Research Article Solution of Point Reactor Neutron Kinetics Equations with Temperature Feedback by Singularly Perturbed Method

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1 Science and Technoogy of Nucear Instaations Voume 213, Artice ID , 6 pages Research Artice Soution of Point Reactor Neutron Kinetics Equations with Temperature Feedback by Singuary Perturbed Method Wenzhen Chen, Jiani Hao, Ling Chen, and Haofeng Li Department of Nucear Energy Science and Engineering, Nava University of Engineering, Facuty 31, Wuhan 4333, China Correspondence shoud be addressed to Wenzhen Chen; cwz2@21cn.com Received 3 May 213; Revised 27 August 213; Accepted 29 August 213 Academic Editor: Arkady Serikov Copyright 213 Wenzhen Chen et a. This is an open access artice distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the origina work is propery cited. The singuary perturbed method (SPM) is proposed to obtain the anaytica soution for the deayed supercritica process of nucear reactor with temperature feedback and sma step reactivity inserted. The reation between the reactivity and time is derived. Aso, the neutron density (or power) and the average density of deayed neutron precursors as the function of reactivity are presented. The variations of neutron density (or power) and temperature with time are cacuated and potted and compared with those by accurate soution and other anaytica methods. It is shown that the resuts by the SPM are vaid and accurate in the arge range and the SPM is simper than those in the previous iterature. 1. Introduction The anaysis of variation of neutron density (or power) and reactivity with time under the different conditions is an important content of nucear reactor physics or neutron kinetics [1 7]. Some important achievements on the supercritica transient with temperature feedback with big (ρ >β) or sma (ρ <β)reactivityinsertedhavebeenapproached through the effort of many schoars [7 12]. The studies on the deayed supercritica transient with sma reactivity insertedandtemperaturefeedbackareintroducedinthe reated iterature [13 15], in which the expicit function of density (or power) and reactivity with respect to time is derived mainy with decouping method, power prompt jump approximation, precursor prompt jump approximation, temperature prompt jump approximation [1, 16], and so forth. From the detaied anaysis and comparison of the resuts in the eary and recent iterature [7, 12, 14], it is found that some resuts have certain imit and rather big error under the particuar conditions. In present work, the variation aw of power, reactivity, and precursor density with respect to time at any eve of initia power is obtained by the singuary perturbed method (SPM). A the resuts are compared with those obtained by the numerica soution which tend to the accurate soution under very sma time step size [17]. It is proved that the SPM is correct and reiabe and is simper than the anaytica methods by the reated iterature. 2. Theoretica Derivation The point reactor neutron kinetics equations with one group of deayed neutrons are [3, 4] dn (t) dt = ρ (t) β n (t) +λc(t), (1) dc (t) = β n (t) λc(t), (2) dt where n is the average neutron density, t is the time, ρ is the reactivity, β is the tota fraction of the deayed neutron, is the prompt neutron ifetime, λ is the radioactive decay constant of deayed neutron precursor, and C is the average density of deayed neutron precursor. When mutipied with a certain coefficient, n represents the power. It is assumed that the reactor has a negative temperature coefficient of reactivity α (α > )whenasmastepreactivityρ (<β) isinserted. Consider the temperature feedback, and the rea reactor reactivity is ρ=ρ αt, (3)

2 2 Science and Technoogy of Nucear Instaations where T is the temperature increment of the reactor, namey, T = T s T, where T s and T are the instantaneous temperature and initia temperature, respectivey. After the reactivity ρ is inserted into the reactor the adiabatic mode is sti empoyed [3, 15]; then we have dt dt =K cn (t), (4) where K c is the reciproca of therma capacity of reactor. Combining (3)and(4) resuts in dt = αk cn (t). (5) Substituting (2) into the derivative of (1)withrespecttot yieds d2 n =(ρ β)dn dt +n dt +λβn λ2 C. (6) Substituting λc obtained from (1) into(6) andsimpifying it yieds d 2 n =(ρ β λ) dn dt + n dt + λρn. (7) The transient process is supposed to begin at t = and ρ() = [15, 17], so the initia conditions can be given as ρ() = ρ, n() = n, (dn/dt) t= =ρ n /, (d 2 n/ ) t= = ((ρ β)/)(ρ n /) (αk c n 2 /),wheren is the initia neutron density (or power). For (ρ β)/ λ, β/ λ [3], (1) and(2) are stiff equations. According to the singuary perturbed method [18], soution of n(t) incudes the inner soution n f (t) in inner part and the outer soution n (t) in outer part. In this paper both inner and outer soutions are approximated to be zero order: n (t) =n f (t) +n (t). (8) The initia conditions are n =n f +n, (9) n () =n f () +n n () =n f () +n () = ρ n, (1) () = ρ β ρ n αk cn 2, (11) where n f is the initia vaue of inner soution and n is the initia vaue of outer soution. Substituting (8)into(7)and(5), respectivey, yieds d 2 (n f +n ) = ( ρ β λ) d(n f +n ) dt + λρ (n f +n ), dt = αk c [n f (t) +n (t)]. + (n f +n ) dt (12) In the outer part the inner soution attenuates to zero, and dn f /dt and d 2 n f / canbenegected,namey,dn f /dt and d 2 n f / ; therefore, in the outer part (12) canbe simpified as foows: d 2 n = ρ β dn dt + n dt λdn dt + λρn, (13) dt = αk cn. (14) Because n (t) varies sowy and 1 4 s, compared to other terms, the term on the eft side of (13)canbenegected, andthenwehave (β ρ + λ) dn dt =( dt +λρ)n. (15) Combining (14)and(15) resuts in dn = αk c n λρ αk c (β ρ + λ). (16) Integrating (16) subjective to the initia conditions ρ() = ρ and n () = n yieds is n = λ(ρ 2 ρ2 ) 2αK c (β ρ + λ) + (β ρ +λ) (β ρ+λ) n. (17) Substituting (17)into(14)eadsto dt = λ(ρ 2 ρ2 )+2αK c (β ρ +λ)n. (18) 2 (β ρ + λ) With the initia conditions ρ() = ρ, the soution of (18) t= β+λ λρ 1 n ( ρ 1 ρ ρ 1 +ρ )(ρ 1 +ρ ρ 1 ρ )+ 1 λ 2 1 n [(ρ ρ2 ) (ρ1 2 ρ2 ) ], (19) where ρ 1 = ρ 2 +2αK c(β ρ +λ)n /λ. In the inner part n (t) is assumed to be constant n (t) = n () = n, and dn (t)/dt (dn (t)/dt) t=, d 2 n (t)/ (d 2 n (t)/ ) t=,so(12) can be simpified as foows, respectivey: d 2 n f +n () = ( ρ β λ)( dn f dt +n ()) + (n f +n ) ( dt +λρ), (2) dt = αk c [n f (t) +n ]. (21)

3 Science and Technoogy of Nucear Instaations 3 In addition, the temperature feedback is not fast enough to affect the reactivity, and in the inner part it is assumed that ρ(t) = ρ, (t)/dt αk c n. From(2)wecanget d2 n f =(ρ β λ) dn f dt +(λρ αk c n )(n f +n ) +(ρ β λ)n The soution of (22)is () n (). n f = n (ρ β λ)n () n () (λρ αk c n ) +C 1 exp ( 1 2 ((ρ β λ) (22) + (ρ β λ) 2 +4(λρ αk c n )) t) The fast varying part n f (t) is assumed to attenuate to zero in the inner part, so C 1 (t) = and n (ρ β λ)n () n () =. (24) (λρ αk c n ) Then we can get the fast varying part n f (t) in the inner part as foows: n f =n f exp ( 1 2 From (25)wehave ((ρ β λ) (ρ β λ) 2 +4(λρ αk c n )) t). (25) +C 2 exp ( 1 2 ((ρ β λ) (ρ β λ) 2 +4(λρ αk c n )) t). (23) n f = n f 2 ((ρ β λ) (ρ β λ) 2 +4(λρ αk c n )). Combining (9), (11), (13), (24), and (26) resuts in (26) n f = 2 (αk c n λρ )n /(ρ β λ) 2 2ρ n /(ρ β λ) 2 (αk c n λρ )/(ρ β λ) 2 (1 1+4(λρ αk c n )/(ρ β λ) 2 ), 2ρ n /(ρ β λ) n (1 1+4(λρ αk c n )/(ρ β λ) 2 ) n =. 2 (αk c n λρ )/(ρ β λ) 2 (1 1+4(λρ αk c n )/(ρ β λ) 2 ) (27) Substituting (27)into(25)and(17)cangetn f (t) and n (t); then the neutron density (or power) wi be obtained by (8). Combining (1)and(2) resuts in d (n+c) = ρ n. (28) dt Eiminating the time variabe in (5)and(28)eadsto d (n+c) = ρ αk c. (29) Integrating (29) with the initia conditions n() = n, C() = βn /λ and ρ() = ρ yieds C (t) = ρ2 ρ2 2αK c + (1+ β λ ) n n(t). (3) Equations (8), (17), (25), (27), and (3) arethenew anaytica expressions derived by this paper. 3. Cacuation and Anaysis The PWR with fue 235 U is taken as an exampe with parameters β =.65, =.1 s, λ =.774 1/s, K c =.5 K/MW s, and α= /K [8, 13]. For the reactor with the initia power 1 MW, whie reactivity ρ =.5β and ρ =.8333β is inserted, respectivey, the variations of reactivity, temperature, power with time, and power with reactivity are presented in Figures 1, 2, 3, and4. Thecurves with smaer change are for ρ =.5β and the curves with arger change are for ρ =.8333β. The soid ine notes theaccuratesoutionbythebestbasicfunctionmethodwith

4 4 Science and Technoogy of Nucear Instaations SmP PPJ Accurate soution/spm/tpj Figure 1: Variation of output power with time whie inserting step reactivity ρ =.5β and ρ =.8333β SmP PPJ 2 ρ 2 Accurate soution/spm/tpj Figure 4: Variation of output power with reactivity whie inserting step reactivity ρ =.5β and ρ =.8333β ρ Accurate soution/spm/tpj Figure 2: Variation of tota reactivity with time whie inserting step reactivity ρ =.5β and ρ =.8333β TPJ PPJ/SPM Accurate soution Figure 5: Variation of output power with time whie inserting step reactivity.995β. 25 ΔT (K) Accurate soution/spm/tpj Figure 3: Variation of temperature rise of reactor with time whie inserting step reactivity ρ =.5β and ρ =.8333β. very sma step size [17]. The resuts of this paper and the accurate soution as we as the temperature prompt jump (TPJ) method in the iterature [1] are amost the same and hard to distinguish in Figures 1 4.The short dashed-dot ine and ong dashed-dot ine represent the resuts of precursor prompt jump () method in the iterature [9]andthesma parameter (SmP) method in the iterature [15], respectivey. The dashed ine notes the resuts of power prompt jump (PPJ) method in the iterature [16]. The difference is caused by the approximate treatment to obey the methods of, SmP, PPJ, and SmP. Furthermore the variation in the vicinity of prompt supercritica process is aso cacuated and is shown in Figures 5 and 6. The correct resuts cannot be obtained by the sma parameter method (SmP) in the vicinity of prompt supercritica process and are not shown in Figures 5 and 6.

5 Science and Technoogy of Nucear Instaations TPJ PPJ/SPM Accurate soution Figure 6: Variation of output power with time whie inserting step reactivity.998β. 4. Concusions The anaytica expressions of power (or neutron density), reactivity, the precursor power (or density), and temperature increase with respect to time are derived for the deayed supercriticaprocesswithsmareactivity(ρ < β)and temperature feedback by the singuary perturbed method. Compared with the resuts by the accurate soution and other methods in the iterature, it is shown that the singuary perturbedmethod(spm)inthispaperisvaidandaccuratein theargerangeandissimperthanthoseinthepreviousiterature. The method in this paper can provide a new theoretica foundation for the anaysis of reactor neutron dynamics. Acknowedgments ThisresearchissupportedbytheNationaNaturaScience Foundation of China (Project no ) and the Natura Science Foundation of Nava University of Engineering. (1) From Figures 1 6 it can be concuded that very good resuts cannot be obtained by the precursor prompt jump () method to cacuate the deayed supercritica progress with sma step reactivity and temperature feedback. (2) For sma step reactivity, the resuts by the sma parameter (SmP) method are cose to those by the power prompt jump (PPJ) method and are better than those by the precursor prompt jump () method, but the accuracy of resuts by the sma parameter method decreases with the increase of the reactivity inserted. The power is negative when the sma parameter method is used to cacuate the transient process in the vicinity of prompt supercritica state. From Figures 1 4, it can be seen that the sma parameter method is more suitabe for the cacuation of reactivity and temperature increasethanforthatofpower. (3) The resuts are quite precise using the power prompt jump (PPJ) method for the deayed supercritica process, but themainprobemcomparedtotheaccuratesoutionisthat some dispacement exists aong time axis. Furthermore it shoud be pointed out that each power peak vaue obtained by the precursor prompt jump () method, power prompt jump (PPJ) method, or sma parameter (SmP) method is ower than that obtained by the accurate soution or singuary perturbed method (SPM) see Figure 1. (4) From Figures 1 4 it can be aso found that the temperaturepromptjumpmethod(tpj)andthesinguary perturbed method (SPM) in this paper are the two most precise methods for the deayed supercritica process with sma step reactivity and temperature feedback. However from Figures 5 and 6 itcanbeseenthatasthereactivity inserted increases to the vicinity of prompt supercritica process, the tota discrepancy of power by the TPJ method is arger than that by the SPM or PPJ method, and the irreevant phenomena that the power jumps at first and then decreases monotonousy from the peak wi appear in the TPJ method as shown in Figure 6. References [1] H. P. Gupta and M. S. Trasi, Asymptoticay stabe soutions of point-reactor kinetics equations in the presence of Newtonian temperature feedback, Annas of Nucear Energy,vo.13,no.4, pp , [2] H. Van Dam, Dynamics of passive reactor shutdown, Progress in Nucear Energy,vo.3,no.3,pp ,1996. [3] D.L.Hetrick,Dynamics of Nucear Reactors,AmericanNucear Society, La Grange Park, I, USA, [4] W. M. Stacey, Nucear Reactors Physics, Wiey-Interscience, New York, NY, USA, 21. [5] A.A.NahaandE.M.E.Zayed, Soutionofthenoninearpoint nucear reactor kinetics equations, Progress in Nucear Energy, vo.52,no.8,pp ,21. [6] G. Espinosa-Paredes, M.-A. Poo-Labarrios, E.-G. Espinosa- Martínez, and E. D. Vae-Gaegos, Fractiona neutron point kinetics equations for nucear reactor dynamics, Annas of Nucear Energy, vo. 38, no. 2-3, pp , 211. [7] A. E. Aboanber and A. A. Naha, Soution of the point kinetics equations in the presence of Newtonian temperature feedback by Padé approximationsviatheanayticainversionmethod, Physics A,vo.35,no.45,pp ,22. [8] A. E. Aboanber and Y. M. Hamada, Power series soution (PWS) of nucear reactor dynamics with newtonian temperature feedback, Annas of Nucear Energy, vo. 3, no. 1, pp , 23. [9]W.Z.Chen,B.Zhu,andH.F.Li, Theanayticasoution of point-reactor neutron-kinetics equation with sma step reactivity, Acta Physica Sinica, vo. 5, no. 8, pp , 24 (Chinese). [1] H.Li,W.Chen,F.Zhang,andL.Luo, Approximatesoutions of point kinetics equations with one deayed neutron group and temperature feedback during deayed supercritica process, Annas of Nucear Energy, vo. 34, no. 6, pp , 27. [11] T. Sathiyasheea, Power series soution method for soving point kinetics equations with umped mode temperature and feedback, Annas of Nucear Energy,vo.36,no.2,pp , 29.

6 6 Science and Technoogy of Nucear Instaations [12] Y. M. Hamada, Confirmation of accuracy of generaized power series method for the soution of point kinetics equations with feedback, Annas of Nucear Energy,vo.55,pp ,213. [13] Q. Zhu, X.-L. Shang, and W.-Z. Chen, Homotopy anaysis soution of point reactor kinetics equations with six-group deayed neutrons, Acta Physica Sinica,vo.61,no.7,ArticeID 721, 212. [14] S. D. Hamieh and M. Saidinezhad, Anaytica soution of the point reactor kinetics equations with temperature feedback, Annas of Nucear Energy,vo.42,pp ,212. [15] A. A. Naha, An anaytica soution for the point reactor kinetics equations with one group of deayed neutrons and the adiabatic feedback mode, Progress in Nucear Energy, vo.51, no. 1, pp , 29. [16]W.Chen,L.Guo,B.Zhu,andH.Li, Accuracyofanaytica methods for obtaining supercritica transients with temperature feedback, Progress in Nucear Energy,vo.49,no.4,pp.29 32, 27. [17] H. Li, W. Chen, L. Luo, and Q. Zhu, A new integra method for soving the point reactor neutron kinetics equations, Annas of Nucear Energy,vo.36,no.4,pp ,29. [18] Z. Q. Huang, Kinetics Base of Nucear Reactor, Peking University Press, Beijing, China, 27 (Chinese).

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