Because of the high temperature and high pressure characteristics of supercritical water-cooled reactor (SCWR), the thermal hydraulic performance of SCWR is greatly different from pressurized water reactor (PWR), which makes the current PWR fuel rod performance analysis codes are no longer applicable to SCWR. In this research, the irradiation swelling, irradiation densification, thermal expansion, thermal creep, plastic deformation, irradiation creep and irradiation hardening of UO2 pellet, and stainless steel cladding were considered; the gas conductance and radiant conductance of gap heat transfer were considered, the forced convective heat transfer on the outer surface of cladding was considered. Meanwhile, the irradiation effects and the thermal effects on the materials parameters such as thermal conductivity, specific heat, and young’s modulus were also considered in this research. With the help of abaqus software, the related user-defined subroutines were developed, and the irradiation effects and thermal effects of SCWR fuel were introduced into the numerical simulation, and then completed the analysis of SCWR fuel rods’ performance under steady power conditions. Some reference suggestions for the design and development of SCWR fuel could be provided by the establishment of this numerical simulation method.

## Introduction

The fuel performance analysis codes play a very important role in the design, safety analysis, in pile and out pile test study of nuclear fuel rods. Supercritical water-cooled reactor (SCWR) owns the high temperature, high pressure characteristics, and there are many important improvements (mainly in the structure of materials, fuel assembly design, etc.) compared with the existing light water-cooled reactor, because of these improvements, its thermal hydraulic characteristics are different from pressurized water reactor (PWR); meantime, the cladding material of SCWR is also different from PWR, which makes the existing PWR fuel performance analysis codes does not suit for SCWR.

In this article, with the help of commercial finite element analysis software abaqus, the irradiation effects and thermal effects of SCWR fuel were introduced into the numerical simulation through the development of related user-defined subroutines, and then completed the analysis of SCWR fuel rods’ performance under the steady power conditions. This simulation method could be helpful for the development of SCWR fuel in China.

## Material Model

abaqus provides a reasonable general framework for fuel performance modeling. Essential for the intended application, abaqus provides a user interface permitting the inclusion of user-developed models (via FORTRAN subroutines) to simulate fuel-specific phenomena. The phenomena considered in this study, include irradiation swelling (both gaseous swelling and solid swelling), densification, thermal expansion, plastic deformation, irradiation hardening of UO2 pellet, and stainless steel cladding. The effects of fast neutron flux (the material performance is mainly affected by fast neutron flux compared with thermal neutron flux) and temperature on the material parameters, such as thermal conductivity, specific heat, and Young's modulus were also considered. All of the above-mentioned models are mainly taken from MATPRO [1].

Volumetric swellings due to solid and gaseous fission products are correlated to burnup empirically as follows [2]:
$Δεsw−s=5.577×10−5ρΔBu$
(1)
$Δεsw−g=1.96×10−31ρΔBu(2800−T)11.73e−0.0162(2800−T)e−0.0178ρBu$
(2)
where $Δεsw−g$ denotes the solid swelling increment, $Δεsw−g$ denotes the gaseous swelling increment, $ρ$ denotes the density (kg/m3), T denotes the local temperature (K), $ΔBu$ and Bu are burnup increment and burnup (fissions/atoms U), respectively. Fuel densification at the early stage of irradiation is expressed as a function of burnup [2]
$εD=Δρ0(e(Buln(0.01)CDBuD)−1)$
(3)
where $εD$ is the densification strain, $Δρ0$ is the total densification equivalent to a fraction of theoretical density, Bu is the burnup (fissions/atoms-U), BuD is the burnup at which densification stops (5 MWD/kgU assumed in this work), and the parameter CD is given by
$CD={1.0 T≥1023K7.2−0.0086(T−298)T<1023K$
(4)
where T is the temperature (K). The thermal expansion coefficient is [2]
$ΔLL=K1T−K2+K3e(−EDKT)$
(5)

where $K1$, $K2$, $K3$, $ED$ are model constants, and $K$ is the Boltzmann constant.

Thermal conductivity of the UO2 pellet is described by the form proposed by Lucuta et al. [3]
$kUO2=k0⋅FD⋅FP⋅FM⋅FR$
(6)
where $kUO2$ is the thermal conductivity (W/m K) of irradiated pellet, $k0$ is the thermal conductivity (W/m K) of as-fabricated UO2 pellet
$k0=10.0375+2.165×10−4T+[4.715×109T2]exp (−16361T)$
(7)

where T is temperature (K).

FP, FD are the dissolution and precipitation impact factor of fission products on the thermal conductivity of as-fabricated UO2 pellet, respectively; FM is the porosity impact factor on the thermal conductivity of as-fabricated UO2 pellet; FR is the irradiation impact factor on the thermal conductivity of as-fabricated UO2 pellet. The calculation expressions as follows:
$FP=[1.09B3.265+0.0643BT]ar tan [11.09B3.265+0.0643BT]$
(8)
$FP=1+(0.019B3−0.019B)[11+exp (1200−T100)]$
(9)
$FP=1−P1+(s−1)P$
(10)
$FR=1−0.21+exp (T−90080)$
(11)
where B is the burnup (atom %), T is the temperature (K), P is the porosity (%), and S is the shape factor. The Young's modulus of UO2 pellet is considered as a constant, 219,000 MPa; Poisson's ratio of UO2 pellet is 0.345, and the density of UO2 pellet is 10.96 g/cm3. Another important model is the gap conductance between pellet and cladding, and it is given by
$hgap=hg+hr$
(12)

where $hgap$ is the total conductance (W/m2 K), $hg$ is the gas conductance (W/m2 K), and $hr$ is the distance conductance (W/m2 K) as a result of radiant heat transfer. Detailed model parameters can be found in Ref. [2].

Thermal conductivity of the 304 stainless steel from the room temperature to the melting point is considered as temperature dependent [4]
$k=15.027+1.287×10−2T$
(13)
where k is thermal conductivity (W/m °C); T is the temperature (°C). The Young's modulus of 304 stainless steel is as follows [4]:
$E=−0.0823T+201$
(14)

where E is the Young's modulus (GPa), and T is the temperature (°C). Poisson's ratio of 304 stainless steel is 0.29. The density of 304 stainless steel is 7.45 g/cm3.

The thermal expansion of 304 stainless steel is considered as temperature dependent [4]
$α=4.277×10−3T+15.981$
(15)

where α is thermal expansion coefficient (10−6/°C), T is the temperature (°C), temperature range: −20°C ≤ T ≤ 760°C.

The yield strength of 304 stainless steel is considered as temperature dependent [4]
$σ={−0.2441T+1176.881 T≤450 °C1657.278−27.830T T>450 °C$
(16)

where $σ$ is the yield strength (MPa), T is temperature (°C).

## Governing Equations

The temperature distribution is determined by the heat conduction equation
$ρCp∂T∂t+∇⋅q−EfḞ=0$
(17)
where T, q, and Cp are the temperature, density, and specific heat, respectively, Ef is the energy released in a single fission event, and $Ḟ$ is the volumetric fission rate. $Ḟ$ can be prescribed as a function of time or input from a separate neutronics calculation. The heat flux is given as
$q=−k∇T$
(18)
For 304 stainless steel, the strain tensor, ε, is the sum of three components; elastic strain, thermal expansion strain, and plastic strain
$ε=εe+εT+εp$
(19)
where ε represents total strain, εe represents elastic strain, εT represents thermal expansion strain, and εp represents plastic strain.
In order to simulate the mechanical behavior of SCWR fuel under the irradiation conditions, some user-defined subroutines were developed to introduce related irradiation effects and thermal effects into simulation. For UO2 pellet, the strain tensor, ε, is the sum of five components: elastic strain, thermal expansion strain, fission gas products induced strain, fission solid products induced strain, and irradiation densification strain.
$ε=εe+εT+εg+εs+εd$
(20)

where ε represents total strain, εe represents elastic strain, εT represents thermal expansion strain, εg represents fission gas products induced strain, εs represents fission solid products induced strain, and εd represents irradiation densification strain.

## Numerical Model

The fuel rod geometry in this simulation as shown in Fig. 1, the activity length of SCWR fuel rod is 4200 mm, all of the pellets are considered as an integral component, there is a gap between the pellet and cladding; the radius of UO2 pellet is 4.13 mm; the cladding diameter and cladding thickness are 5.1 mm and 0.63 mm, respectively; the total length of SCWR fuel rod is 4300 mm; the power history in this simulation as follows: power increase from 0 W/cm to 180 W/cm in 1 day, steady operation for 1000 days with the power of 180 W/cm. During the steady power conditions, the fast neutron flux rate of fuel rods remain unchanged, its value is 9.5 × 1023 n/m2·s. The heat on the outer surface of cladding was transferred by forced convective effect, and the outer surface temperature remained unchanged during the simulation process; the temperature linear increasing from 280 °C (bottom) to 510 °C (top), and coolant pressure was supposed as 25.0 MPa during the simulation process. The pellet-cladding gap is filled with helium with an initial pressure of 2 MPa; the simplified gap pressure model was used in this simulation, supposed the fuel rod internal pressure reaches 12 MPa when the steady power operation time is up to 1200 days, and internal pressure was introduced into the simulation through the application of pressure boundary. The fuel rod was assumed symmetry along the axial direction, so a symmetry plane was created at the middle axial position of fuel rods. Symmetric model was created in this simulation based on the symmetry conditions. The finite element model of SCWR fuel rod was shown in Fig. 2.

Fig. 1
Fig. 1
Close modal
Fig. 2
Fig. 2
Close modal

## Results and Discussion

According to the simulation results, the center temperature, pellet surface temperature, inside cladding surface temperature, outside cladding surface temperature (middle axial location of the fuel rod) varies with time as shown in Fig. 3. Figure 3 shows that the temperature of the fuel rods will raise rapidly due to the power increase in the power increase phase; the center temperature and the surface temperature of pellet will drop rapidly in the initial steady power phase and then drop slowly in the following steady power phase. The main reasons for this phenomenon as follows: gas swelling is more drastic than the densification because of the high temperature of fuel rods in the initial steady power phase, the gap distance of fuel rods decreases rapidly, and then, the gap conduction ability increases, and pellet temperature decreases rapidly. In the following steady power phase, the pellet temperature is lower than the temperature in the initial steady power phase, gas swelling is not so drastic as previous, and the gap distance of fuel rods and pellet temperature decreases slowly. The inner surface temperature and outer surface temperature of 304 stainless steel cladding remain unchanged during the whole steady power phase.

Fig. 3
Fig. 3
Close modal

According to the simulation results, the temperature distribution on the pellet center axial path (path 1), pellet surface axial path (path 2), cladding inner surface axial path (path 3), and cladding outer surface axial path (path 4) (1000th day in the steady power phase) are shown in Fig. 4. Figure 4 shows that the temperature on the four paths increases with the axial distance increase, meantime, the temperature difference between the pellet center and pellet surface is relatively large, and the gap temperature difference is relatively large, the temperature difference between the cladding inner surface and cladding outer surface is relatively small. Gap distance variations with the operation time are shown in Fig. 5.

Fig. 4
Fig. 4
Close modal
Fig. 5
Fig. 5
Close modal

Pellet diameter and cladding inner diameter variations with the operating time are shown in Fig. 6. Figure 6 shows that pellet diameter and cladding inner diameter increase drastically because of the thermal expansion during the power increase phase. During the steady power phase, the inner diameter of cladding is almost unchanged, while the pellet diameter increases rapidly at the initial steady power phase, and the growth rate decreases as the steady operation time increase. The temperature distribution and stress distribution (the unit of temperature is K, and the unit of stress is MPa) of fuel rod at 1000th day are shown in Figs. 7 and 8. Figures 7 and 8 show that the maximum temperature is located in the pellet center, and the maximum stress is located on the pellet outer surface. It is needed to note that the creep effect and pellet crack effect are not considered in this simulation, and the reality stress of pellet will be much smaller than the current simulation results because the creep effect and pellet crack effect could cause stress release. The temperature distribution (1000th day, and the unit of temperature is K) of pellet and the stainless steel cladding are shown in Figs. 9 and 10.

Fig. 6
Fig. 6
Close modal
Fig. 7
Fig. 7
Close modal
Fig. 8
Fig. 8
Close modal
Fig. 9
Fig. 9
Close modal
Fig. 10
Fig. 10
Close modal

According to the simulation results, the Mises stress distribution contour (the unit of stress is MPa) at 100th day, 500th day, and 1000th day during the steady power phase are shown in Fig. 11. Figure 11 shows that the stress of 304 stainless steel cladding decreases as the operation time increases in the steady power phase. The main reason for this phenomenon as follows: the gap pressure increases as the steady operation time increases, the pressure difference between the inner cladding surface and outer cladding surface reduces gradually, and the cladding stress decreases gradually.

Fig. 11
Fig. 11
Close modal

## Conclusions and Outlook

In this research, with the help of commercial finite element analysis software abaqus, the irradiation effects and thermal effects of SCWR fuel, gap heat conduction model were introduced into the numerical simulation through the development of related user-defined subroutines, and then completed the analysis of SCWR fuel rods’ performance under the steady-state and transient power conditions. The following conclusions were obtained through this study:

1. (1)

The numerical simulation method for the irradiation–thermal–mechanical coupling behaviors of SCWR fuel rods was preliminary established;

2. (2)

The pellet temperature is sensitive to the change of the gap distance, and the cladding temperature is not very sensitive to the change of the gap distance.

3. (3)

Because of the high temperature in SCWR fuel, the fission gas swelling effect is more drastic than the PWR fuel; also, the pellet expansion of SCWR is larger than PWR fuel, so it is needed to set up a larger pellet-cladding gap distance to avoid pellet cladding mechanical interaction.

However, it should be noted that there are some simplification in this research, and some effects have not been fully considered, such as the realistic fission gas release, realistic neutron distribution, and forced convective heat transfer variation with the axial location. In the following, research will continue to improve the models.

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