research article solution of point reactor neutron...
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Hindawi Publishing CorporationScience and Technology of Nuclear InstallationsVolume 2013 Article ID 261327 6 pageshttpdxdoiorg1011552013261327
Research ArticleSolution of Point Reactor Neutron Kinetics Equations withTemperature Feedback by Singularly Perturbed Method
Wenzhen Chen Jianli Hao Ling Chen and Haofeng Li
Department of Nuclear Energy Science and Engineering Naval University of Engineering Faculty 301 Wuhan 430033 China
Correspondence should be addressed to Wenzhen Chen cwz221cncom
Received 30 May 2013 Revised 27 August 2013 Accepted 29 August 2013
Academic Editor Arkady Serikov
Copyright copy 2013 Wenzhen Chen et alThis 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
The singularly perturbedmethod (SPM) is proposed to obtain the analytical solution for the delayed supercritical process of nuclearreactor with temperature feedback and small step reactivity inserted The relation between the reactivity and time is derived Alsothe neutron density (or power) and the average density of delayed neutron precursors as the function of reactivity are presentedThe variations of neutron density (or power) and temperature with time are calculated and plotted and compared with those byaccurate solution and other analytical methods It is shown that the results by the SPM are valid and accurate in the large range andthe SPM is simpler than those in the previous literature
1 Introduction
The analysis of variation of neutron density (or power) andreactivity with time under the different conditions is animportant content of nuclear reactor physics or neutronkinetics [1ndash7] Some important achievements on the super-critical transient with temperature feedback with big (120588
0gt 120573)
or small (1205880lt 120573) reactivity inserted have been approached
through the effort of many scholars [7ndash12] The studieson the delayed supercritical transient with small reactivityinserted and temperature feedback are introduced in therelated literature [13ndash15] in which the explicit function ofdensity (or power) and reactivity with respect to time isderived mainly with decoupling method power promptjump approximation precursor prompt jump approximationtemperature prompt jump approximation [10 16] and soforth From the detailed analysis and comparison of theresults in the early and recent literature [7 12 14] it is foundthat some results have certain limit and rather big error underthe particular conditions In present work the variation lawof power reactivity and precursor density with respect totime at any level of initial power is obtained by the singularlyperturbed method (SPM) All the results are compared withthose obtained by the numerical solution which tend to theaccurate solution under very small time step size [17] It is
proved that the SPM is correct and reliable and is simpler thanthe analytical methods by the related literature
2 Theoretical Derivation
The point reactor neutron kinetics equations with one groupof delayed neutrons are [3 4]
119889119899 (119905)
119889119905=120588 (119905) minus 120573
119897119899 (119905) + 120582119862 (119905) (1)
119889119862 (119905)
119889119905=120573
119897119899 (119905) minus 120582119862 (119905) (2)
where 119899 is the average neutron density 119905 is the time 120588 isthe reactivity 120573 is the total fraction of the delayed neutron119897 is the prompt neutron lifetime 120582 is the radioactive decayconstant of delayed neutron precursor and 119862 is the averagedensity of delayed neutron precursor When multiplied witha certain coefficient 119899 represents the power It is assumed thatthe reactor has a negative temperature coefficient of reactivity120572 (120572 gt 0) when a small step reactivity 120588
0(lt120573) is insert-
ed Consider the temperature feedback and the real reactorreactivity is
120588 = 1205880minus 120572119879 (3)
2 Science and Technology of Nuclear Installations
where 119879 is the temperature increment of the reactor namely119879 = 119879
119904minus 1198790 where 119879
119904and 119879
0are the instantaneous
temperature and initial temperature respectively After thereactivity 120588
0is inserted into the reactor the adiabatic model is
still employed [3 15] then we have
119889119879
119889119905= 119870119888119899 (119905) (4)
where119870119888is the reciprocal of thermal capacity of reactor
Combining (3) and (4) results in
119889120588
119889119905= minus120572119870
119888119899 (119905) (5)
Substituting (2) into the derivative of (1) with respect to 119905yields
1198971198892119899
1198891199052= (120588 minus 120573)
119889119899
119889119905+ 119899119889120588
119889119905+ 120582120573119899 minus 119897120582
2119862 (6)
Substituting 120582119862 obtained from (1) into (6) and simplify-ing it yields
1198892119899
1198891199052= (120588 minus 120573
119897minus 120582)119889119899
119889119905+119899
119897
119889120588
119889119905+120582120588119899
119897 (7)
The transient process is supposed to begin at 119905 = 0 and120588(0) = 0 [15 17] so the initial conditions can be given as120588(0) = 120588
0 119899(0) = 119899
0 (119889119899119889119905)|
119905=0= 12058801198990119897 (11988921198991198891199052)|
119905=0=
((1205880minus120573)119897)(120588
01198990119897)minus(120572119870
1198881198992
0119897) where 119899
0is the initial neutron
density (or power)For |(120588 minus 120573)119897| ≫ 120582 120573119897 ≫ 120582 [3] (1) and (2) are
stiff equations According to the singularly perturbedmethod[18] solution of 119899(119905) includes the inner solution 119899
119891(119905) in inner
part and the outer solution 119899119897(119905) in outer part In this paper
both inner and outer solutions are approximated to be zeroorder
119899 (119905) = 119899119891(119905) + 119899
119897(119905) (8)
The initial conditions are
1198990= 1198991198910+ 1198991198970 (9)
1198991015840(0) = 119899
1015840
119891(0) + 119899
1015840
119897(0) =12058801198990
119897 (10)
11989910158401015840(0) = 119899
10158401015840
119891(0) + 119899
10158401015840
119897(0) =1205880minus 120573
119897
12058801198990
119897minus1205721198701198881198992
0
119897 (11)
where 1198991198910
is the initial value of inner solution and 1198991198970is the
initial value of outer solutionSubstituting (8) into (7) and (5) respectively yields
1198892(119899119891+ 119899119897)
1198891199052= (120588 minus 120573
119897minus 120582)119889 (119899119891+ 119899119897)
119889119905+(119899119891+ 119899119897)
119897
119889120588
119889119905
+120582120588 (119899119891+ 119899119897)
119897
119889120588
119889119905= minus120572119870
119888[119899119891(119905) + 119899
119897(119905)]
(12)
In the outer part the inner solution attenuates to zero and119889119899119891119889119905 and 1198892119899
1198911198891199052 can be neglected namely 119889119899
119891119889119905 asymp 0
and 11988921198991198911198891199052asymp 0 therefore in the outer part (12) can be
simplified as follows
1198892119899119897
1198891199052=120588 minus 120573
119897
119889119899119897
119889119905+119899119897
119897
119889120588
119889119905minus 120582119889119899119897
119889119905+120582120588119899119897
119897 (13)
119889120588
119889119905= minus120572119870
119888119899119897 (14)
Because 119899119897(119905) varies slowly and 119897 asymp 10minus4 s compared to
other terms the term on the left side of (13) can be neglectedand then we have
(120573 minus 120588 + 120582119897)119889119899119897
119889119905= (119889120588
119889119905+ 120582120588) 119899
119897 (15)
Combining (14) and (15) results in
119889119899119897
119889120588=120572119870119888119899119897minus 120582120588
120572119870119888(120573 minus 120588 + 120582119897)
(16)
Integrating (16) subjective to the initial conditions 120588(0) =1205880and 119899119897(0) = 119899
1198970yields
119899119897=120582 (1205882
0minus 1205882)
2120572119870119888(120573 minus 120588 + 120582119897)
+(120573 minus 120588
0+ 120582119897)
(120573 minus 120588 + 120582119897)1198991198970 (17)
Substituting (17) into (14) leads to
minus119889120588
119889119905=120582 (1205882
0minus 1205882) + 2120572119870
119888(120573 minus 120588
0+ 120582119897) 119899
1198970
2 (120573 minus 120588 + 120582119897) (18)
With the initial conditions 120588(0) = 1205880 the solution of (18)
is
119905 =120573 + 120582119897
1205821205881
ln(1205881minus 120588
1205881+ 120588)(1205881+ 1205880
1205881minus 1205880
) +1
120582ln[(1205882
1minus 1205882
0)
(12058821minus 1205882)]
(19)
where 1205881= radic12058820+ 2120572119870
119888(120573 minus 120588
0+ 120582119897)119899
1198970120582
In the inner part 119899119897(119905) is assumed to be constant 119899
119897(119905) =
119899119897(0) = 119899
1198970 and 119889119899
119897(119905)119889119905 asymp (119889119899
119897(119905)119889119905)|
119905=0 1198892119899119897(119905)119889119905
2asymp
(1198892119899119897(119905)119889119905
2)|119905=0
so (12) can be simplified as follows respec-tively
1198892119899119891
1198891199052+ 11989910158401015840
119897(0) = (
120588 minus 120573
119897minus 120582)(119889119899119891
119889119905+ 1198991015840
119897(0))
+(119899119891+ 1198991198970)
119897(119889120588
119889119905+ 120582120588)
(20)
119889120588
119889119905= minus120572119870
119888[119899119891(119905) + 119899
1198970] (21)
Science and Technology of Nuclear Installations 3
In addition the temperature feedback is not fast enoughto affect the reactivity and in the inner part it is assumed that120588(119905) = 120588
0 119889120588(119905)119889119905 asymp minus120572119870
1198881198990 From (20) we can get
1198971198892119899119891
1198891199052= (1205880minus 120573 minus 120582119897)
119889119899119891
119889119905+ (1205821205880minus 1205721198701198881198990) (119899119891+ 1198991198970)
+ (1205880minus 120573 minus 119897120582) 119899
1015840
119897(0) minus 119899
10158401015840
119897(0) 119897
(22)
The solution of (22) is
119899119891= minus1198991198970minus(1205880minus 120573 minus 119897120582) 119899
1015840
119897(0) minus 119899
10158401015840
119897(0) 119897
(1205821205880minus 1205721198701198881198990)
+ 1198621exp ( 12119897
times ((1205880minus 120573 minus 119897120582)
+radic(1205880minus 120573 minus 119897120582)
2+4119897 (120582120588
0minus 1205721198701198881198990)) 119905)
+ 1198622exp ( 12119897
times ((1205880minus 120573 minus 119897120582)
minusradic(1205880minus 120573 minus 119897120582)
2+4119897 (120582120588
0minus 1205721198701198881198990)) 119905)
(23)
The fast varying part 119899119891(119905) is assumed to attenuate to zero
in the inner part so 1198621(119905) = 0 and
minus1198991198970minus(1205880minus 120573 minus 119897120582) 119899
1015840
119897(0) minus 119899
10158401015840
119897(0) 119897
(1205821205880minus 1205721198701198881198990)
= 0 (24)
Then we can get the fast varying part 119899119891(119905) in the inner
part as follows
119899119891= 1198991198910
exp ( 12119897
times ((1205880minus 120573 minus 119897120582)
minusradic(1205880minus 120573 minus 119897120582)
2+ 4119897 (120582120588
0minus 1205721198701198881198990)) 119905)
(25)
From (25) we have
1198991015840
1198910=1198991198910
2119897((1205880minus 120573 minus 119897120582)
minusradic(1205880minus 120573 minus 119897120582)
2+ 4119897 (120582120588
0minus 1205721198701198881198990))
(26)
Combining (9) (11) (13) (24) and (26) results in
1198991198910=
2119897 (1205721198701198881198990minus 1205821205880) 1198990(1205880minus 120573 minus 119897120582)
2minus 212058801198990 (1205880minus 120573 minus 119897120582)
2119897 (1205721198701198881198990minus 1205821205880) (1205880minus 120573 minus 119897120582)
2minus (1 minus radic1 + 4119897 (120582120588
0minus 1205721198701198881198990) (1205880minus 120573 minus 119897120582)
2)
1198991198970=
212058801198990 (1205880minus 120573 minus 119897120582) minus 119899
0(1 minus radic1 + 4119897 (120582120588
0minus 1205721198701198881198990) (1205880minus 120573 minus 119897120582)
2)
2119897 (1205721198701198881198990minus 1205821205880) (1205880minus 120573 minus 119897120582)
2minus (1 minus radic1 + 4119897 (120582120588
0minus 1205721198701198881198990) (1205880minus 120573 minus 119897120582)
2)
(27)
Substituting (27) into (25) and (17) can get 119899119891(119905) and 119899
119897(119905)
then the neutron density (or power) will be obtained by (8)Combining (1) and (2) results in
119889 (119899 + 119862)
119889119905=120588
119897119899 (28)
Eliminating the time variable in (5) and (28) leads to
119889 (119899 + 119862)
119889120588= minus120588
120572119870119888119897 (29)
Integrating (29) with the initial conditions 119899(0) = 1198990
119862(0) = 1205731198990120582119897 and 120588(0) = 120588
0yields
119862 (119905) =1205882
0minus 1205882
2120572119870119888119897+ (1 +120573
120582119897) 1198990minus 119899 (119905) (30)
Equations (8) (17) (25) (27) and (30) are the newanalytical expressions derived by this paper
3 Calculation and Analysis
The PWR with fuel 235U is taken as an example withparameters 120573 = 00065 119897 = 00001 s 120582 = 00774 1s 119870
119888=
005KMWsdots and 120572 = 5 times 10minus5 1K [8 13] For the reactorwith the initial power 1MW while reactivity 120588
0= 05120573
and 1205880= 08333120573 is inserted respectively the variations
of reactivity temperature power with time and power withreactivity are presented in Figures 1 2 3 and 4 The curveswith smaller change are for 120588
0= 05120573 and the curves with
larger change are for 1205880= 08333120573 The solid line notes
the accurate solution by the best basic function method with
4 Science and Technology of Nuclear Installations
SmPPPJ
Accurate solutionSPMTPJPrPJ
SmPPPJ
t (s)0 50 100 150 200
n(M
W)
100
80
60
40
20
0
Figure 1 Variation of output power with time while inserting stepreactivity 120588
0= 05120573 and 120588
0= 08333120573
Accurate solutionSPMTPJPrPJ
SmPPPJ
t (s)0 50 100 150 200
times10minus3
6
4
2
0
minus2
minus4
minus6
120588
Figure 2 Variation of total reactivity with time while inserting stepreactivity 120588
0= 05120573 and 120588
0= 08333120573
Accurate solutionSPMTPJPrPJ
t (s)0 50 100 150 200
250
200
150
100
50
0
ΔT
(K)
SmPPPJ
Figure 3 Variation of temperature rise of reactor with time whileinserting step reactivity 120588
0= 05120573 and 120588
0= 08333120573
SmPPPJ
times10minus36420minus2minus4minus6
120588
n(M
W)
100
80
60
40
20
0
SmPPPJ
Accurate solutionSPMTPJPrPJ
Figure 4 Variation of output power with reactivity while insertingstep reactivity 120588
0= 05120573 and 120588
0= 08333120573
200
150
100
50
0
n(M
W)
0 20 40 60t (s)
TPJPPJSPM
Accurate solutionPrPJ
Figure 5 Variation of output power with time while inserting stepreactivity 0995120573
very small step size [17] The results of this paper and theaccurate solution as well as the temperature prompt jump(TPJ) method in the literature [10] are almost the same andhard to distinguish in Figures 1ndash4 The short dashed-dot lineand long dashed-dot line represent the results of precursorprompt jump (PrPJ)method in the literature [9] and the smallparameter (SmP) method in the literature [15] respectivelyThe dashed line notes the results of power prompt jump (PPJ)method in the literature [16] The difference is caused by theapproximate treatment to obey the methods of PrPJ SmPPPJ and SmP Furthermore the variation in the vicinity ofprompt supercritical process is also calculated and is shownin Figures 5 and 6 The correct results cannot be obtained bythe small parameter method (SmP) in the vicinity of promptsupercritical process and are not shown in Figures 5 and 6
Science and Technology of Nuclear Installations 5
0 20 40 60t (s)
TPJPPJSPM
Accurate solutionPrPJ
350
300
250
200
150
100
50
0
n(M
W)
Figure 6 Variation of output power with time while inserting stepreactivity 0998120573
(1) From Figures 1ndash6 it can be concluded that very goodresults cannot be obtained by the precursor prompt jump(PrPJ) method to calculate the delayed supercritical progresswith small step reactivity and temperature feedback(2) For small step reactivity the results by the small
parameter (SmP) method are close to those by the powerprompt jump (PPJ) method and are better than those by theprecursor prompt jump (PrPJ) method but the accuracy ofresults by the small parameter method decreases with theincrease of the reactivity insertedThepower is negativewhenthe small parameter method is used to calculate the transientprocess in the vicinity of prompt supercritical state FromFigures 1ndash4 it can be seen that the small parameter method ismore suitable for the calculation of reactivity and temperatureincrease than for that of power(3)The results are quite precise using the power prompt
jump (PPJ) method for the delayed supercritical process butthe main problem compared to the accurate solution is thatsome displacement exists along time axis Furthermore itshould be pointed out that each power peak value obtainedby the precursor prompt jump (PrPJ) method power promptjump (PPJ) method or small parameter (SmP) methodis lower than that obtained by the accurate solution orsingularly perturbed method (SPM) see Figure 1(4) From Figures 1ndash4 it can be also found that the
temperature prompt jump method (TPJ) and the singularlyperturbed method (SPM) in this paper are the two mostprecise methods for the delayed supercritical process withsmall step reactivity and temperature feedback Howeverfrom Figures 5 and 6 it can be seen that as the reactivityinserted increases to the vicinity of prompt supercriticalprocess the total discrepancy of power by the TPJ method islarger than that by the SPM or PPJmethod and the irrelevantphenomena that the power jumps at first and then decreasesmonotonously from the peak will appear in the TPJ methodas shown in Figure 6
4 Conclusions
The analytical expressions of power (or neutron density)reactivity the precursor power (or density) and temperatureincrease with respect to time are derived for the delayedsupercritical process with small reactivity (120588
0lt 120573) and
temperature feedback by the singularly perturbed methodCompared with the results by the accurate solution and othermethods in the literature it is shown that the singularlyperturbedmethod (SPM) in this paper is valid and accurate inthe large range and is simpler than those in the previous liter-atureThemethod in this paper can provide a new theoreticalfoundation for the analysis of reactor neutron dynamics
Acknowledgments
This research is supported by the National Natural ScienceFoundation of China (Project no 11301540) and the NaturalScience Foundation of Naval University of Engineering
References
[1] H P Gupta and M S Trasi ldquoAsymptotically stable solutions ofpoint-reactor kinetics equations in the presence of Newtoniantemperature feedbackrdquo Annals of Nuclear Energy vol 13 no 4pp 203ndash207 1986
[2] H Van Dam ldquoDynamics of passive reactor shutdownrdquo Progressin Nuclear Energy vol 30 no 3 pp 255ndash264 1996
[3] D L HetrickDynamics of Nuclear Reactors American NuclearSociety La Grange Park Ill USA 1993
[4] WM StaceyNuclear Reactors PhysicsWiley-Interscience NewYork NY USA 2001
[5] A A Nahla and EM E Zayed ldquoSolution of the nonlinear pointnuclear reactor kinetics equationsrdquo Progress in Nuclear Energyvol 52 no 8 pp 743ndash746 2010
[6] G Espinosa-Paredes M-A Polo-Labarrios E-G Espinosa-Martınez and E D Valle-Gallegos ldquoFractional neutron pointkinetics equations for nuclear reactor dynamicsrdquo Annals ofNuclear Energy vol 38 no 2-3 pp 307ndash330 2011
[7] A E Aboanber and A A Nahla ldquoSolution of the point kineticsequations in the presence of Newtonian temperature feedbackby Pade approximations via the analytical inversion methodrdquoJournal of Physics A vol 35 no 45 pp 9609ndash9627 2002
[8] A E Aboanber and Y M Hamada ldquoPower series solution(PWS) of nuclear reactor dynamics with newtonian tempera-ture feedbackrdquoAnnals of Nuclear Energy vol 30 no 10 pp 1111ndash1122 2003
[9] W Z Chen B Zhu and H F Li ldquoThe analytical solutionof point-reactor neutron-kinetics equation with small stepreactivityrdquo Acta Physica Sinica vol 50 no 8 pp 2486ndash24892004 (Chinese)
[10] H Li W Chen F Zhang and L Luo ldquoApproximate solutionsof point kinetics equations with one delayed neutron groupand temperature feedback during delayed supercritical processrdquoAnnals of Nuclear Energy vol 34 no 6 pp 521ndash526 2007
[11] T Sathiyasheela ldquoPower series solution method for solvingpoint kinetics equations with lumped model temperature andfeedbackrdquo Annals of Nuclear Energy vol 36 no 2 pp 246ndash2502009
6 Science and Technology of Nuclear Installations
[12] YM Hamada ldquoConfirmation of accuracy of generalized powerseries method for the solution of point kinetics equations withfeedbackrdquo Annals of Nuclear Energy vol 55 pp 184ndash193 2013
[13] Q Zhu X-L Shang and W-Z Chen ldquoHomotopy analysissolution of point reactor kinetics equations with six-groupdelayed neutronsrdquo Acta Physica Sinica vol 61 no 7 Article ID070201 2012
[14] S D Hamieh and M Saidinezhad ldquoAnalytical solution of thepoint reactor kinetics equations with temperature feedbackrdquoAnnals of Nuclear Energy vol 42 pp 148ndash152 2012
[15] A A Nahla ldquoAn analytical solution for the point reactorkinetics equations with one group of delayed neutrons and theadiabatic feedback modelrdquo Progress in Nuclear Energy vol 51no 1 pp 124ndash128 2009
[16] W Chen L Guo B Zhu and H Li ldquoAccuracy of analyticalmethods for obtaining supercritical transients with temperaturefeedbackrdquoProgress inNuclear Energy vol 49 no 4 pp 290ndash3022007
[17] H Li W Chen L Luo and Q Zhu ldquoA new integral method forsolving the point reactor neutron kinetics equationsrdquo Annals ofNuclear Energy vol 36 no 4 pp 427ndash432 2009
[18] Z QHuangKinetics Base of Nuclear Reactor PekingUniversityPress Beijing China 2007 (Chinese)
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2 Science and Technology of Nuclear Installations
where 119879 is the temperature increment of the reactor namely119879 = 119879
119904minus 1198790 where 119879
119904and 119879
0are the instantaneous
temperature and initial temperature respectively After thereactivity 120588
0is inserted into the reactor the adiabatic model is
still employed [3 15] then we have
119889119879
119889119905= 119870119888119899 (119905) (4)
where119870119888is the reciprocal of thermal capacity of reactor
Combining (3) and (4) results in
119889120588
119889119905= minus120572119870
119888119899 (119905) (5)
Substituting (2) into the derivative of (1) with respect to 119905yields
1198971198892119899
1198891199052= (120588 minus 120573)
119889119899
119889119905+ 119899119889120588
119889119905+ 120582120573119899 minus 119897120582
2119862 (6)
Substituting 120582119862 obtained from (1) into (6) and simplify-ing it yields
1198892119899
1198891199052= (120588 minus 120573
119897minus 120582)119889119899
119889119905+119899
119897
119889120588
119889119905+120582120588119899
119897 (7)
The transient process is supposed to begin at 119905 = 0 and120588(0) = 0 [15 17] so the initial conditions can be given as120588(0) = 120588
0 119899(0) = 119899
0 (119889119899119889119905)|
119905=0= 12058801198990119897 (11988921198991198891199052)|
119905=0=
((1205880minus120573)119897)(120588
01198990119897)minus(120572119870
1198881198992
0119897) where 119899
0is the initial neutron
density (or power)For |(120588 minus 120573)119897| ≫ 120582 120573119897 ≫ 120582 [3] (1) and (2) are
stiff equations According to the singularly perturbedmethod[18] solution of 119899(119905) includes the inner solution 119899
119891(119905) in inner
part and the outer solution 119899119897(119905) in outer part In this paper
both inner and outer solutions are approximated to be zeroorder
119899 (119905) = 119899119891(119905) + 119899
119897(119905) (8)
The initial conditions are
1198990= 1198991198910+ 1198991198970 (9)
1198991015840(0) = 119899
1015840
119891(0) + 119899
1015840
119897(0) =12058801198990
119897 (10)
11989910158401015840(0) = 119899
10158401015840
119891(0) + 119899
10158401015840
119897(0) =1205880minus 120573
119897
12058801198990
119897minus1205721198701198881198992
0
119897 (11)
where 1198991198910
is the initial value of inner solution and 1198991198970is the
initial value of outer solutionSubstituting (8) into (7) and (5) respectively yields
1198892(119899119891+ 119899119897)
1198891199052= (120588 minus 120573
119897minus 120582)119889 (119899119891+ 119899119897)
119889119905+(119899119891+ 119899119897)
119897
119889120588
119889119905
+120582120588 (119899119891+ 119899119897)
119897
119889120588
119889119905= minus120572119870
119888[119899119891(119905) + 119899
119897(119905)]
(12)
In the outer part the inner solution attenuates to zero and119889119899119891119889119905 and 1198892119899
1198911198891199052 can be neglected namely 119889119899
119891119889119905 asymp 0
and 11988921198991198911198891199052asymp 0 therefore in the outer part (12) can be
simplified as follows
1198892119899119897
1198891199052=120588 minus 120573
119897
119889119899119897
119889119905+119899119897
119897
119889120588
119889119905minus 120582119889119899119897
119889119905+120582120588119899119897
119897 (13)
119889120588
119889119905= minus120572119870
119888119899119897 (14)
Because 119899119897(119905) varies slowly and 119897 asymp 10minus4 s compared to
other terms the term on the left side of (13) can be neglectedand then we have
(120573 minus 120588 + 120582119897)119889119899119897
119889119905= (119889120588
119889119905+ 120582120588) 119899
119897 (15)
Combining (14) and (15) results in
119889119899119897
119889120588=120572119870119888119899119897minus 120582120588
120572119870119888(120573 minus 120588 + 120582119897)
(16)
Integrating (16) subjective to the initial conditions 120588(0) =1205880and 119899119897(0) = 119899
1198970yields
119899119897=120582 (1205882
0minus 1205882)
2120572119870119888(120573 minus 120588 + 120582119897)
+(120573 minus 120588
0+ 120582119897)
(120573 minus 120588 + 120582119897)1198991198970 (17)
Substituting (17) into (14) leads to
minus119889120588
119889119905=120582 (1205882
0minus 1205882) + 2120572119870
119888(120573 minus 120588
0+ 120582119897) 119899
1198970
2 (120573 minus 120588 + 120582119897) (18)
With the initial conditions 120588(0) = 1205880 the solution of (18)
is
119905 =120573 + 120582119897
1205821205881
ln(1205881minus 120588
1205881+ 120588)(1205881+ 1205880
1205881minus 1205880
) +1
120582ln[(1205882
1minus 1205882
0)
(12058821minus 1205882)]
(19)
where 1205881= radic12058820+ 2120572119870
119888(120573 minus 120588
0+ 120582119897)119899
1198970120582
In the inner part 119899119897(119905) is assumed to be constant 119899
119897(119905) =
119899119897(0) = 119899
1198970 and 119889119899
119897(119905)119889119905 asymp (119889119899
119897(119905)119889119905)|
119905=0 1198892119899119897(119905)119889119905
2asymp
(1198892119899119897(119905)119889119905
2)|119905=0
so (12) can be simplified as follows respec-tively
1198892119899119891
1198891199052+ 11989910158401015840
119897(0) = (
120588 minus 120573
119897minus 120582)(119889119899119891
119889119905+ 1198991015840
119897(0))
+(119899119891+ 1198991198970)
119897(119889120588
119889119905+ 120582120588)
(20)
119889120588
119889119905= minus120572119870
119888[119899119891(119905) + 119899
1198970] (21)
Science and Technology of Nuclear Installations 3
In addition the temperature feedback is not fast enoughto affect the reactivity and in the inner part it is assumed that120588(119905) = 120588
0 119889120588(119905)119889119905 asymp minus120572119870
1198881198990 From (20) we can get
1198971198892119899119891
1198891199052= (1205880minus 120573 minus 120582119897)
119889119899119891
119889119905+ (1205821205880minus 1205721198701198881198990) (119899119891+ 1198991198970)
+ (1205880minus 120573 minus 119897120582) 119899
1015840
119897(0) minus 119899
10158401015840
119897(0) 119897
(22)
The solution of (22) is
119899119891= minus1198991198970minus(1205880minus 120573 minus 119897120582) 119899
1015840
119897(0) minus 119899
10158401015840
119897(0) 119897
(1205821205880minus 1205721198701198881198990)
+ 1198621exp ( 12119897
times ((1205880minus 120573 minus 119897120582)
+radic(1205880minus 120573 minus 119897120582)
2+4119897 (120582120588
0minus 1205721198701198881198990)) 119905)
+ 1198622exp ( 12119897
times ((1205880minus 120573 minus 119897120582)
minusradic(1205880minus 120573 minus 119897120582)
2+4119897 (120582120588
0minus 1205721198701198881198990)) 119905)
(23)
The fast varying part 119899119891(119905) is assumed to attenuate to zero
in the inner part so 1198621(119905) = 0 and
minus1198991198970minus(1205880minus 120573 minus 119897120582) 119899
1015840
119897(0) minus 119899
10158401015840
119897(0) 119897
(1205821205880minus 1205721198701198881198990)
= 0 (24)
Then we can get the fast varying part 119899119891(119905) in the inner
part as follows
119899119891= 1198991198910
exp ( 12119897
times ((1205880minus 120573 minus 119897120582)
minusradic(1205880minus 120573 minus 119897120582)
2+ 4119897 (120582120588
0minus 1205721198701198881198990)) 119905)
(25)
From (25) we have
1198991015840
1198910=1198991198910
2119897((1205880minus 120573 minus 119897120582)
minusradic(1205880minus 120573 minus 119897120582)
2+ 4119897 (120582120588
0minus 1205721198701198881198990))
(26)
Combining (9) (11) (13) (24) and (26) results in
1198991198910=
2119897 (1205721198701198881198990minus 1205821205880) 1198990(1205880minus 120573 minus 119897120582)
2minus 212058801198990 (1205880minus 120573 minus 119897120582)
2119897 (1205721198701198881198990minus 1205821205880) (1205880minus 120573 minus 119897120582)
2minus (1 minus radic1 + 4119897 (120582120588
0minus 1205721198701198881198990) (1205880minus 120573 minus 119897120582)
2)
1198991198970=
212058801198990 (1205880minus 120573 minus 119897120582) minus 119899
0(1 minus radic1 + 4119897 (120582120588
0minus 1205721198701198881198990) (1205880minus 120573 minus 119897120582)
2)
2119897 (1205721198701198881198990minus 1205821205880) (1205880minus 120573 minus 119897120582)
2minus (1 minus radic1 + 4119897 (120582120588
0minus 1205721198701198881198990) (1205880minus 120573 minus 119897120582)
2)
(27)
Substituting (27) into (25) and (17) can get 119899119891(119905) and 119899
119897(119905)
then the neutron density (or power) will be obtained by (8)Combining (1) and (2) results in
119889 (119899 + 119862)
119889119905=120588
119897119899 (28)
Eliminating the time variable in (5) and (28) leads to
119889 (119899 + 119862)
119889120588= minus120588
120572119870119888119897 (29)
Integrating (29) with the initial conditions 119899(0) = 1198990
119862(0) = 1205731198990120582119897 and 120588(0) = 120588
0yields
119862 (119905) =1205882
0minus 1205882
2120572119870119888119897+ (1 +120573
120582119897) 1198990minus 119899 (119905) (30)
Equations (8) (17) (25) (27) and (30) are the newanalytical expressions derived by this paper
3 Calculation and Analysis
The PWR with fuel 235U is taken as an example withparameters 120573 = 00065 119897 = 00001 s 120582 = 00774 1s 119870
119888=
005KMWsdots and 120572 = 5 times 10minus5 1K [8 13] For the reactorwith the initial power 1MW while reactivity 120588
0= 05120573
and 1205880= 08333120573 is inserted respectively the variations
of reactivity temperature power with time and power withreactivity are presented in Figures 1 2 3 and 4 The curveswith smaller change are for 120588
0= 05120573 and the curves with
larger change are for 1205880= 08333120573 The solid line notes
the accurate solution by the best basic function method with
4 Science and Technology of Nuclear Installations
SmPPPJ
Accurate solutionSPMTPJPrPJ
SmPPPJ
t (s)0 50 100 150 200
n(M
W)
100
80
60
40
20
0
Figure 1 Variation of output power with time while inserting stepreactivity 120588
0= 05120573 and 120588
0= 08333120573
Accurate solutionSPMTPJPrPJ
SmPPPJ
t (s)0 50 100 150 200
times10minus3
6
4
2
0
minus2
minus4
minus6
120588
Figure 2 Variation of total reactivity with time while inserting stepreactivity 120588
0= 05120573 and 120588
0= 08333120573
Accurate solutionSPMTPJPrPJ
t (s)0 50 100 150 200
250
200
150
100
50
0
ΔT
(K)
SmPPPJ
Figure 3 Variation of temperature rise of reactor with time whileinserting step reactivity 120588
0= 05120573 and 120588
0= 08333120573
SmPPPJ
times10minus36420minus2minus4minus6
120588
n(M
W)
100
80
60
40
20
0
SmPPPJ
Accurate solutionSPMTPJPrPJ
Figure 4 Variation of output power with reactivity while insertingstep reactivity 120588
0= 05120573 and 120588
0= 08333120573
200
150
100
50
0
n(M
W)
0 20 40 60t (s)
TPJPPJSPM
Accurate solutionPrPJ
Figure 5 Variation of output power with time while inserting stepreactivity 0995120573
very small step size [17] The results of this paper and theaccurate solution as well as the temperature prompt jump(TPJ) method in the literature [10] are almost the same andhard to distinguish in Figures 1ndash4 The short dashed-dot lineand long dashed-dot line represent the results of precursorprompt jump (PrPJ)method in the literature [9] and the smallparameter (SmP) method in the literature [15] respectivelyThe dashed line notes the results of power prompt jump (PPJ)method in the literature [16] The difference is caused by theapproximate treatment to obey the methods of PrPJ SmPPPJ and SmP Furthermore the variation in the vicinity ofprompt supercritical process is also calculated and is shownin Figures 5 and 6 The correct results cannot be obtained bythe small parameter method (SmP) in the vicinity of promptsupercritical process and are not shown in Figures 5 and 6
Science and Technology of Nuclear Installations 5
0 20 40 60t (s)
TPJPPJSPM
Accurate solutionPrPJ
350
300
250
200
150
100
50
0
n(M
W)
Figure 6 Variation of output power with time while inserting stepreactivity 0998120573
(1) From Figures 1ndash6 it can be concluded that very goodresults cannot be obtained by the precursor prompt jump(PrPJ) method to calculate the delayed supercritical progresswith small step reactivity and temperature feedback(2) For small step reactivity the results by the small
parameter (SmP) method are close to those by the powerprompt jump (PPJ) method and are better than those by theprecursor prompt jump (PrPJ) method but the accuracy ofresults by the small parameter method decreases with theincrease of the reactivity insertedThepower is negativewhenthe small parameter method is used to calculate the transientprocess in the vicinity of prompt supercritical state FromFigures 1ndash4 it can be seen that the small parameter method ismore suitable for the calculation of reactivity and temperatureincrease than for that of power(3)The results are quite precise using the power prompt
jump (PPJ) method for the delayed supercritical process butthe main problem compared to the accurate solution is thatsome displacement exists along time axis Furthermore itshould be pointed out that each power peak value obtainedby the precursor prompt jump (PrPJ) method power promptjump (PPJ) method or small parameter (SmP) methodis lower than that obtained by the accurate solution orsingularly perturbed method (SPM) see Figure 1(4) From Figures 1ndash4 it can be also found that the
temperature prompt jump method (TPJ) and the singularlyperturbed method (SPM) in this paper are the two mostprecise methods for the delayed supercritical process withsmall step reactivity and temperature feedback Howeverfrom Figures 5 and 6 it can be seen that as the reactivityinserted increases to the vicinity of prompt supercriticalprocess the total discrepancy of power by the TPJ method islarger than that by the SPM or PPJmethod and the irrelevantphenomena that the power jumps at first and then decreasesmonotonously from the peak will appear in the TPJ methodas shown in Figure 6
4 Conclusions
The analytical expressions of power (or neutron density)reactivity the precursor power (or density) and temperatureincrease with respect to time are derived for the delayedsupercritical process with small reactivity (120588
0lt 120573) and
temperature feedback by the singularly perturbed methodCompared with the results by the accurate solution and othermethods in the literature it is shown that the singularlyperturbedmethod (SPM) in this paper is valid and accurate inthe large range and is simpler than those in the previous liter-atureThemethod in this paper can provide a new theoreticalfoundation for the analysis of reactor neutron dynamics
Acknowledgments
This research is supported by the National Natural ScienceFoundation of China (Project no 11301540) and the NaturalScience Foundation of Naval University of Engineering
References
[1] H P Gupta and M S Trasi ldquoAsymptotically stable solutions ofpoint-reactor kinetics equations in the presence of Newtoniantemperature feedbackrdquo Annals of Nuclear Energy vol 13 no 4pp 203ndash207 1986
[2] H Van Dam ldquoDynamics of passive reactor shutdownrdquo Progressin Nuclear Energy vol 30 no 3 pp 255ndash264 1996
[3] D L HetrickDynamics of Nuclear Reactors American NuclearSociety La Grange Park Ill USA 1993
[4] WM StaceyNuclear Reactors PhysicsWiley-Interscience NewYork NY USA 2001
[5] A A Nahla and EM E Zayed ldquoSolution of the nonlinear pointnuclear reactor kinetics equationsrdquo Progress in Nuclear Energyvol 52 no 8 pp 743ndash746 2010
[6] G Espinosa-Paredes M-A Polo-Labarrios E-G Espinosa-Martınez and E D Valle-Gallegos ldquoFractional neutron pointkinetics equations for nuclear reactor dynamicsrdquo Annals ofNuclear Energy vol 38 no 2-3 pp 307ndash330 2011
[7] A E Aboanber and A A Nahla ldquoSolution of the point kineticsequations in the presence of Newtonian temperature feedbackby Pade approximations via the analytical inversion methodrdquoJournal of Physics A vol 35 no 45 pp 9609ndash9627 2002
[8] A E Aboanber and Y M Hamada ldquoPower series solution(PWS) of nuclear reactor dynamics with newtonian tempera-ture feedbackrdquoAnnals of Nuclear Energy vol 30 no 10 pp 1111ndash1122 2003
[9] W Z Chen B Zhu and H F Li ldquoThe analytical solutionof point-reactor neutron-kinetics equation with small stepreactivityrdquo Acta Physica Sinica vol 50 no 8 pp 2486ndash24892004 (Chinese)
[10] H Li W Chen F Zhang and L Luo ldquoApproximate solutionsof point kinetics equations with one delayed neutron groupand temperature feedback during delayed supercritical processrdquoAnnals of Nuclear Energy vol 34 no 6 pp 521ndash526 2007
[11] T Sathiyasheela ldquoPower series solution method for solvingpoint kinetics equations with lumped model temperature andfeedbackrdquo Annals of Nuclear Energy vol 36 no 2 pp 246ndash2502009
6 Science and Technology of Nuclear Installations
[12] YM Hamada ldquoConfirmation of accuracy of generalized powerseries method for the solution of point kinetics equations withfeedbackrdquo Annals of Nuclear Energy vol 55 pp 184ndash193 2013
[13] Q Zhu X-L Shang and W-Z Chen ldquoHomotopy analysissolution of point reactor kinetics equations with six-groupdelayed neutronsrdquo Acta Physica Sinica vol 61 no 7 Article ID070201 2012
[14] S D Hamieh and M Saidinezhad ldquoAnalytical solution of thepoint reactor kinetics equations with temperature feedbackrdquoAnnals of Nuclear Energy vol 42 pp 148ndash152 2012
[15] A A Nahla ldquoAn analytical solution for the point reactorkinetics equations with one group of delayed neutrons and theadiabatic feedback modelrdquo Progress in Nuclear Energy vol 51no 1 pp 124ndash128 2009
[16] W Chen L Guo B Zhu and H Li ldquoAccuracy of analyticalmethods for obtaining supercritical transients with temperaturefeedbackrdquoProgress inNuclear Energy vol 49 no 4 pp 290ndash3022007
[17] H Li W Chen L Luo and Q Zhu ldquoA new integral method forsolving the point reactor neutron kinetics equationsrdquo Annals ofNuclear Energy vol 36 no 4 pp 427ndash432 2009
[18] Z QHuangKinetics Base of Nuclear Reactor PekingUniversityPress Beijing China 2007 (Chinese)
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High Energy PhysicsAdvances in
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Science and Technology of Nuclear Installations 3
In addition the temperature feedback is not fast enoughto affect the reactivity and in the inner part it is assumed that120588(119905) = 120588
0 119889120588(119905)119889119905 asymp minus120572119870
1198881198990 From (20) we can get
1198971198892119899119891
1198891199052= (1205880minus 120573 minus 120582119897)
119889119899119891
119889119905+ (1205821205880minus 1205721198701198881198990) (119899119891+ 1198991198970)
+ (1205880minus 120573 minus 119897120582) 119899
1015840
119897(0) minus 119899
10158401015840
119897(0) 119897
(22)
The solution of (22) is
119899119891= minus1198991198970minus(1205880minus 120573 minus 119897120582) 119899
1015840
119897(0) minus 119899
10158401015840
119897(0) 119897
(1205821205880minus 1205721198701198881198990)
+ 1198621exp ( 12119897
times ((1205880minus 120573 minus 119897120582)
+radic(1205880minus 120573 minus 119897120582)
2+4119897 (120582120588
0minus 1205721198701198881198990)) 119905)
+ 1198622exp ( 12119897
times ((1205880minus 120573 minus 119897120582)
minusradic(1205880minus 120573 minus 119897120582)
2+4119897 (120582120588
0minus 1205721198701198881198990)) 119905)
(23)
The fast varying part 119899119891(119905) is assumed to attenuate to zero
in the inner part so 1198621(119905) = 0 and
minus1198991198970minus(1205880minus 120573 minus 119897120582) 119899
1015840
119897(0) minus 119899
10158401015840
119897(0) 119897
(1205821205880minus 1205721198701198881198990)
= 0 (24)
Then we can get the fast varying part 119899119891(119905) in the inner
part as follows
119899119891= 1198991198910
exp ( 12119897
times ((1205880minus 120573 minus 119897120582)
minusradic(1205880minus 120573 minus 119897120582)
2+ 4119897 (120582120588
0minus 1205721198701198881198990)) 119905)
(25)
From (25) we have
1198991015840
1198910=1198991198910
2119897((1205880minus 120573 minus 119897120582)
minusradic(1205880minus 120573 minus 119897120582)
2+ 4119897 (120582120588
0minus 1205721198701198881198990))
(26)
Combining (9) (11) (13) (24) and (26) results in
1198991198910=
2119897 (1205721198701198881198990minus 1205821205880) 1198990(1205880minus 120573 minus 119897120582)
2minus 212058801198990 (1205880minus 120573 minus 119897120582)
2119897 (1205721198701198881198990minus 1205821205880) (1205880minus 120573 minus 119897120582)
2minus (1 minus radic1 + 4119897 (120582120588
0minus 1205721198701198881198990) (1205880minus 120573 minus 119897120582)
2)
1198991198970=
212058801198990 (1205880minus 120573 minus 119897120582) minus 119899
0(1 minus radic1 + 4119897 (120582120588
0minus 1205721198701198881198990) (1205880minus 120573 minus 119897120582)
2)
2119897 (1205721198701198881198990minus 1205821205880) (1205880minus 120573 minus 119897120582)
2minus (1 minus radic1 + 4119897 (120582120588
0minus 1205721198701198881198990) (1205880minus 120573 minus 119897120582)
2)
(27)
Substituting (27) into (25) and (17) can get 119899119891(119905) and 119899
119897(119905)
then the neutron density (or power) will be obtained by (8)Combining (1) and (2) results in
119889 (119899 + 119862)
119889119905=120588
119897119899 (28)
Eliminating the time variable in (5) and (28) leads to
119889 (119899 + 119862)
119889120588= minus120588
120572119870119888119897 (29)
Integrating (29) with the initial conditions 119899(0) = 1198990
119862(0) = 1205731198990120582119897 and 120588(0) = 120588
0yields
119862 (119905) =1205882
0minus 1205882
2120572119870119888119897+ (1 +120573
120582119897) 1198990minus 119899 (119905) (30)
Equations (8) (17) (25) (27) and (30) are the newanalytical expressions derived by this paper
3 Calculation and Analysis
The PWR with fuel 235U is taken as an example withparameters 120573 = 00065 119897 = 00001 s 120582 = 00774 1s 119870
119888=
005KMWsdots and 120572 = 5 times 10minus5 1K [8 13] For the reactorwith the initial power 1MW while reactivity 120588
0= 05120573
and 1205880= 08333120573 is inserted respectively the variations
of reactivity temperature power with time and power withreactivity are presented in Figures 1 2 3 and 4 The curveswith smaller change are for 120588
0= 05120573 and the curves with
larger change are for 1205880= 08333120573 The solid line notes
the accurate solution by the best basic function method with
4 Science and Technology of Nuclear Installations
SmPPPJ
Accurate solutionSPMTPJPrPJ
SmPPPJ
t (s)0 50 100 150 200
n(M
W)
100
80
60
40
20
0
Figure 1 Variation of output power with time while inserting stepreactivity 120588
0= 05120573 and 120588
0= 08333120573
Accurate solutionSPMTPJPrPJ
SmPPPJ
t (s)0 50 100 150 200
times10minus3
6
4
2
0
minus2
minus4
minus6
120588
Figure 2 Variation of total reactivity with time while inserting stepreactivity 120588
0= 05120573 and 120588
0= 08333120573
Accurate solutionSPMTPJPrPJ
t (s)0 50 100 150 200
250
200
150
100
50
0
ΔT
(K)
SmPPPJ
Figure 3 Variation of temperature rise of reactor with time whileinserting step reactivity 120588
0= 05120573 and 120588
0= 08333120573
SmPPPJ
times10minus36420minus2minus4minus6
120588
n(M
W)
100
80
60
40
20
0
SmPPPJ
Accurate solutionSPMTPJPrPJ
Figure 4 Variation of output power with reactivity while insertingstep reactivity 120588
0= 05120573 and 120588
0= 08333120573
200
150
100
50
0
n(M
W)
0 20 40 60t (s)
TPJPPJSPM
Accurate solutionPrPJ
Figure 5 Variation of output power with time while inserting stepreactivity 0995120573
very small step size [17] The results of this paper and theaccurate solution as well as the temperature prompt jump(TPJ) method in the literature [10] are almost the same andhard to distinguish in Figures 1ndash4 The short dashed-dot lineand long dashed-dot line represent the results of precursorprompt jump (PrPJ)method in the literature [9] and the smallparameter (SmP) method in the literature [15] respectivelyThe dashed line notes the results of power prompt jump (PPJ)method in the literature [16] The difference is caused by theapproximate treatment to obey the methods of PrPJ SmPPPJ and SmP Furthermore the variation in the vicinity ofprompt supercritical process is also calculated and is shownin Figures 5 and 6 The correct results cannot be obtained bythe small parameter method (SmP) in the vicinity of promptsupercritical process and are not shown in Figures 5 and 6
Science and Technology of Nuclear Installations 5
0 20 40 60t (s)
TPJPPJSPM
Accurate solutionPrPJ
350
300
250
200
150
100
50
0
n(M
W)
Figure 6 Variation of output power with time while inserting stepreactivity 0998120573
(1) From Figures 1ndash6 it can be concluded that very goodresults cannot be obtained by the precursor prompt jump(PrPJ) method to calculate the delayed supercritical progresswith small step reactivity and temperature feedback(2) For small step reactivity the results by the small
parameter (SmP) method are close to those by the powerprompt jump (PPJ) method and are better than those by theprecursor prompt jump (PrPJ) method but the accuracy ofresults by the small parameter method decreases with theincrease of the reactivity insertedThepower is negativewhenthe small parameter method is used to calculate the transientprocess in the vicinity of prompt supercritical state FromFigures 1ndash4 it can be seen that the small parameter method ismore suitable for the calculation of reactivity and temperatureincrease than for that of power(3)The results are quite precise using the power prompt
jump (PPJ) method for the delayed supercritical process butthe main problem compared to the accurate solution is thatsome displacement exists along time axis Furthermore itshould be pointed out that each power peak value obtainedby the precursor prompt jump (PrPJ) method power promptjump (PPJ) method or small parameter (SmP) methodis lower than that obtained by the accurate solution orsingularly perturbed method (SPM) see Figure 1(4) From Figures 1ndash4 it can be also found that the
temperature prompt jump method (TPJ) and the singularlyperturbed method (SPM) in this paper are the two mostprecise methods for the delayed supercritical process withsmall step reactivity and temperature feedback Howeverfrom Figures 5 and 6 it can be seen that as the reactivityinserted increases to the vicinity of prompt supercriticalprocess the total discrepancy of power by the TPJ method islarger than that by the SPM or PPJmethod and the irrelevantphenomena that the power jumps at first and then decreasesmonotonously from the peak will appear in the TPJ methodas shown in Figure 6
4 Conclusions
The analytical expressions of power (or neutron density)reactivity the precursor power (or density) and temperatureincrease with respect to time are derived for the delayedsupercritical process with small reactivity (120588
0lt 120573) and
temperature feedback by the singularly perturbed methodCompared with the results by the accurate solution and othermethods in the literature it is shown that the singularlyperturbedmethod (SPM) in this paper is valid and accurate inthe large range and is simpler than those in the previous liter-atureThemethod in this paper can provide a new theoreticalfoundation for the analysis of reactor neutron dynamics
Acknowledgments
This research is supported by the National Natural ScienceFoundation of China (Project no 11301540) and the NaturalScience Foundation of Naval University of Engineering
References
[1] H P Gupta and M S Trasi ldquoAsymptotically stable solutions ofpoint-reactor kinetics equations in the presence of Newtoniantemperature feedbackrdquo Annals of Nuclear Energy vol 13 no 4pp 203ndash207 1986
[2] H Van Dam ldquoDynamics of passive reactor shutdownrdquo Progressin Nuclear Energy vol 30 no 3 pp 255ndash264 1996
[3] D L HetrickDynamics of Nuclear Reactors American NuclearSociety La Grange Park Ill USA 1993
[4] WM StaceyNuclear Reactors PhysicsWiley-Interscience NewYork NY USA 2001
[5] A A Nahla and EM E Zayed ldquoSolution of the nonlinear pointnuclear reactor kinetics equationsrdquo Progress in Nuclear Energyvol 52 no 8 pp 743ndash746 2010
[6] G Espinosa-Paredes M-A Polo-Labarrios E-G Espinosa-Martınez and E D Valle-Gallegos ldquoFractional neutron pointkinetics equations for nuclear reactor dynamicsrdquo Annals ofNuclear Energy vol 38 no 2-3 pp 307ndash330 2011
[7] A E Aboanber and A A Nahla ldquoSolution of the point kineticsequations in the presence of Newtonian temperature feedbackby Pade approximations via the analytical inversion methodrdquoJournal of Physics A vol 35 no 45 pp 9609ndash9627 2002
[8] A E Aboanber and Y M Hamada ldquoPower series solution(PWS) of nuclear reactor dynamics with newtonian tempera-ture feedbackrdquoAnnals of Nuclear Energy vol 30 no 10 pp 1111ndash1122 2003
[9] W Z Chen B Zhu and H F Li ldquoThe analytical solutionof point-reactor neutron-kinetics equation with small stepreactivityrdquo Acta Physica Sinica vol 50 no 8 pp 2486ndash24892004 (Chinese)
[10] H Li W Chen F Zhang and L Luo ldquoApproximate solutionsof point kinetics equations with one delayed neutron groupand temperature feedback during delayed supercritical processrdquoAnnals of Nuclear Energy vol 34 no 6 pp 521ndash526 2007
[11] T Sathiyasheela ldquoPower series solution method for solvingpoint kinetics equations with lumped model temperature andfeedbackrdquo Annals of Nuclear Energy vol 36 no 2 pp 246ndash2502009
6 Science and Technology of Nuclear Installations
[12] YM Hamada ldquoConfirmation of accuracy of generalized powerseries method for the solution of point kinetics equations withfeedbackrdquo Annals of Nuclear Energy vol 55 pp 184ndash193 2013
[13] Q Zhu X-L Shang and W-Z Chen ldquoHomotopy analysissolution of point reactor kinetics equations with six-groupdelayed neutronsrdquo Acta Physica Sinica vol 61 no 7 Article ID070201 2012
[14] S D Hamieh and M Saidinezhad ldquoAnalytical solution of thepoint reactor kinetics equations with temperature feedbackrdquoAnnals of Nuclear Energy vol 42 pp 148ndash152 2012
[15] A A Nahla ldquoAn analytical solution for the point reactorkinetics equations with one group of delayed neutrons and theadiabatic feedback modelrdquo Progress in Nuclear Energy vol 51no 1 pp 124ndash128 2009
[16] W Chen L Guo B Zhu and H Li ldquoAccuracy of analyticalmethods for obtaining supercritical transients with temperaturefeedbackrdquoProgress inNuclear Energy vol 49 no 4 pp 290ndash3022007
[17] H Li W Chen L Luo and Q Zhu ldquoA new integral method forsolving the point reactor neutron kinetics equationsrdquo Annals ofNuclear Energy vol 36 no 4 pp 427ndash432 2009
[18] Z QHuangKinetics Base of Nuclear Reactor PekingUniversityPress Beijing China 2007 (Chinese)
TribologyAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FuelsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofPetroleum Engineering
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
CombustionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Renewable Energy
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
StructuresJournal of
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear InstallationsScience and Technology of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solar EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Wind EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear EnergyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
4 Science and Technology of Nuclear Installations
SmPPPJ
Accurate solutionSPMTPJPrPJ
SmPPPJ
t (s)0 50 100 150 200
n(M
W)
100
80
60
40
20
0
Figure 1 Variation of output power with time while inserting stepreactivity 120588
0= 05120573 and 120588
0= 08333120573
Accurate solutionSPMTPJPrPJ
SmPPPJ
t (s)0 50 100 150 200
times10minus3
6
4
2
0
minus2
minus4
minus6
120588
Figure 2 Variation of total reactivity with time while inserting stepreactivity 120588
0= 05120573 and 120588
0= 08333120573
Accurate solutionSPMTPJPrPJ
t (s)0 50 100 150 200
250
200
150
100
50
0
ΔT
(K)
SmPPPJ
Figure 3 Variation of temperature rise of reactor with time whileinserting step reactivity 120588
0= 05120573 and 120588
0= 08333120573
SmPPPJ
times10minus36420minus2minus4minus6
120588
n(M
W)
100
80
60
40
20
0
SmPPPJ
Accurate solutionSPMTPJPrPJ
Figure 4 Variation of output power with reactivity while insertingstep reactivity 120588
0= 05120573 and 120588
0= 08333120573
200
150
100
50
0
n(M
W)
0 20 40 60t (s)
TPJPPJSPM
Accurate solutionPrPJ
Figure 5 Variation of output power with time while inserting stepreactivity 0995120573
very small step size [17] The results of this paper and theaccurate solution as well as the temperature prompt jump(TPJ) method in the literature [10] are almost the same andhard to distinguish in Figures 1ndash4 The short dashed-dot lineand long dashed-dot line represent the results of precursorprompt jump (PrPJ)method in the literature [9] and the smallparameter (SmP) method in the literature [15] respectivelyThe dashed line notes the results of power prompt jump (PPJ)method in the literature [16] The difference is caused by theapproximate treatment to obey the methods of PrPJ SmPPPJ and SmP Furthermore the variation in the vicinity ofprompt supercritical process is also calculated and is shownin Figures 5 and 6 The correct results cannot be obtained bythe small parameter method (SmP) in the vicinity of promptsupercritical process and are not shown in Figures 5 and 6
Science and Technology of Nuclear Installations 5
0 20 40 60t (s)
TPJPPJSPM
Accurate solutionPrPJ
350
300
250
200
150
100
50
0
n(M
W)
Figure 6 Variation of output power with time while inserting stepreactivity 0998120573
(1) From Figures 1ndash6 it can be concluded that very goodresults cannot be obtained by the precursor prompt jump(PrPJ) method to calculate the delayed supercritical progresswith small step reactivity and temperature feedback(2) For small step reactivity the results by the small
parameter (SmP) method are close to those by the powerprompt jump (PPJ) method and are better than those by theprecursor prompt jump (PrPJ) method but the accuracy ofresults by the small parameter method decreases with theincrease of the reactivity insertedThepower is negativewhenthe small parameter method is used to calculate the transientprocess in the vicinity of prompt supercritical state FromFigures 1ndash4 it can be seen that the small parameter method ismore suitable for the calculation of reactivity and temperatureincrease than for that of power(3)The results are quite precise using the power prompt
jump (PPJ) method for the delayed supercritical process butthe main problem compared to the accurate solution is thatsome displacement exists along time axis Furthermore itshould be pointed out that each power peak value obtainedby the precursor prompt jump (PrPJ) method power promptjump (PPJ) method or small parameter (SmP) methodis lower than that obtained by the accurate solution orsingularly perturbed method (SPM) see Figure 1(4) From Figures 1ndash4 it can be also found that the
temperature prompt jump method (TPJ) and the singularlyperturbed method (SPM) in this paper are the two mostprecise methods for the delayed supercritical process withsmall step reactivity and temperature feedback Howeverfrom Figures 5 and 6 it can be seen that as the reactivityinserted increases to the vicinity of prompt supercriticalprocess the total discrepancy of power by the TPJ method islarger than that by the SPM or PPJmethod and the irrelevantphenomena that the power jumps at first and then decreasesmonotonously from the peak will appear in the TPJ methodas shown in Figure 6
4 Conclusions
The analytical expressions of power (or neutron density)reactivity the precursor power (or density) and temperatureincrease with respect to time are derived for the delayedsupercritical process with small reactivity (120588
0lt 120573) and
temperature feedback by the singularly perturbed methodCompared with the results by the accurate solution and othermethods in the literature it is shown that the singularlyperturbedmethod (SPM) in this paper is valid and accurate inthe large range and is simpler than those in the previous liter-atureThemethod in this paper can provide a new theoreticalfoundation for the analysis of reactor neutron dynamics
Acknowledgments
This research is supported by the National Natural ScienceFoundation of China (Project no 11301540) and the NaturalScience Foundation of Naval University of Engineering
References
[1] H P Gupta and M S Trasi ldquoAsymptotically stable solutions ofpoint-reactor kinetics equations in the presence of Newtoniantemperature feedbackrdquo Annals of Nuclear Energy vol 13 no 4pp 203ndash207 1986
[2] H Van Dam ldquoDynamics of passive reactor shutdownrdquo Progressin Nuclear Energy vol 30 no 3 pp 255ndash264 1996
[3] D L HetrickDynamics of Nuclear Reactors American NuclearSociety La Grange Park Ill USA 1993
[4] WM StaceyNuclear Reactors PhysicsWiley-Interscience NewYork NY USA 2001
[5] A A Nahla and EM E Zayed ldquoSolution of the nonlinear pointnuclear reactor kinetics equationsrdquo Progress in Nuclear Energyvol 52 no 8 pp 743ndash746 2010
[6] G Espinosa-Paredes M-A Polo-Labarrios E-G Espinosa-Martınez and E D Valle-Gallegos ldquoFractional neutron pointkinetics equations for nuclear reactor dynamicsrdquo Annals ofNuclear Energy vol 38 no 2-3 pp 307ndash330 2011
[7] A E Aboanber and A A Nahla ldquoSolution of the point kineticsequations in the presence of Newtonian temperature feedbackby Pade approximations via the analytical inversion methodrdquoJournal of Physics A vol 35 no 45 pp 9609ndash9627 2002
[8] A E Aboanber and Y M Hamada ldquoPower series solution(PWS) of nuclear reactor dynamics with newtonian tempera-ture feedbackrdquoAnnals of Nuclear Energy vol 30 no 10 pp 1111ndash1122 2003
[9] W Z Chen B Zhu and H F Li ldquoThe analytical solutionof point-reactor neutron-kinetics equation with small stepreactivityrdquo Acta Physica Sinica vol 50 no 8 pp 2486ndash24892004 (Chinese)
[10] H Li W Chen F Zhang and L Luo ldquoApproximate solutionsof point kinetics equations with one delayed neutron groupand temperature feedback during delayed supercritical processrdquoAnnals of Nuclear Energy vol 34 no 6 pp 521ndash526 2007
[11] T Sathiyasheela ldquoPower series solution method for solvingpoint kinetics equations with lumped model temperature andfeedbackrdquo Annals of Nuclear Energy vol 36 no 2 pp 246ndash2502009
6 Science and Technology of Nuclear Installations
[12] YM Hamada ldquoConfirmation of accuracy of generalized powerseries method for the solution of point kinetics equations withfeedbackrdquo Annals of Nuclear Energy vol 55 pp 184ndash193 2013
[13] Q Zhu X-L Shang and W-Z Chen ldquoHomotopy analysissolution of point reactor kinetics equations with six-groupdelayed neutronsrdquo Acta Physica Sinica vol 61 no 7 Article ID070201 2012
[14] S D Hamieh and M Saidinezhad ldquoAnalytical solution of thepoint reactor kinetics equations with temperature feedbackrdquoAnnals of Nuclear Energy vol 42 pp 148ndash152 2012
[15] A A Nahla ldquoAn analytical solution for the point reactorkinetics equations with one group of delayed neutrons and theadiabatic feedback modelrdquo Progress in Nuclear Energy vol 51no 1 pp 124ndash128 2009
[16] W Chen L Guo B Zhu and H Li ldquoAccuracy of analyticalmethods for obtaining supercritical transients with temperaturefeedbackrdquoProgress inNuclear Energy vol 49 no 4 pp 290ndash3022007
[17] H Li W Chen L Luo and Q Zhu ldquoA new integral method forsolving the point reactor neutron kinetics equationsrdquo Annals ofNuclear Energy vol 36 no 4 pp 427ndash432 2009
[18] Z QHuangKinetics Base of Nuclear Reactor PekingUniversityPress Beijing China 2007 (Chinese)
TribologyAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FuelsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofPetroleum Engineering
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
CombustionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Renewable Energy
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
StructuresJournal of
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear InstallationsScience and Technology of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solar EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Wind EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear EnergyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Science and Technology of Nuclear Installations 5
0 20 40 60t (s)
TPJPPJSPM
Accurate solutionPrPJ
350
300
250
200
150
100
50
0
n(M
W)
Figure 6 Variation of output power with time while inserting stepreactivity 0998120573
(1) From Figures 1ndash6 it can be concluded that very goodresults cannot be obtained by the precursor prompt jump(PrPJ) method to calculate the delayed supercritical progresswith small step reactivity and temperature feedback(2) For small step reactivity the results by the small
parameter (SmP) method are close to those by the powerprompt jump (PPJ) method and are better than those by theprecursor prompt jump (PrPJ) method but the accuracy ofresults by the small parameter method decreases with theincrease of the reactivity insertedThepower is negativewhenthe small parameter method is used to calculate the transientprocess in the vicinity of prompt supercritical state FromFigures 1ndash4 it can be seen that the small parameter method ismore suitable for the calculation of reactivity and temperatureincrease than for that of power(3)The results are quite precise using the power prompt
jump (PPJ) method for the delayed supercritical process butthe main problem compared to the accurate solution is thatsome displacement exists along time axis Furthermore itshould be pointed out that each power peak value obtainedby the precursor prompt jump (PrPJ) method power promptjump (PPJ) method or small parameter (SmP) methodis lower than that obtained by the accurate solution orsingularly perturbed method (SPM) see Figure 1(4) From Figures 1ndash4 it can be also found that the
temperature prompt jump method (TPJ) and the singularlyperturbed method (SPM) in this paper are the two mostprecise methods for the delayed supercritical process withsmall step reactivity and temperature feedback Howeverfrom Figures 5 and 6 it can be seen that as the reactivityinserted increases to the vicinity of prompt supercriticalprocess the total discrepancy of power by the TPJ method islarger than that by the SPM or PPJmethod and the irrelevantphenomena that the power jumps at first and then decreasesmonotonously from the peak will appear in the TPJ methodas shown in Figure 6
4 Conclusions
The analytical expressions of power (or neutron density)reactivity the precursor power (or density) and temperatureincrease with respect to time are derived for the delayedsupercritical process with small reactivity (120588
0lt 120573) and
temperature feedback by the singularly perturbed methodCompared with the results by the accurate solution and othermethods in the literature it is shown that the singularlyperturbedmethod (SPM) in this paper is valid and accurate inthe large range and is simpler than those in the previous liter-atureThemethod in this paper can provide a new theoreticalfoundation for the analysis of reactor neutron dynamics
Acknowledgments
This research is supported by the National Natural ScienceFoundation of China (Project no 11301540) and the NaturalScience Foundation of Naval University of Engineering
References
[1] H P Gupta and M S Trasi ldquoAsymptotically stable solutions ofpoint-reactor kinetics equations in the presence of Newtoniantemperature feedbackrdquo Annals of Nuclear Energy vol 13 no 4pp 203ndash207 1986
[2] H Van Dam ldquoDynamics of passive reactor shutdownrdquo Progressin Nuclear Energy vol 30 no 3 pp 255ndash264 1996
[3] D L HetrickDynamics of Nuclear Reactors American NuclearSociety La Grange Park Ill USA 1993
[4] WM StaceyNuclear Reactors PhysicsWiley-Interscience NewYork NY USA 2001
[5] A A Nahla and EM E Zayed ldquoSolution of the nonlinear pointnuclear reactor kinetics equationsrdquo Progress in Nuclear Energyvol 52 no 8 pp 743ndash746 2010
[6] G Espinosa-Paredes M-A Polo-Labarrios E-G Espinosa-Martınez and E D Valle-Gallegos ldquoFractional neutron pointkinetics equations for nuclear reactor dynamicsrdquo Annals ofNuclear Energy vol 38 no 2-3 pp 307ndash330 2011
[7] A E Aboanber and A A Nahla ldquoSolution of the point kineticsequations in the presence of Newtonian temperature feedbackby Pade approximations via the analytical inversion methodrdquoJournal of Physics A vol 35 no 45 pp 9609ndash9627 2002
[8] A E Aboanber and Y M Hamada ldquoPower series solution(PWS) of nuclear reactor dynamics with newtonian tempera-ture feedbackrdquoAnnals of Nuclear Energy vol 30 no 10 pp 1111ndash1122 2003
[9] W Z Chen B Zhu and H F Li ldquoThe analytical solutionof point-reactor neutron-kinetics equation with small stepreactivityrdquo Acta Physica Sinica vol 50 no 8 pp 2486ndash24892004 (Chinese)
[10] H Li W Chen F Zhang and L Luo ldquoApproximate solutionsof point kinetics equations with one delayed neutron groupand temperature feedback during delayed supercritical processrdquoAnnals of Nuclear Energy vol 34 no 6 pp 521ndash526 2007
[11] T Sathiyasheela ldquoPower series solution method for solvingpoint kinetics equations with lumped model temperature andfeedbackrdquo Annals of Nuclear Energy vol 36 no 2 pp 246ndash2502009
6 Science and Technology of Nuclear Installations
[12] YM Hamada ldquoConfirmation of accuracy of generalized powerseries method for the solution of point kinetics equations withfeedbackrdquo Annals of Nuclear Energy vol 55 pp 184ndash193 2013
[13] Q Zhu X-L Shang and W-Z Chen ldquoHomotopy analysissolution of point reactor kinetics equations with six-groupdelayed neutronsrdquo Acta Physica Sinica vol 61 no 7 Article ID070201 2012
[14] S D Hamieh and M Saidinezhad ldquoAnalytical solution of thepoint reactor kinetics equations with temperature feedbackrdquoAnnals of Nuclear Energy vol 42 pp 148ndash152 2012
[15] A A Nahla ldquoAn analytical solution for the point reactorkinetics equations with one group of delayed neutrons and theadiabatic feedback modelrdquo Progress in Nuclear Energy vol 51no 1 pp 124ndash128 2009
[16] W Chen L Guo B Zhu and H Li ldquoAccuracy of analyticalmethods for obtaining supercritical transients with temperaturefeedbackrdquoProgress inNuclear Energy vol 49 no 4 pp 290ndash3022007
[17] H Li W Chen L Luo and Q Zhu ldquoA new integral method forsolving the point reactor neutron kinetics equationsrdquo Annals ofNuclear Energy vol 36 no 4 pp 427ndash432 2009
[18] Z QHuangKinetics Base of Nuclear Reactor PekingUniversityPress Beijing China 2007 (Chinese)
TribologyAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FuelsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofPetroleum Engineering
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
CombustionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Renewable Energy
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
StructuresJournal of
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear InstallationsScience and Technology of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solar EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Wind EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear EnergyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
6 Science and Technology of Nuclear Installations
[12] YM Hamada ldquoConfirmation of accuracy of generalized powerseries method for the solution of point kinetics equations withfeedbackrdquo Annals of Nuclear Energy vol 55 pp 184ndash193 2013
[13] Q Zhu X-L Shang and W-Z Chen ldquoHomotopy analysissolution of point reactor kinetics equations with six-groupdelayed neutronsrdquo Acta Physica Sinica vol 61 no 7 Article ID070201 2012
[14] S D Hamieh and M Saidinezhad ldquoAnalytical solution of thepoint reactor kinetics equations with temperature feedbackrdquoAnnals of Nuclear Energy vol 42 pp 148ndash152 2012
[15] A A Nahla ldquoAn analytical solution for the point reactorkinetics equations with one group of delayed neutrons and theadiabatic feedback modelrdquo Progress in Nuclear Energy vol 51no 1 pp 124ndash128 2009
[16] W Chen L Guo B Zhu and H Li ldquoAccuracy of analyticalmethods for obtaining supercritical transients with temperaturefeedbackrdquoProgress inNuclear Energy vol 49 no 4 pp 290ndash3022007
[17] H Li W Chen L Luo and Q Zhu ldquoA new integral method forsolving the point reactor neutron kinetics equationsrdquo Annals ofNuclear Energy vol 36 no 4 pp 427ndash432 2009
[18] Z QHuangKinetics Base of Nuclear Reactor PekingUniversityPress Beijing China 2007 (Chinese)
TribologyAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FuelsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofPetroleum Engineering
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
CombustionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Renewable Energy
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
StructuresJournal of
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear InstallationsScience and Technology of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solar EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Wind EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear EnergyInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
High Energy PhysicsAdvances in
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
TribologyAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
FuelsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofPetroleum Engineering
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Power ElectronicsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Advances in
CombustionJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Renewable Energy
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
StructuresJournal of
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Hindawi Publishing Corporation httpwwwhindawicom Volume 2014
International Journal ofPhotoenergy
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear InstallationsScience and Technology of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Solar EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Wind EnergyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nuclear EnergyInternational Journal of
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014