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

The singularly perturbedmethod (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.


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][2][3][4][5][6][7].Some important achievements on the supercritical transient with temperature feedback with big ( 0 > ) or small ( 0 < ) reactivity inserted have been approached through the effort of many scholars [7][8][9][10][11][12].The studies on the delayed supercritical transient with small reactivity inserted and temperature feedback are introduced in the related literature [13][14][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.

Theoretical Derivation
The point reactor neutron kinetics equations with one group of delayed neutrons are [3,4] where  is the average neutron density,  is the time,  is the reactivity,  is the total fraction of the delayed neutron,  is the prompt neutron lifetime,  is the radioactive decay constant of delayed neutron precursor, and  is the average density of delayed neutron precursor.When multiplied with a certain coefficient,  represents the power.It is assumed that the reactor has a negative temperature coefficient of reactivity  ( > 0) when a small step reactivity  0 (<) is inserted.Consider the temperature feedback, and the real reactor reactivity is where  is the temperature increment of the reactor, namely,  =   −  0 , where   and  0 are the instantaneous temperature and initial temperature, respectively.After the reactivity  0 is inserted into the reactor the adiabatic model is still employed [3,15]; then we have where   is the reciprocal of thermal capacity of reactor.Combining (3) and ( 4) results in Substituting (2) into the derivative of (1) with respect to  yields Substituting  obtained from (1) into ( 6) and simplifying it yields The transient process is supposed to begin at  = 0 and (0) ̸ = 0 [15,17], so the initial conditions can be given as , where  0 is the initial neutron density (or power).
For |( − )/| ≫ , / ≫  [3], (1) and ( 2) are stiff equations.According to the singularly perturbed method [18], solution of () includes the inner solution   () in inner part and the outer solution   () in outer part.In this paper both inner and outer solutions are approximated to be zero order: The initial conditions are where  0 is the initial value of inner solution and  0 is the initial value of outer solution.Substituting ( 8) into ( 7) and ( 5), respectively, yields In the outer part the inner solution attenuates to zero, and   / and  2   / 2 can be neglected, namely,   / ≈ 0 and  2   / 2 ≈ 0; therefore, in the outer part (12) can be simplified as follows: Because   () varies slowly and  ≈ 10 −4 s, compared to other terms, the term on the left side of ( 13) can be neglected, and then we have Combining ( 14) and ( 15) results in Integrating ( 16) subjective to the initial conditions (0) =  0 and   (0) =  0 yields Substituting ( 17) into ( 14) leads to With the initial conditions (0) =  0 , the solution of ( 18) is where (12) can be simplified as follows, respectively: In addition, the temperature feedback is not fast enough to affect the reactivity, and in the inner part it is assumed that () =  0 , ()/ ≈ −   0 .From (20) we can get The solution of ( 22) is The fast varying part   () is assumed to attenuate to zero in the inner part, so  1 () = 0 and Then we can get the fast varying part   () in the inner part as follows: From (25) we have Combining ( 9), ( 11), ( 13), (24), and (26) results in Substituting ( 27) into ( 25) and ( 17) can get   () and   (); then the neutron density (or power) will be obtained by (8).

Calculation and Analysis
The PWR with fuel 235 U is taken as an example with parameters  = 0.0065,  = 0.0001 s,  = 0.0774 1/s,   = 0.05 K/MW⋅s, and  = 5 × 10 −5 1/K [8,13].For the reactor with the initial power 1 MW, while reactivity  0 = 0.5 and  0 = 0.8333 is inserted, respectively, the variations of reactivity, temperature, power with time, and power with reactivity are presented in Figures 1, 2  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. (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.

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 ( 0 < ) 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.

Figure 5 :
Figure 5: Variation of output power with time while inserting step reactivity 0.995.

Figure 6 :
Figure 6: Variation of output power with time while inserting step reactivity 0.998.