Research Article Solution of Point Reactor Neutron...

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)


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


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



2+ 4119897 (120582120588

0minus 1205721198701198881198990)) 119905)

(25)

From (25) we have

1198991015840

1198910=1198991198910

2119897((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


2)




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


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


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


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


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


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


0 20 40 60t (s)

TPJPPJSPM


350

300

250

200

150

100

50

0

n(M

W)


(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


[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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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


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)


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)



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)



1015840

119897(0) minus 119899

10158401015840

119897(0) 119897

(1205821205880minus 1205721198701198881198990)

+ 1198621exp ( 12119897



2+4119897 (120582120588

0minus 1205721198701198881198990)) 119905)

+ 1198622exp ( 12119897



2+4119897 (120582120588

0minus 1205721198701198881198990)) 119905)

(23)




1015840

119897(0) minus 119899

10158401015840

119897(0) 119897

(1205821205880minus 1205721198701198881198990)

= 0 (24)


part as follows

119899119891= 1198991198910

exp ( 12119897



2+ 4119897 (120582120588

0minus 1205721198701198881198990)) 119905)

(25)

From (25) we have

1198991015840

1198910=1198991198910

2119897((1205880minus 120573 minus 119897120582)


2+ 4119897 (120582120588

0minus 1205721198701198881198990))

(26)


1198991198910=






2)

1198991198970=


0(1 minus radic1 + 4119897 (120582120588


2)




2)

(27)


119897(119905)


119889 (119899 + 119862)

119889119905=120588

119897119899 (28)


119889 (119899 + 119862)

119889120588= minus120588

120572119870119888119897 (29)


119862(0) = 1205731198990120582119897 and 120588(0) = 120588

0yields

119862 (119905) =1205882

0minus 1205882

2120572119870119888119897+ (1 +120573

120582119897) 1198990minus 119899 (119905) (30)




119888=


0= 05120573







SmPPPJ


SmPPPJ

t (s)0 50 100 150 200

n(M

W)

100

80

60

40

20

0


0= 05120573 and 120588

0= 08333120573


SmPPPJ

t (s)0 50 100 150 200

times10minus3

6

4

2

0

minus2

minus4

minus6

120588


0= 05120573 and 120588

0= 08333120573


t (s)0 50 100 150 200

250

200

150

100

50

0

ΔT

(K)

SmPPPJ


0= 05120573 and 120588

0= 08333120573

SmPPPJ


120588

n(M

W)

100

80

60

40

20

0

SmPPPJ



0= 05120573 and 120588

0= 08333120573

200

150

100

50

0

n(M

W)

0 20 40 60t (s)

TPJPPJSPM





0 20 40 60t (s)

TPJPPJSPM


350

300

250

200

150

100

50

0

n(M

W)






4 Conclusions


0lt 120573) and


Acknowledgments


References
























FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of



















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)



1015840

119897(0) minus 119899

10158401015840

119897(0) 119897

(1205821205880minus 1205721198701198881198990)

+ 1198621exp ( 12119897



2+4119897 (120582120588

0minus 1205721198701198881198990)) 119905)

+ 1198622exp ( 12119897



2+4119897 (120582120588

0minus 1205721198701198881198990)) 119905)

(23)




1015840

119897(0) minus 119899

10158401015840

119897(0) 119897

(1205821205880minus 1205721198701198881198990)

= 0 (24)


part as follows

119899119891= 1198991198910

exp ( 12119897



2+ 4119897 (120582120588

0minus 1205721198701198881198990)) 119905)

(25)

From (25) we have

1198991015840

1198910=1198991198910

2119897((1205880minus 120573 minus 119897120582)


2+ 4119897 (120582120588

0minus 1205721198701198881198990))

(26)


1198991198910=






2)

1198991198970=


0(1 minus radic1 + 4119897 (120582120588


2)




2)

(27)


119897(119905)


119889 (119899 + 119862)

119889119905=120588

119897119899 (28)


119889 (119899 + 119862)

119889120588= minus120588

120572119870119888119897 (29)


119862(0) = 1205731198990120582119897 and 120588(0) = 120588

0yields

119862 (119905) =1205882

0minus 1205882

2120572119870119888119897+ (1 +120573

120582119897) 1198990minus 119899 (119905) (30)




119888=


0= 05120573







SmPPPJ


SmPPPJ

t (s)0 50 100 150 200

n(M

W)

100

80

60

40

20

0


0= 05120573 and 120588

0= 08333120573


SmPPPJ

t (s)0 50 100 150 200

times10minus3

6

4

2

0

minus2

minus4

minus6

120588


0= 05120573 and 120588

0= 08333120573


t (s)0 50 100 150 200

250

200

150

100

50

0

ΔT

(K)

SmPPPJ


0= 05120573 and 120588

0= 08333120573

SmPPPJ


120588

n(M

W)

100

80

60

40

20

0

SmPPPJ



0= 05120573 and 120588

0= 08333120573

200

150

100

50

0

n(M

W)

0 20 40 60t (s)

TPJPPJSPM





0 20 40 60t (s)

TPJPPJSPM


350

300

250

200

150

100

50

0

n(M

W)






4 Conclusions


0lt 120573) and


Acknowledgments


References
























FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of


















SmPPPJ


SmPPPJ

t (s)0 50 100 150 200

n(M

W)

100

80

60

40

20

0


0= 05120573 and 120588

0= 08333120573


SmPPPJ

t (s)0 50 100 150 200

times10minus3

6

4

2

0

minus2

minus4

minus6

120588


0= 05120573 and 120588

0= 08333120573


t (s)0 50 100 150 200

250

200

150

100

50

0

ΔT

(K)

SmPPPJ


0= 05120573 and 120588

0= 08333120573

SmPPPJ


120588

n(M

W)

100

80

60

40

20

0

SmPPPJ



0= 05120573 and 120588

0= 08333120573

200

150

100

50

0

n(M

W)

0 20 40 60t (s)

TPJPPJSPM





0 20 40 60t (s)

TPJPPJSPM


350

300

250

200

150

100

50

0

n(M

W)






4 Conclusions


0lt 120573) and


Acknowledgments


References
























FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of


















0 20 40 60t (s)

TPJPPJSPM


350

300

250

200

150

100

50

0

n(M

W)






4 Conclusions


0lt 120573) and


Acknowledgments


References
























FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of





























FuelsJournal of







Advances in



Journal of


Renewable Energy





RotatingMachinery


EnergyJournal of

















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