In nuclear reactors such as LWRs, in which the coolant serves as moderator, the moderator density varies with temperature change and boiling of coolant, and subsequently the reactivity and reactor power vary. The power distribution also varies with the moderator density change. The heat transfer calculation is performed to evaluate those core characteristics using the heat generation distribution in the core acquired from the nuclear calculation. The nuclear calculation is performed again using the macroscopic cross section changed by the coolant or moderator density distribution obtained in the heat transfer calculation. These calculations are repeated until a convergence. This is the nuclear and thermal-hydraulic coupled core calculation (the N-TH coupled core calculation) mentioned earlier.

A 2D core model cannot precisely handle power and burnup in each part of the core. This usually needs a 3D nuclear calculation, which is combined with the heat transfer calculation in the single channel model for each fuel assembly. Figure 2.29 shows a 3D N-TH coupled core calculation model which is a symmetric quadrant. Since the fuel assembly burnup differs with the power

Homogenized ruel hlements

Single Channel

I hermal-Hydraulic

1/4 Core

Fuel

Assembly

Fig. 2.29 3D nuclear and thermal-hydraulic coupled core calculation model

history at each position, the fuel assembly is axially segmented and the macro­scopic cross section at each section is prepared for the core calculation. Each fuel assembly, which consists of a large number of fuel rods, is described as the single-channel thermal-hydraulic calculation model. The heat transfer calculation is carried out with the axial power distribution of each assembly obtained from the nuclear calculation and it provides an axial moderator density distribution.

The N-TH coupled core calculation above is repeatedly continued until it satisfies a convergence criterion and then it gives the macroscopic cross section which is used for calculations such as the core power distribution. This process is repeated at each burnup step and is referred to as the 3D nuclear and thermal- hydraulic coupled core calculation (3D N-TH coupled core calculation). In the practical calculation, as mentioned in Sect. 2.1, the macroscopic cross sections are tabulated in advance with respect to parameters such as fuel burnup and enrichment, moderator density, and so on. Each corresponding cross section is used in the N-TH coupled core calculation.

New Set of Design Parameters

Burnup Distribution
Converge?

Design Criteria?

END

Fig. 2.30 Flow chart of equilibrium code design

Figure 2.30 shows a flow chart of an equilibrium core design using the 3D N-TH coupled core calculation [25]. Macroscopic cross section data are previ­ously prepared from the lattice and/or assembly burnup calculation and then tabulated with fuel burnup and enrichment, moderator density, and so on. The macroscopic cross section is used in the 3D multi-group diffusion calculation to obtain the core power distribution, which is employed again in the thermal — hydraulic calculation.

Usually several different types of fuel assemblies are loaded into the core according to the purpose such as radial power distribution flattening or neutron leakage reduction. In the first operation of a reactor, all fresh fuel assemblies are loaded but they may have several different enrichments of fuel. This fresh core is called the initial loading core. After one cycle operation, low reactivity assemblies representing about 1/4 to 1/3 of the loaded assemblies are discharged from the reactor. The other assemblies are properly rearranged in the core considering the burnup distribution and other new fresh ones are loaded.

In the end of each cycle, the core is refueled as in the above way and then the reactor restarts in the next cycle. While such a cycle is being repeated, the characteristics of the core vary but gradually approach equilibrium quantities. When finally the characteristics at N +1 the cycle have little change in comparison with those at the N cycle, the core is called the equilibrium core. In certain cases, it shows equilibrium characteristics at N and N +2 cycles and these are also generally referred to as the equilibrium core. By contrast, the core at each cycle is called the “transition core” until it reaches the equilibrium core.

In the equilibrium core design, first a proper burnup distribution of the core is guessed at the beginning of the cycle (BOC), and a control rod and fuel loading pattern is assumed. Then, the N-TH coupled core calculation is performed for one cycle based on the assumed core design with the burnup distribution. After the core calculation, the refueling according to the fuel loading pattern is followed by evaluation of the new burnup distribution at BOC of the next cycle. Such evaluation after one cycle calculation is repeated until the burnup distribution at BOC converges, at which time it is regarded as attaining the equilibrium core.

The core design criteria, such as shutdown margin, moderator density coef­ficient, Doppler coefficient, fuel cladding temperature, and maximum linear power density, are applied to the equilibrium core. If the equilibrium core does not satisfy the core design criteria, the equilibrium core design is performed again after reviewing fuel enrichment zoning, concentration and number of burnable poison rods, fuel loading and control rod pattern, coolant flow rate distribution, and so on.

A 3D N-TH coupled core calculation is also performed for fuel management such as fuel loading and reloading. In this case, the equilibrium core is searched from an initial loading core. Some fuel assemblies are replaced with new fresh ones after the first cycle calculation and the new core configuration is consid­ered in the next cycle calculation. The calculation procedure is also similar to Fig. 2.30.

It is noted again that operation management such as fuel loading pattern and reloading, and control rod insertion can be considered in the 3D N-TH coupled core calculation. Figure 2.31 shows an example of 3D core power distribution and coolant outlet temperature distribution.

In the core design and operation management, it is necessary to confirm that maximum linear power density (or maximum linear heat generation rate, MLHGR) is less than a limiting value. Figure 2.32 shows variation in MLHGR and maximum cladding surface temperature (MCST) with cycle burnup, which was from a 3D N-TH coupled core calculation for a supercritical water-cooled reactor (SCWR) as an example [26]. The limiting values at normal operation are 39 kW/m and 650 °C respectively in this example. The core calculation was performed according to the fuel loading and reloading pattern as described in Fig. 2.33. It is noted that the third cycle fuel assemblies are loaded in the core peripheral region to reduce neutron leakage (low leakage loading pattern). The control rod pattern is shown in Fig. 2.34.

The 3D N-TH coupled core calculation has been done for BWRs for many years. Since there are not many vapor bubbles in PWRs due to subcooled boiling, the N-TH coupled core calculation has not always been necessary, but it is being employed recently for accuracy improvement.

Outlet Temperature Distribution

Fuel Assembly

Fuel Loading Pattern

Power Distribution

Fig. 2.31 3D core power distribution and coolant outlet temperature distribution

Fig. 2.32 Example of Variation in maximum linear power density and maximum cladding surface temperature with cycle burnup

Fig. 2.33 Example of fuel loading and reloading pattern

In the core calculation, one fuel assembly is modeled not as a bundle of its constituent fuel rods but a homogenized assembly, which is represented as a single channel in the thermal-hydraulic calculation. However, the power of each fuel rod is actually a little different and hence it leads to a different coolant flow rate and a different coolant enthalpy in the flow path (subchannel) between fuel rods. This also causes a difference such as in cladding temperature which is particular to a single-phase flow cooled reactor. The heat transfer analysis for coolant flow paths formed by many fuel rods is called subchannel analysis, and it is used in evaluation of cladding temperature in liquid sodium-cooled fast reactors and so on.

In computational fluid dynamics (CFD), the behavior of fluid is evaluated through numerical analysis of the simultaneous equations: Navier-Stokes equations, continuity equation, and in some cases perhaps the energy conser­vation equation and state equation. It subsequently requires a huge computing time, but a detailed heat transfer analysis can be made. The CFD analysis can be used to calculate a circumferential distribution of fuel rod temperature in a single-phase flow cooled reactor or to analyze detailed fluid behavior inside the reactor vessel and pipes. Figure 2.35 depicts the three heat transfer analysis models mentioned above including the CFD analysis model.

Analysis

Fig. 2.35 Heat transfer analysis models