Since the time variation of nuclear characteristics in the core is relatively slow with burnup, the normal N-TH coupled core calculation is carried out using the time-independent equation at each time step as shown in Fig. 2.26.

*N = *0 + ^

—

}— HUn)

H : Historical Parameters I : Instantaneous Parameters

Fig. 2.26 Renewal of historical parameters in core burnup calculation

~(t0) represents the distribution of historical parameters in the core at time t0. The burnup E(r, t0) and the historical moderator density p(r, t0) are the typical historical parameters, and other ones can be used depending on reactor design code. The burnup in conventional LWRs is usually expressed in the unit of (MWd/t), which is measured as the energy (MWd) produced per metric ton of heavy metal initially contained in the fuel. That is, the burnup is the time — integrated quantity of the thermal power.

For the historical moderator density, a spatial distribution of burnup E(r, t0) at time t0 and a spatial distribution of thermal power density q //0(r, t0) obtained from the N-TH coupled core calculation are considered next. The thermal power density is assumed not to change much through the interval to the next time step (t0 < t < t0+ ^t). Then, the burnup distribution at time tN (=t0 + ^t) can be given by

E(r, tN)~E(?, to)+C’q"'(r, to)At (2-101)

where C is a constant for unit conversion.

Next, the distribution of the historical moderator density is recalculated. Since it is the burnup-weighted average value of the instantaneous moderator density, the historical moderator density at time tN is given by

where E0 and EN are abbreviations of E(r, t0) and E(r, tN), respectively. Here, if the distribution of instantaneous moderator density obtained from the N-TH coupled core calculation at time t0 remains almost constant during the time interval (t0 < t < tN), ~(r, tN) can be expressed as Eq. (2.103).

Hence, the historical moderator density at the next time step tN can be calculated using the historical moderator density and the instantaneous mod­erator density obtained from the N-TH coupled core calculation, at time t0. Thus, the core burnup calculation can proceed until the target burnup by renewing the historical parameters with the burnup step.

[8] Space-dependent kinetics calculation

A space-dependent transient analysis for a short time is made using the time— dependent diffusion equation as

(2.104)

where в is the delayed neutron fraction, Xg is the prompt neutron spectrum, and jdp is the neutron spectrum of delayed neutron group i. Ci is the precursor concentration of delayed neutron group i and Xi is its decay constant. Com­parison with the normal multi-group diffusion equation [see Eq. (2.90)] shows that the time derivative term in the LHS is added and a different expression of the fission source is given by the fourth and fifth terms of the RHS. Both terms represent the prompt and delayed neutron sources which are classified by time behavior. The precursor concentration balance equation can be given by

(2.105)

For an extremely short time difference of 10 4 to 10 3 (t = told + ^t), the following approximations are introduced.

і дф„ ^ і /ф„-ф^

Vg dt VgV At /

The macroscopic cross section in Eq. (2.104) is the feedback cross section similar to that in the N-TH coupled core calculation. Since At is very short, the macroscopic cross section at time t can be substituted by the macroscopic cross section corresponding to the instantaneous parameters at t = told.

Zx, g(r, *)~Х*,Др(г, tM Tm(r, tMX Tf(r, tM -) (2-108)

Substitution of Eqs. (2.106) and (2.107) into Eqs. (2.104) and (2.105) gives Eq. (2.109).[6]

V’DgVtpg (sr, j,+ J фд ^ Xy—y фд

VgAt / д’фд

(2-109)

І і 1+XiAt )Xj ko It 1 +kiAt VgAt J

=0

This is in the same form as that of the steady-state multi-group diffusion equation for a fixed-source problem and it can be solved for фя (r, t) by the numerical solutions mentioned until now. Then, the thermal-hydraulic calcu­lation is performed and the instantaneous parameters are recalculated, simi­larly to the steady-state N-TH coupled calculation. The feedback cross section at the next time step is in turn prepared from Eq. (2.108) and the diffusion equation of Eq. (2.109) is solved again. These repeated calculations give the neutron flux distribution and its corresponding thermal power distribution at each time step.

Since it is hard to use a fine time interval less than 10_3 s in a practical code for the space-dependent kinetics calculation, a large number of considerations have been introduced for high-speed and high-accuracy calculation within a practical time [19]; two examples are the method to express the time variation in neutron flux as an exponential function and the method to describe the neutron flux distribution as the product of the amplitude component with fast time-variation and the space component with relatively smooth variation (the improved quasi-static method). The fast nodal diffusion method is a general solution in the analysis code for LWRs.

The point kinetics model widely used is in principle based on the first order perturbation theory where it is assumed that the spatial distribution of neutron flux does not change much even though the cross section varies by some external perturbation. Hence, when the neutron flux distribution is consider­ably distorted due to such an event as a control rod ejection accident, the space-dependent kinetics analysis must accurately predict the time behavior of the reactor.