In the nuclear reactor design calculation, the thermal-hydraulic calculation is performed based on information on the heat generation distribution acquired from the nuclear calculation of the reactor core. In LWRs, the parameters such as moderator temperature, moderator density or void fraction, and fuel tem­perature obtained from the thermal-hydraulic calculation have a large effect on nuclear characteristics (nuclear and thermal-hydraulic feedback). The nuclear and thermal-hydraulic calculations should be mutually repeated until parame­ters of both calculations converge. This coupled calculation shown in Fig. 2.25 is referred to as the nuclear and thermal-hydraulic coupled core calculation (hereafter the N-TH coupled core calculation). The procedure of the N-TH coupled core calculation using the macroscopic cross section table prepared from lattice burnup calculations is discussed for a BWR example.

Two types of parameters are used in the coupled calculation. One is histor­ical parameters related to the fuel burnup and the other is instantaneous parameters without a direct relation to it. The historical parameters such as burnup are obtained from the core burnup calculation discussed in the next section. Here it is assumed that all of the historical parameters are known. The macroscopic cross section is first required for the nuclear calculation of the

reactor core. The space and time-dependent macroscopic cross section of a LWR is, for example, represented as Eq. (2.92).

^(r, t) = ZXia(F, R,E, p,p, Tf, Tm, NXe, NSm, SB, fCR,-‘) (2.92)

F: fuel type (initial enrichment, Gd concentration, structure type, etc.)

R: control rod type (absorber, number of rods, concentration, structure type, etc.) E: burnup (GWd/t)

p: historical moderator density (the burnup-weighted average moderator density)

p=j*pdE lf0EdE (2.93)

p: moderator density (also called the instantaneous moderator density against p) in which the void fraction a in BWRs is calculated from two densities p/ and pg in liquid and vapor phases, respectively

p = Pi(l — а) + рда (2.94)

Tf fuel temperature (the average fuel temperature in the fuel assembly)

Tm: moderator temperature (the average moderator temperature in the fuel assembly)

Nl: homogenized atomic number density of nuclide i (e. g., 135Xe or 149Sm) for which the atomic number density changes independently of the burnup or the historical moderator density and which has an effect on the core reactivity depending on the operation condition such as reactor startup or shutdown SB: concentration of soluble boron (e. g., boron in the chemical shim of PWRs) fCR: control rod insertion fraction (the fraction of control rod insertion depth:

0 < fCR < 1)

These parameters are given as operation conditions, initial guess values, or iterative calculation values, and then the macroscopic cross section for the core calculation is prepared as follows.

• E and R: Use the corresponding macroscopic cross section table.

• E and p: Interpolate the macroscopic cross section table in two dimensions.

• p, Tf, and Tm: Use the function fitting Eqs. (2.34), (2.35), and (2.36) based on the branch-off calculation.

• Nl and SB: Correct the macroscopic cross section in the following equations, using the changes from the condition, in which the homogenized atomic number density was prepared, and the homogenized microscopic cross section:

ANl(r, 0=№(г, t)~Ni(E, p) (2-96)

where X0,x, g (r, t) is the macroscopic cross section before the correction and aX g (r, t) is the homogenized microscopic cross section prepared in the same way as Eq. (2.92). N0 (E, p) is the atomic number density homogenized in the lattice burnup calculation and Nl (r, t) is the homogenized atomic number density in the core calculation. For example, the atomic number density of 135Xe after a long shutdown is zero and it reaches an equilibrium concen­tration depending on the neutron flux level after startup. The concentration of soluble boron is similarly corrected changing the homogenized atomic number density of 10B

• fcR: Weight and average the macroscopic cross sections X Xng and X O>gt at control rod insertion and withdrawal respectively with the control rod insertion fraction.

Since the cross section prepared in the iteration of the N-TH coupled core calculation reflects the feedback of instantaneous parameters, it is hereafter referred to as the feedback cross section. The nuclear calculation is performed with the feedback cross section by the nodal diffusion method or the finite difference method and it provides the effective multiplication factor, neutron flux distribution, power distribution, and so on. The distribution of homoge­nized atomic number density of nuclides such as 135Xe is calculated from necessity. For example, since 135Xe has an equilibrium concentration in a short time after startup as below, it is provided as a homogenized atomic number density used for the correction of microscopic cross section:

(2:98)

where yXe is the cumulative fission yield of 135Xe and AXe is the decay constant of 135Xe.

The thermal-hydraulic calculation is performed using the power distribution acquired from the nuclear calculation and gives the instantaneous moderator density p and the moderator temperature Tm. A BWR fuel assembly is enclosed in a channel box and the coolant flow inside the assembly is described as a 1D flow in a single channel with a hydraulic equivalent diameter (the “single channel model”). The core is modeled as a bundle of single channels, which are connected at the inlet and outlet, corresponding to each fuel assembly. This is referred to as the 1D multi-channel model. The calculation procedure for p and Tm in the 1D multi-channel model is as below.

(i) The total flow rate at the inlet is constant and the inlet flow rate W* for channel i is distributed (flow rate distribution). A guessed value is given to Wi if the flow rate distribution is unknown.

(ii) The axial heat generation distribution qi(z) (assembly linear power) of the fuel assembly from the nuclear calculation and the enthalpy at inlet

are used to calculate the enthalpy rise in each channel.

hi(z)=hfN + ^r f‘q’i (z)dz (2.99)

(iii) The physical property values at an arbitrary position r(i, z) such as the fluid temperature Tm(r) and p(r) are obtained from the steam table.

(iv) The void fraction distribution a(r) is acquired using qi(z), hi(z), physical property values, and some correlations based on the subcooled boiling model. The distribution of the instantaneous moderator density p(r) is calculated using the void fraction distribution.

p (r)=pi {l — a(r)}+pga(r) (2.100)

(v) The pressure drop APt by the channel is calculated using the information from (i) to (iv) including a(r) and correlations on pressure drop

(vi) The inlet flow rate is determined so that the pressure drop in all channels is equal. A new flow rate distribution W"ew(z) is decided from the flow rate distribution Wi in (i) and the pressure drop APi in (v).

(vii) The calculations from (i) to (vi) are repeated until the flow rate distribution is in agreement with the previously calculated one and the pressure drop of each channel is coincident (iterative calculation for flow rate adjustment). p(r) and Tm (r) at a convergence of the iterative calculation are taken and then the next fuel temperature calculation is done.

In a case such as the space-dependent kinetics calculation, the heat convection and conduction calculation is done using the same iteration approach of the N-TH coupled core calculation and then the temperature distribution in the fuel rod is calculated. In a case such as the steady-state calculation in which operation conditions are limited, it is common to calculate the fuel temperatures at representative linear power and fuel burnup in advance and then interpolate the linear power and burnup obtained from the core calculation into the fuel tem­perature table to simplify calculation of the fuel temperature Tf (r).

The calculations mentioned above, the feedback cross section calculation, the nuclear calculation, the thermal-hydraulic calculation, and the fuel temper­ature calculation, are repeatedly performed until the effective multiplication factor and power distribution reach convergence.