The singularly perturbed method (SPM) is proposed to obtain the analytical solution for the delayed supercritical process of nuclear reactor with temperature feedback and small step reactivity inserted. The relation between the reactivity and time is derived. Also, the neutron density (or power) and the average density of delayed neutron precursors as the function of reactivity are presented. The variations of neutron density (or power) and temperature with time are calculated and plotted and compared with those by accurate solution and other analytical methods. It is shown that the results by the SPM are valid and accurate in the large range and the SPM is simpler than those in the previous literature.

1. Introduction

The analysis of variation of neutron density (or power) and reactivity with time under the different conditions is an important content of nuclear reactor physics or neutron kinetics [1–7]. Some important achievements on the supercritical transient with temperature feedback with big (ρ0>β) or small (ρ0<β) reactivity inserted have been approached through the effort of many scholars [7–12]. The studies on the delayed supercritical transient with small reactivity inserted and temperature feedback are introduced in the related literature [13–15], in which the explicit function of density (or power) and reactivity with respect to time is derived mainly with decoupling method, power prompt jump approximation, precursor prompt jump approximation, temperature prompt jump approximation [10, 16], and so forth. From the detailed analysis and comparison of the results in the early and recent literature [7, 12, 14], it is found that some results have certain limit and rather big error under the particular conditions. In present work, the variation law of power, reactivity, and precursor density with respect to time at any level of initial power is obtained by the singularly perturbed method (SPM). All the results are compared with those obtained by the numerical solution which tend to the accurate solution under very small time step size [17]. It is proved that the SPM is correct and reliable and is simpler than the analytical methods by the related literature.

2. Theoretical Derivation

The point reactor neutron kinetics equations with one group of delayed neutrons are [3, 4]
(1)dn(t)dt=ρ(t)-βln(t)+λC(t),(2)dC(t)dt=βln(t)-λC(t),
where n is the average neutron density, t is the time, ρ is the reactivity, β is the total fraction of the delayed neutron, l is the prompt neutron lifetime, λ is the radioactive decay constant of delayed neutron precursor, and C is the average density of delayed neutron precursor. When multiplied with a certain coefficient, n represents the power. It is assumed that the reactor has a negative temperature coefficient of reactivity α (α>0) when a small step reactivity ρ0(<β) is inserted. Consider the temperature feedback, and the real reactor reactivity is
(3)ρ=ρ0-αT,
where T is the temperature increment of the reactor, namely, T=Ts-T0, where Ts and T0 are the instantaneous temperature and initial temperature, respectively. After the reactivity ρ0 is inserted into the reactor the adiabatic model is still employed [3, 15]; then we have
(4)dTdt=Kcn(t),
where Kc is the reciprocal of thermal capacity of reactor.

Combining (3) and (4) results in
(5)dρdt=-αKcn(t).

Substituting (2) into the derivative of (1) with respect to t yields
(6)ld2ndt2=(ρ-β)dndt+ndρdt+λβn-lλ2C.

Substituting λC obtained from (1) into (6) and simplifying it yields
(7)d2ndt2=(ρ-βl-λ)dndt+nldρdt+λρnl.
The transient process is supposed to begin at t=0 and ρ(0)≠0 [15, 17], so the initial conditions can be given as ρ(0)=ρ0, n(0)=n0, (dn/dt)|t=0=ρ0n0/l, (d2n/dt2)|t=0=((ρ0-β)/l)(ρ0n0/l)-(αKcn02/l), where n0 is the initial neutron density (or power).

For |(ρ-β)/l|≫λ, β/l≫λ [3], (1) and (2) are stiff equations. According to the singularly perturbed method [18], solution of n(t) includes the inner solution nf(t) in inner part and the outer solution nl(t) in outer part. In this paper both inner and outer solutions are approximated to be zero order:
(8)n(t)=nf(t)+nl(t).
The initial conditions are
(9)n0=nf0+nl0,(10)n′(0)=nf′(0)+nl′(0)=ρ0n0l,(11)n′′(0)=nf′′(0)+nl′′(0)=ρ0-βlρ0n0l-αKcn02l,
where nf0 is the initial value of inner solution and nl0 is the initial value of outer solution.

Substituting (8) into (7) and (5), respectively, yields
(12)d2(nf+nl)dt2=(ρ-βl-λ)d(nf+nl)dt+(nf+nl)ldρdt+λρ(nf+nl)l,dρdt=-αKc[nf(t)+nl(t)].

In the outer part the inner solution attenuates to zero, and dnf/dt and d2nf/dt2 can be neglected, namely, dnf/dt≈0 and d2nf/dt2≈0; therefore, in the outer part (12) can be simplified as follows:
(13)d2nldt2=ρ-βldnldt+nlldρdt-λdnldt+λρnll,(14)dρdt=-αKcnl.

Because nl(t) varies slowly and l≈10-4 s, compared to other terms, the term on the left side of (13) can be neglected, and then we have
(15)(β-ρ+λl)dnldt=(dρdt+λρ)nl.

Combining (14) and (15) results in
(16)dnldρ=αKcnl-λραKc(β-ρ+λl).

Integrating (16) subjective to the initial conditions ρ(0)=ρ0 and nl(0)=nl0 yields
(17)nl=λ(ρ02-ρ2)2αKc(β-ρ+λl)+(β-ρ0+λl)(β-ρ+λl)nl0.

Substituting (17) into (14) leads to
(18)-dρdt=λ(ρ02-ρ2)+2αKc(β-ρ0+λl)nl02(β-ρ+λl).

With the initial conditions ρ(0)=ρ0, the solution of (18) is
(19)t=β+λlλρ1ln(ρ1-ρρ1+ρ)(ρ1+ρ0ρ1-ρ0)+1λln[(ρ12-ρ02)(ρ12-ρ2)],
where ρ1=ρ02+2αKc(β-ρ0+λl)nl0/λ.

In the inner part nl(t) is assumed to be constant nl(t)=nl(0)=nl0, and dnl(t)/dt≈(dnl(t)/dt)|t=0, d2nl(t)/dt2≈(d2nl(t)/dt2)|t=0, so (12) can be simplified as follows, respectively:
(20)d2nfdt2+nl′′(0)=(ρ-βl-λ)(dnfdt+nl′(0))+(nf+nl0)l(dρdt+λρ),(21)dρdt=-αKc[nf(t)+nl0].

In addition, the temperature feedback is not fast enough to affect the reactivity, and in the inner part it is assumed that ρ(t)=ρ0, dρ(t)/dt≈-αKcn0. From (20) we can get
(22)ld2nfdt2=(ρ0-β-λl)dnfdt+(λρ0-αKcn0)(nf+nl0)+(ρ0-β-lλ)nl′(0)-nl′′(0)l.

The solution of (22) is
(23)nf=-nl0-(ρ0-β-lλ)nl′(0)-nl′′(0)l(λρ0-αKcn0)+C1exp(12l×((ρ0-β-lλ)2+4l(λρ0-αKcn0)(ρ0-β-lλ)+(ρ0-β-lλ)2+4l(λρ0-αKcn0))t12l)+C2exp(12l×((ρ0-β-lλ)2+4l(λρ0-αKcn0)(ρ0-β-lλ)-(ρ0-β-lλ)2+4l(λρ0-αKcn0))t12l).

The fast varying part nf(t) is assumed to attenuate to zero in the inner part, so C1(t)=0 and
(24)-nl0-(ρ0-β-lλ)nl′(0)-nl′′(0)l(λρ0-αKcn0)=0.

Then we can get the fast varying part nf(t) in the inner part as follows:
(25)nf=nf0exp(12l×((ρ0-β-lλ)2+4l(λρ0-αKcn0)(ρ0-β-lλ)-(ρ0-β-lλ)2+4l(λρ0-αKcn0))t12l).

From (25) we have
(26)nf0′=nf02l((ρ0-β-lλ)2+4l(λρ0-αKcn0)(ρ0-β-lλ)-(ρ0-β-lλ)2+4l(λρ0-αKcn0)).

Combining (9), (11), (13), (24), and (26) results in(27)nf0=2l(αKcn0-λρ0)n0/(ρ0-β-lλ)2-2ρ0n0/(ρ0-β-lλ)2l(αKcn0-λρ0)/(ρ0-β-lλ)2-(1-1+4l(λρ0-αKcn0)/(ρ0-β-lλ)2),nl0=2ρ0n0/(ρ0-β-lλ)-n0(1-1+4l(λρ0-αKcn0)/(ρ0-β-lλ)2)2l(αKcn0-λρ0)/(ρ0-β-lλ)2-(1-1+4l(λρ0-αKcn0)/(ρ0-β-lλ)2).

Substituting (27) into (25) and (17) can get nf(t) and nl(t); then the neutron density (or power) will be obtained by (8).

Combining (1) and (2) results in
(28)d(n+C)dt=ρln.

Eliminating the time variable in (5) and (28) leads to
(29)d(n+C)dρ=-ραKcl.

Integrating (29) with the initial conditions n(0)=n0, C(0)=βn0/λl and ρ(0)=ρ0 yields
(30)C(t)=ρ02-ρ22αKcl+(1+βλl)n0-n(t).

Equations (8), (17), (25), (27), and (30) are the new analytical expressions derived by this paper.

3. Calculation and Analysis

The PWR with fuel ^{235}U is taken as an example with parameters β=0.0065, l=0.0001 s, λ=0.0774 1/s, Kc=0.05 K/MW·s, and α=5×10-5 1/K [8, 13]. For the reactor with the initial power 1 MW, while reactivity ρ0=0.5β and ρ0=0.8333β is inserted, respectively, the variations of reactivity, temperature, power with time, and power with reactivity are presented in Figures 1, 2, 3, and 4. The curves with smaller change are for ρ0=0.5β and the curves with larger change are for ρ0=0.8333β. The solid line notes the accurate solution by the best basic function method with very small step size [17]. The results of this paper and the accurate solution as well as the temperature prompt jump (TPJ) method in the literature [10] are almost the same and hard to distinguish in Figures 1–4. The short dashed-dot line and long dashed-dot line represent the results of precursor prompt jump (PrPJ) method in the literature [9] and the small parameter (SmP) method in the literature [15], respectively. The dashed line notes the results of power prompt jump (PPJ) method in the literature [16]. The difference is caused by the approximate treatment to obey the methods of PrPJ, SmP, PPJ, and SmP. Furthermore the variation in the vicinity of prompt supercritical process is also calculated and is shown in Figures 5 and 6. The correct results cannot be obtained by the small parameter method (SmP) in the vicinity of prompt supercritical process and are not shown in Figures 5 and 6.

Variation of output power with time while inserting step reactivity ρ0=0.5β and ρ0=0.8333β.

Variation of total reactivity with time while inserting step reactivity ρ0=0.5β and ρ0=0.8333β.

Variation of temperature rise of reactor with time while inserting step reactivity ρ0=0.5β and ρ0=0.8333β.

Variation of output power with reactivity while inserting step reactivity ρ0=0.5β and ρ0=0.8333β.

Variation of output power with time while inserting step reactivity 0.995β.

Variation of output power with time while inserting step reactivity 0.998β.

(1) From Figures 1–6 it can be concluded that very good results cannot be obtained by the precursor prompt jump (PrPJ) method to calculate the delayed supercritical progress with 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 power prompt jump (PPJ) method and are better than those by the precursor prompt jump (PrPJ) method, but the accuracy of results by the small parameter method decreases with the increase of the reactivity inserted. The power is negative when the small parameter method is used to calculate the transient process in the vicinity of prompt supercritical state. From Figures 1–4, it can be seen that the small parameter method is more suitable for the calculation of reactivity and temperature increase than for that of power.

(3) The results are quite precise using the power prompt jump (PPJ) method for the delayed supercritical process, but the main problem compared to the accurate solution is that some displacement exists along time axis. Furthermore it should be pointed out that each power peak value obtained by the precursor prompt jump (PrPJ) method, power prompt jump (PPJ) method, or small parameter (SmP) method is lower than that obtained by the accurate solution or singularly perturbed method (SPM) see Figure 1.

(4) From Figures 1–4 it can be also found that the temperature prompt jump method (TPJ) and the singularly perturbed method (SPM) in this paper are the two most precise methods for the delayed supercritical process with small step reactivity and temperature feedback. However from Figures 5 and 6 it can be seen that as the reactivity inserted increases to the vicinity of prompt supercritical process, the total discrepancy of power by the TPJ method is larger than that by the SPM or PPJ method, and the irrelevant phenomena that the power jumps at first and then decreases monotonously from the peak will appear in the TPJ method as shown in Figure 6.

4. Conclusions

The analytical expressions of power (or neutron density), reactivity, the precursor power (or density), and temperature increase with respect to time are derived for the delayed supercritical process with small reactivity (ρ0<β) and temperature feedback by the singularly perturbed method. Compared with the results by the accurate solution and other methods in the literature, it is shown that the singularly perturbed method (SPM) in this paper is valid and accurate in the large range and is simpler than those in the previous literature. The method in this paper can provide a new theoretical foundation for the analysis of reactor neutron dynamics.

Acknowledgments

This research is supported by the National Natural Science Foundation of China (Project no. 11301540) and the Natural Science Foundation of Naval University of Engineering.

GuptaH. P.TrasiM. S.Asymptotically stable solutions of point-reactor kinetics equations in the presence of Newtonian temperature feedbackVan DamH.Dynamics of passive reactor shutdownHetrickD. L.StaceyW. M.NahlaA. A.ZayedE. M. E.Solution of the nonlinear point nuclear reactor kinetics equationsEspinosa-ParedesG.Polo-LabarriosM.-A.Espinosa-MartínezE.-G.Valle-GallegosE. D.Fractional neutron point kinetics equations for nuclear reactor dynamicsAboanberA. E.NahlaA. A.Solution of the point kinetics equations in the presence of Newtonian temperature feedback by Padé approximations via the analytical inversion methodAboanberA. E.HamadaY. M.Power series solution (PWS) of nuclear reactor dynamics with newtonian temperature feedbackChenW. Z.ZhuB.LiH. F.The analytical solution of point-reactor neutron-kinetics equation with small step reactivityLiH.ChenW.ZhangF.LuoL.Approximate solutions of point kinetics equations with one delayed neutron group and temperature feedback during delayed supercritical processSathiyasheelaT.Power series solution method for solving point kinetics equations with lumped model temperature and feedbackHamadaY. M.Confirmation of accuracy of generalized power series method for the solution of point kinetics equations with feedbackZhuQ.ShangX.-L.ChenW.-Z.Homotopy analysis solution of point reactor kinetics equations with six-group delayed neutronsHamiehS. D.SaidinezhadM.Analytical solution of the point reactor kinetics equations with temperature feedbackNahlaA. A.An analytical solution for the point reactor kinetics equations with one group of delayed neutrons and the adiabatic feedback modelChenW.GuoL.ZhuB.LiH.Accuracy of analytical methods for obtaining supercritical transients with temperature feedbackLiH.ChenW.LuoL.ZhuQ.A new integral method for solving the point reactor neutron kinetics equationsHuangZ. Q.