stiff re -ac tor point ki net ics equa tions with New to nian tem per -a ture feed back ef fects very ef fec tively by avoid ing theover com pli cated schemes pro posed in the pre vi ousworks.

AP PLI CA TION OF MATLAB FORSO LU TION OF POINT KI NET ICSEQUA TIONS

The ba sic space in de pend ent re ac tor point ki net -ics equa tions with m group of de layed neu trons can bewrit ten as fol lows

d

di i

N t

t

t NN t C t q t

i

m( ) ( , )( ) ( ) ( )=

-é

ëêù

ûú+ +å

=

r bl

L 1(1)

d

dwherei i

C t

tN t C t i mi i( )

( ) ( ) , , ,= - =b

lL

1 2 3K (2)

by in tro duc ing the New to nian feed back, re ac tiv ity can be rep re sented as

r r( , ) ( ) ( )t N t B t N tt

= - ¢ ¢ò00

d (3)

Here N (t) is the neu tron den sity or re ac tor power at time t, r – the re ac tiv ity, b – the to tal de layed neu tron frac tion, bi – the frac tion of the i-group of de layed neu -trons, L – the prompt neu tron gen er a tion time, Li – thepre cur sor de cay con stant of the ith group, q – the ex ter -nal source, r0 – the ini tial re ac tiv ity, and B – the ab so -lute value of the tem per a ture co ef fi cient of re ac tiv ity.The above stiff non-lin ear or di nary dif fer en tial equa -tions are solved by MATLAB for five bench markprob lems with step, ramp and si nu soi dal re ac tiv ity in -ser tions with and with out tem per a ture feed back.These bench mark prob lems are the same, which weresolved by Ganapol and Nahla [1, 25].

MATLAB ODE suite for solv ingstiff dif fer en tial equa tions

For sci en tific com put ing MATLAB soft ware hasinbuilt solv ers to han dle dif fer ent classes of or di narydif fer en tial equa tions (ODE). A broad over view on theMATLAB ODE solv ers along with the nu mer i cal pro -ce dures is given by Shampine and Reichelt [26]. In anini tial value prob lem, the so lu tion of in ter est is for y' = =.F (t, y) which sat is fies a spe cific ini tial con di tion y ==.y0 at a given ini tial time t0. The solv ers for stiff prob -lems al low the more gen eral form M(t) y' = f (t, y) with amass ma trix M(t) that is non-sin gu lar and usu allysparse. There are four spe cial solv ers de signeddedicatedly to solve the stiff prob lems namely ODE15s, ODE23s, ODE23t, and ODE23tb. In this work theODE15s solver has been used for solv ing the point ki -net ics equa tions. It is a vari able or der solver based onthe nu mer i cal dif fer en ti a tion for mu las (NDF), which isa mod i fied form of the back ward dif fer en ti a tion for mu -las (BDF). The var i ous op tions used in the solver aremen tioned in the fol low ing sec tions while deal ing withthe in di vid ual bench mark prob lems.

NU MER I CAL RE SULTS OFBENCH MARK PROB LEMS

As men tioned ear lier, to sub stan ti ate the ac cu -racy of the MATLAB ODE solver, five benchmarksare an a lyzed and the nu mer i cal re sults are com paredmainly with those ob tained by Ganapol through BEFD method. The re sults of Ganapol (hence for ward re -ferred to as BEFD re sults) are of spe cific in ter est as the nu mer i cal val ues are of very high pre ci sion (in somecases up to 19 dec i mal places for neu tron den sity pre -dic tion). The ba sic ki net ics data used in thefive-bench mark cases per tain ing to the three re ac torsys tems are pre sented in tab. 1.

Bench mark-1: step re ac tiv ityin ser tion with out feed back

In the first bench mark the vari a tion of neu tronden sity start ing with an ini tial equi lib rium value

S. S. Singh, et al.: So lu tion of the Re ac tor Point Ki net ics Equa tions by ...12 Nu clear Tech nol ogy & Ra di a tion Pro tec tion: Year 2015, Vol. 30, No. 1, pp. 11-17

Ta ble 1. Re ac tor ki net ics pa ram e ters used in thebenchmarks

Re ac tor-I Re ac tor-II Re ac tor-III

i bi li [s–1] bi li [s

–1] bi li [s–1]

1 0.000285 0.0127 0.000266 0.0127 0.00021 0.0124

2 0.0015975 0.0317 0.001491 0.0317 0.00141 0.0305

3 0.001410 0.115 0.001316 0.115 0.00127 0.111

4 0.0030525 0.311 0.002849 0.311 0.00255 0.301

5 0.00096 1.40 0.000896 1.40 0.00074 1.13

6 0.000195 3.87 0.000182 3.87 0.00027 3.00

b 0.0075 0.007 0.00645

L 5×10–4 s 2×10–5 s 5×10–5 s

N0 = 1.0 is de ter mined with time. The neu tron den sityvari a tion with time for 100 sec onds in case of Re ac -tor-I is cal cu lated for step re ac tiv ity in ser tions of –1 $, –0.5 $, +0.5 $, and +1 $. It may be re called here that,when re ac tiv ity r is ex pressed in units of ef fec tive de -layed neu tron frac tion (beff) the unit is called “dol lar”($). Thus as per the SI unit, one dol lar (1 $) of re ac tiv -ity is equiv a lent to the ef fec tive de layed neu tron frac -tion.

The re sults ob tained by MATLAB are com paredwith the BEFD val ues in tabs. 2 and 3. It can be ob -served that for re ac tiv ity in ser tions of –1 $, –0.5 $,+0.5 $, the neu tron den si ties cal cu lated by MATLABmatch ex actly (up to 9 dec i mal places) with the BEFDre sults (tab. 2). In the case of re ac tiv ity in ser tion of 1 $, the re sults by BEFD method have been pro duced up to19 dec i mal places by qua dru ple pre ci sion cal cu la tion.Due to lim i ta tions, in the pres ent study the cal cu la tions are car ried out by dou ble pre ci sion only. There fore,the re sults match up to 11-15 dec i mal places (em bold -ened in tab. 3). The rel a tive and ab so lute tol er ancescon sid ered in the MATLAB cal cu la tions are 10–12 and10–8, re spec tively.

Bench mark-2: ramp re ac tiv ityin ser tion with out feed back

In the sec ond bench mark a com par i son is madefor ramp re ac tiv ity in ser tion rate of 0.1 $/s with outfeed back in Re ac tor-II. The vari a tion of neu tron den -sity with time, start ing from N0 = 1.0 are pre sented intab. 4. In this case also a fair match ing is ob served be -tween the MATLAB and BEFD re sults. Only at t = 11 s the MATLAB re sult dif fers at the 9th dec i mal place.

Bench mark-3: si nu soi dal re ac tiv ityin ser tion with out feed back

The third bench mark deals with a si nu soi dal re -ac tiv ity in ser tion with out feed back in a fast re ac tor

with b = 0.0079 (one de layed group), l = 0.077 s–1, L = =.10–8 s and a si nu soi dal re ac tiv ity r(t) = r0sin(pt/T).In this bench mark, two cases are ana lysed. In the firstcase the neu tron den sity with an ini tial value of N0 ==.1.0 is cal cu lated for r0 = 0.0053333 and T = 50 s, there sults of which are given in tab. 5. For com par i son,the em bold ened dig its in di cate the match ing be tweenthe BEFD and MATLAB cal cu la tions. The sec ondcase is de signed to de ter mine ac cu rately the time to the first peak (tp) and the peak neu tron den sity (Np) forfour dif fer ent si nu soi dal re ac tiv ity in ser tions given intab. 6. In ad di tion to BEFD and MATLAB meth ods,the re sults ob tained by Nahla [25] through an ef fi cienttech nique (re ferred to as ET in tab. 6) and also bypower se ries so lu tion (re ferred to as PWS in tab. 6) inan ear lier work of Aboanber and Ha ma da [27] are alsocom pared. In ter est ingly the peak neu tron den sity val -ues cal cu lated by MATLAB match ex actly with theBEFD val ues and the time to at tain the first peak alsomatches fairly well vis- a-vis other meth ods. The vari -a tion of neu tron den si ties over 1000 sec onds for all thefour val ues of T are shown in fig. 1 which was alsoposed as a chal lenge by Ganapol.

S. S. Singh, et al.: So lu tion of the Re ac tor Point Ki net ics Equa tions by ....Nu clear Tech nol ogy & Ra di a tion Pro tec tion: Year 2015, Vol. 30, No. 1, pp. 11-17 13

Ta ble 2. Neu tron den sity for step re ac tiv ity in ser tions

Time [s]r = –1 $ r = –0.5 $ r = +0.5 $

BEFD MATLAB BEFD MATLAB BEFD MATLAB

1.0E–01 5.205642866E-01 5.205642866E-01 6.989252256E-01 6.989252256E-01 1.533112646E+00 1.533112646E+00

1.0E+00 4.333334453E-01 4.333334453E-01 6.070535656E-01 6.070535656E-01 2.511494291E+00 2.511494291E+00

1.0E+01 2.361106508E-01 2.361106508E-01 3.960776907E-01 3.960776907E-01 1.421502524E+01 1.421502524E+01

1.0E+02 2.866764245E-02 2.866764245E-02 7.158285444E-02 7.158285444E-02 8.006143562E+07 8.006143562E+07

Ta ble 3. Neu tron den sity for step re ac tiv ity in ser tion r = 1 $

Time [s] BEFD MATLAB

1.0E-01 2.5157661414043723001E+00 2.515766141404372E+00

5.0E-01 1.0362533810640214680E+01 1.036253381064022E+01

1.0E+00 3.2183540945534212174E+01 3.218354094553435E+01

1.0E+01 3.2469788980305281366E+09 3.246978898030739E+09

1.0E+02 2.5964846465508730749E+89 2.596484646556433E+89

Ta ble 4. Ramp re ac tiv ity in ser tion of 0.1 $/s

Time [s]Neu tron den sity

BEFD MATLAB

2.0E+00 1.338200050E+00 1.338200050E+00

4.0E+00 2.228441897E+00 2.228441897E+00

6.0E+00 5.582052449E+00 5.582052449E+00

8.0E+00 4.278629573E+01 4.278629573E+01

1.0E+01 4.511636239E+05 4.511636239E+05

1.1E+01 1.792213607E+16 1.792213608E+16

Ta ble 5. Neu tron den si ties for si nu soi dal re ac tiv ityvari a tion

Time [s] BEFD MATLAB

10.0 2.065383519E+00 2.065383519E+00

20.0 8.854133921E+00 8.854133925E+00

30.0 4.064354222E+01 4.064354222E+01

40.0 6.135607517E+01 6.135607517E+01

50.0 4.610628770E+01 4.610628771E+01

60.0 2.912634840E+01 2.912634841E+01

70.0 1.895177042E+01 1.895177053E+01

80.0 1.393829211E+01 1.393829229E+01

90.0 1.253353406E+01 1.253353422E+01

100.0 1.544816514E+01 1.544816534E+01

Bench mark-4: step re ac tiv ityin ser tion with feed back

In the fourth bench mark, step re ac tiv ity in ser -tion with tem per a ture feed back is con sid ered for Re -ac tor-III. Like the pre vi ous cases, here the neu tronden sity vari a tion with time in steps of 10 s is de ter -mined for three step re ac tiv ity ad di tions of 1.0 $, 1.5 $,and 2 $ with a neg a tive re ac tiv ity feed back (B ==.2.5×10–6) as given in eq. 3. The re sults are com paredin tab. 7. Fur ther, the peak neu tron den sity along withthe time to at tain the peak den sity for sev eral step in -ser tions with the same feed back is given in tab. 8. Itmay be noted that for r0 = 0.1 $, 0.2 $, and 0.5 $ themax i mum time step taken is 10–5 s, where as for r0 ==.1.0 $, 1.2 $, 1.5 $, and 2 $ the max i mum time steptaken is 10–6 s. As a com plete ness of the bench markprob lem, the neu tron den sity trace for sev eral step re -ac tiv ity in ser tions with feed back is also dis played infig. 2. The more or less ex act match ing of the

MATLAB re sults with that of BEFD is also ob servedin this bench mark.

Bench mark-5: ramp re ac tiv ityin ser tion with feed back

Bench mark-5 deals with a spe cial case of re ac -tiv ity feed back called power ex cur sions with com pen -

S. S. Singh, et al.: So lu tion of the Re ac tor Point Ki net ics Equa tions by ...14 Nu clear Tech nol ogy & Ra di a tion Pro tec tion: Year 2015, Vol. 30, No. 1, pp. 11-17

Ta ble 6. Time to first peak and peak den si ties

T [s] r0×103 BEFD MATLAB ET PWS

tp [s] Np tp [s] Np tp [s] Np tp [s] Np

50 5.3333 3.910712E+01 6.153015E+01 3.910719E+01 6.153015E+01 3.9111E+01 6.1531E+01 3.9112E+01 6.1530E+01

150 3.2327 1.373198E+02 9.581150E+01 1.373197E+02 9.581150E+01 1.3732E+02 9.5811E+01 1.3729E+02 9.5811E+01

250 2.3193 2.371265E+02 1.134679E+02 2.371265E+02 1.134679E+02 2.3712E+02 1.1346E+02 2.3715E+02 1.1346E+02

350 1.8083 3.370713E+02 1.238209E+02 3.370714E+02 1.238209E+02 3.3707E+02 1.2382E+02 3.3705E+02 1.2382E+02

Fig ure 1. Neu tron den sity vari a tion for var i ousval ues of T

Ta ble 7. Neu tron den sity with re ac tiv ity feed back (B = 2.5×10–6)

Time [s]N

(for r = 1 $)N

(for r = 1.5 $)N

(for r = 2 $)

BEFD MATLAB BEFD MATLAB BEFD MATLAB

10 132.0385964 132.0385964 107.9116832 107.9116830 103.3808535 103.3808534

20 51.69986094 41.60428128 41.60428123 39.13886903 39.13886899 39.13886899

30 28.17468536 28.17468536 23.29893150 23.29893148 22.00377721 22.00377719

40 18.14633000 18.14632999 15.30342749 15.30342749 14.49367193 14.49367192

50 12.77957703 12.77957703 10.89014315 10.89014314 10.31861108 10.31861108

60 9.474932501 9.474932501 8.101031859 8.101031856 7.663319203 7.663319201

70 7.244477494 7.244477494 6.182690459 6.182690457 5.829395378 5.829395376

80 5.646289700 5.646289700 4.793307820 4.793307818 4.499427073 4.499427071

90 4.456834255 4.456834255 3.755614629 3.755614628 3.507422663 3.507422662

100 3.550102766 3.550102766 2.966074952 2.966074951 2.755126886 2.755126886

Ta ble 8. Time to first peak and peak den sity

r[$]tp [s] Np

BEFD MATLAB BEFD MATLAB

0.1 1.089694E+02 1.089694E+02 2.110490E+00 2.110490E+00

0.2 9.236818E+01 9.236818E+01 5.699818E+00 5.699818E+00

0.5 2.829469E+01 2.829469E+01 4.575243E+01 4.575243E+01

1.0 9.534776E-01 9.534770E-01 8.078681E+02 8.078681E+02

1.2 3.165970E-01 3.165970E-01 8.021025E+03 8.021025E+03

1.5 1.682894E-01 1.682890E-01 4.302461E+04 4.302460E+04

2.0 9.839055E-02 9.839040E-02 1.678457E+05 1.678457E+05

Fig ure 2. Neu tron den sity vari a tion for step re ac tiv ityin ser tions with feed back

sated re sponse. This is a fa mous ex am ple of com pen -sated re ac tor shut down as dem on strated by Keepin [3,28]. In this case a tran sient is ini ti ated by ramp re ac tiv -ity in ser tion with a ramp rate “a” where the re ac tiv itycom pen sa tion orig i nates from strong Dopp ler feed -back and ther mal ex pan sion. Us ing the ki net ics datafor Re ac tor-III, com pre hen sive feed back of a com pen -sated re ac tor shut down is ana lysed. The tran sient isini ti ated by ramp re ac tiv ity in ser tion with a ramp rateof “a” and strong Dopp ler shut down “B”. The peakneu tron den sity and time to reach the peak neu tronden sity is com pared in tab. 9. The neu tron den sity pro -files for dif fer ent ramp rates and re ac tiv ity feed backco ef fi cients are shown in fig. 3 that per fectly matchwith the same ob tained by the BEFD scheme. For de -ter min ing the time to peak den sity ac cu rately, the max -i mum time step con sid ered in the cal cu la tion is 10–6 s.While ana lys ing this bench mark Ganapol has pointedout the poor agree ments be tween the BEFD re sults and that ob tained by the PWS method in the work ofAboanber and Ha ma da [27]. As com pared to the PWSmethod, a fair agree ment of the BEFD and MATLABre sults are ob served in this work.

CON CLU SIONS

Al though a pleth ora of nu mer i cal meth ods havebeen in tro duced to solve the point ki net ics equa tions

over the years, some of the sim ple and straight for wardmeth ods still work very ef fi ciently with ex traor di naryac cu racy. As an ex am ple, it has been shown re centlythat the fun da men tal BEFD al go rithm with its sim plic -ity has proven to be one of the most ef fec tive leg acymeth ods. Com ple ment ing the BEFD scheme, in thepres ent work ad vanced or di nary dif fer en tial equa tionsolv ers based on the NDF avail able in the MATLABsoft ware pack age are ap plied to solve the stiff re ac torpoint ki net ics equa tions with New to nian tem per a turefeed back ef fects. Five clas si cal bench mark cases in -volv ing step, ramp and si nu soi dal re ac tiv ity in ser tionhave been an a lyzed and the re sults are com pared withthe ear lier pub lished BEFD re sults. Fair ac cu racy ofthe re sults im plies the ef fi cient ap pli ca tion ofMATLAB ODE suite for solv ing the re ac tor point ki -net ics equa tions as an al ter nate method for fu ture ap -pli ca tions.

AC KNOWL EDGE MENT

The au thors greatly ac knowl edge the keen in ter -est shown by Mr. V. Balasubramaniyan, Di rec tor,Safety Re search In sti tute for pub li ca tion of this work.He has also whole heart edly sup ported the au thors intheir endeavour.

AU THOR CON TRI BU TIONS

The re search idea was orig i nated by D.Mohapatra. The com pu ta tional part was car ried out byS. S. Singh. Both the au thors to gether dis cussed andan a lyzed the re sults pre sented in the pa per. The manu -script was writ ten by D. Mohapatra and the fig ureswere pre pared by S. S. Singh.

REF ER ENCES

[1] Ganapol, B. D., A Highly Ac cu rate Al go rithm for theSo lu tion of the Point Ki net ics Equa tions, An nals ofNu clear En ergy, 62 (2013), 12, pp. 564-571

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S. S. Singh, et al.: So lu tion of the Re ac tor Point Ki net ics Equa tions by ....Nu clear Tech nol ogy & Ra di a tion Pro tec tion: Year 2015, Vol. 30, No. 1, pp. 11-17 15

Ta ble 9. Time to peak and peak neu tron den sity for ramp ini ti ated adi a batic com pen sa tion with Doppler

a [s–1] B [cm3s–1]tpeak [s] Npeak

BEFD MATLAB BEFD MATLAB

0.110–11 2.2466344E-01 2.2466300E-01 2.4203815E+11 2.4203803E+11

10–13 2.3890693E-01 2.3890700E-01 2.8986741E+13 2.8986728E+13

0.0110–11 1.1060774E+00 1.1060769E+00 2.0123519E+10 2.0123513E+10

10–13 1.1551476E+00 1.1551479E+00 2.4911782E+12 2.4911777E+12

0.00310–11 2.9105821E+00 2.9105820E+00 5.1141599E+09 5.1141589E+09

10–13 3.0076015E+00 3.0076010E+00 6.5344738E+11 6.5344729E+11

0.00110–11 7.4887663E+00 7.4887660E+00 1.2740752E+09 1.2740750E+09

10–13 7.6835876E+00 7.6835880E+00 1.7210080E+11 1.7210078E+11

Fig ure 3. Com pen sated re ac tor shut down by var i ousramp re ac tiv ity

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Re ceived on April 28, 2014Ac cepted on Feb ru ary 10, 2015

S. S. Singh, et al.: So lu tion of the Re ac tor Point Ki net ics Equa tions by ...16 Nu clear Tech nol ogy & Ra di a tion Pro tec tion: Year 2015, Vol. 30, No. 1, pp. 11-17

S. S. Singh, et al.: So lu tion of the Re ac tor Point Ki net ics Equa tions by ....Nu clear Tech nol ogy & Ra di a tion Pro tec tion: Year 2015, Vol. 30, No. 1, pp. 11-17 17

Sudhansu S. SING, Dinakru{na K. MOHAPATRA

RE[AVAWE JEDNA^INA TA^KASTE KINETIKEREAKTORA SOFTVERSKIM PAKETOM MATLAB

Numeri~ko re{avawe jedna~ina ta~kaste kinetike sa wutnovskom toplotnom povratnomspregom izazovno je pitawe u analizi reaktorskih prelaznih stawa. Jedna~ine ta~kaste kinetikereaktora predstavqaju sistem strogih ordinarnih diferencijalnih jedna~ina koje zahtevajuposebne numeri~ke postupke. Mada je tokom godina uvedeno obiqe slo`enih numeri~kih postupakaradi re{avawa jedna~ina ta~kaste kinetike, jo{ uvek su neke jednostavne i direktne metodedelotvorne sa izuzetnom ta~no{}u. Na primer, u skorije vreme pokazano je da jednostavanfundamentalni Ojlerov algoritam kona~nih razlika predstavqa jedan od najefikasnijihnasle|enih metoda. Dopuwuju}i ovaj algoritam, u ovom radu se analizom razli~itih klasi~nih testslu~ajeva demonstrira izuzetna primenqivost pro ce dure za obi~ne diferencijalne jedna~ine(ODE Suite), raspolo`ive u softverskom paketu MATLAB, na re{avawe strogih jedna~ina ta~kastekinetike reaktora sa uticajima wutnovske toplotne povratne sprege. Prili~na ta~nost rezultataukazuje na efikasnu upotrebu MATLAB paketa za re{avawe jedna~ina ta~kaste kinetike reaktorakao alternativne metode za budu}e primene.

Kqu~ne re~i: jedna~ina ta~kaste kinetike, Ojlerov algoritam kona~nih razlika, MATLAB,.........................numeri~ka diferencijacija