Thermodynamic properties and equations of state for fast reactor safety analysis: Part I: Analytic equation-of-state model

Numerical simulation of postulated severe-accident sequences in liquid-metal fast reactors (LMFRs) requires thermodynamic properties of reactor-core materials over wide temperature and pressure ranges. Here an improved analytic equation-of-state (EOS) model using flexible thermodynamic functions is newly developed for a multiphase, multicomponent fluid-dynamics code for LMFR safety analysis. Extension of a van der Waals type equation for vapor phase is also proposed to include the dimerization process of sodium vapor and thereby enables us to predict sodium vapor properties with adequate accuracy. The present EOS model is designed to have adequate accuracy at high temperature and high pressure and to consistently satisfy basic thermodynamic relationships over a wide temperature range without deterioration of the computing efficiency. The newly introduced fluid-dynamic algorithm for pressure iteration consistently coupled with the EOS model is also described in the present paper.     

Fission gas swelling and long-range migration at low temperatures

Fission gas behavior at temperatures below ~ 1100°C is assumed to consist of gas bubble nucleation and coalescence through random motion of the bubbles and then complete bubble destruction by subsequent fission events. A model is proposed describing the gas behavior based on a modified form of Van der Waals gas law for very small bubbles. A bubble is assumed to move inversely to the cube of its radius. Long range migration of gas bubbles to grain boundaries is also predicted and the swelling due to gas motion in the grain boundaries is calculated.     

不同状态方程对超临界二氧化碳强迫对流传热中流动加速因子的影响

明显的流动加速效应对超临界二氧化碳强迫对流传热有重要影响。流动加速因子是表征流动加速效应强度的重要无量纲数，在建立流动加速因子过程中需要用到二氧化碳状态方程。理论分析了选取理想气体状态方程和van der Waals方程对建立的流动加速因子合理性的影响，并结合实验数据进行了评估。结果表明，基于van der Waals 方程建立的流动加速因子可以较好地预测流动加速效应引起的传热恶化区域，而基于理想气体状态方程建立的流动加速因子与实验结果存在较大偏差。分析表明体积膨胀系数和体积压缩系数反映了流动加速现象的本质，以体积膨胀系数和体积压缩系数来建立流动加速因子更合理。      

Thermodynamic properties and equations of state for fast reactor safety analysis: Part II: Properties of fast reactor materials

In this paper, thermodynamic properties of reactor-core materials are newly evaluated for an analytic equation-of-state (EOS) model with flexible thermodynamic functions. This model has been developed for a multiphase, multicomponent fluid-dynamics code for fast reactor safety analysis. The most up-to-date and reliable sources for uranium dioxide, mixed-oxide fuel, stainless steel, and sodium available at present are first compiled, with particular emphasis on including new thermodynamic data of type 316 stainless steel. This compilation is used to develop the EOS data up to the critical point using basic thermodynamic relationships. The EOS model for use in the accident analysis code is then completed by determining all the EOS parameters in the thermodynamic functions. The thermodynamic consistency and accuracy of the evaluated EOS data are also discussed by comparison with the available sources. The thermodynamic properties of sodium vapor are successively reproduced using a van der Waals type equation extended to include the dimerization process of sodium vapor.     

Fission gas induced swelling in uranium at high temperatures and pressures

The effect of externally applied hydrostatic pressures from 0–1000 bar at 900 °C on the fission gas induced swelling in uranium has been studied. The swelling is a sensitive function of pressures below 110 bar and relatively insensitive to pressures above 110 bar. Metallography of the samples show that the average diameter of the bubbles is reduced from 3300 to 1800 Å and the total number increased from 0.9 to 4.2 × 10¹²/cm³ as the annealing pressure is increased from 0–110 bar respectively. Further increases in pressure from 110–1000 bar have little effect on the average bubble size and density. The mechanism of pressure reduced swelling is discussed in terms of the equilibrium number of vacancies that can be associated with fission gas atoms under stress. It is also shown, both experimentally and theoretically, that there is a loss of surface area when bubbles coalesce under pressure.     

Behaviour of fission gas products in single crystal UO2 during intermediate temperature irradiation

A new mathematical interpretation is presented of fission gas release from monocrystalline uranium dioxide fuel during intermediate temperature irradiation in terms of a defect trap model, knock-out process and diffusion of bubbles. In the present model it is assumed that gas in an intermediate state exists side by side with the dissolved fission gas and that trapped in bubbles. It is assumed also that the isolated gas atoms, being re-dissolved, are immobile.The present model gives a satisfactory interpretation of the relative proportions of isotopes in the steady state fission gas release at diffrent temperatures. The dependence of fractional fission gas release on fission rate is also interpreted; regimes either proportional to fission rate or inversely proportional to fission rate are predicted depending on the fission rate interval considered. Both temperature dependent and temperature independent fission gas release can arise.The presented dynamic method of studying the release of fission gases during irradiadion provides a further test beside the static method of the veracity of the assumed mechanisms. Calculations show that fission gas behaviour becomes more complex for oscillated fission rate in the regime where the fractional release is inversely proportional to the fission rate for the steady state.     

体积法测量储氢量的状态方程研究

体积法是表征储氢量最主要的技术手段之一,状态方程能否真实描述氢气的p-V-T特性是影响测量精度的关键因素。本文基于芶清泉的物理思想,提出了状态方程新模型。从碳气凝胶储氢量测量过程中得到的实验数据出发,分别应用理想气体状态方程、范德华耳斯方程以及状态方程新模型计算间苯二酚-甲醛碳气凝胶在不同温度、压力下的储氢吸附量,分析了造成结果差别的物理原因。结果显示,利用新的气体状态方程,可明显改善体积法测量结果的可信度。     

Theoretical study of flashing and water hammer in a supercritical water cycle during pressure drop

During a loss of coolant accident (LOCA) the pressure of the coolant can drop significantly in the vicinity of the leak. It will be shown that unlike in pressurized water reactors (PWRs) where this pressure drop can cause only sudden vaporization - also called flashing - in supercritical water cooled reactors (SCWRs) it can cause sudden condensation (condensation-induced water hammer), too. The reason is that from supercritical state the system can go to metastable liquid as well as to metastable vapour state after LOCA. Relaxation from metastable fluid states is a fast process, followed by a local positive or negative pressure-jump, which might increase the damage around the leak. Conservative estimation will be given for the magnitude of these pressure jumps caused by the flashing or water hammer by assuming various initial pressure losses. In our calculations, three different equations of state are used: the simple van der Waals EoS; the Redlich-Kwong as an empirical development; and the more sophisticated non-cubic Deiters equation of state. These equations are able to describe metastable states qualitatively but with different accuracy. These calculations can help us to map the local immediate effect of any sudden pressure drop and therefore it can help to design better safety protocols.     

Radiation re-solution of fission gas in uranium dioxide and carbide

Calculations have been performed to estimate the removal rate of fission gas atoms from bubbles due to collisions with energetic fission fragments and recoil cascades. The efficiency of this process was found to be higher than estimated earlier, but is still too low to be responsible for the experimental observations of fission gas bubble destruction during irradiation of oxide fuel. An irradiation experiment to investigate the interaction of fission spikes with free surfaces has enabled a simple theory to be developed which can explain the shrinkage of bubbles and pores by the surface relaxation of a shock wave produced by the passage of a fission fragment. This mechanism occurs in oxides but not carbides because of the faster dispersion of the fission fragment energy and provides the major reason for the difference in gas bubble distributions in oxide and carbide fuel. This process, however, does not remove gas atoms from the bubbles. Since high levels of apparently diffusive fission gas release are observed in oxides, the “effective solubility” of the fission gases required for this release must be sought in phenomena other than the fission spike.     

Grain boundaries as vacancy sources during the growth of creep voids and gas bubbles

Irradiation-induced creep embrittlement of metals and alloys may be a consequence of the diffusional growth, and eventual impingement, of intergranular fission gas bubbles. It is proposed that a suitable array of intergranular precipitates will prevent the grain boundaries acting as vacancy sources and thus inhibit the growth of the gas bubbles and prevent them evolving into creep voids. A model is developed enabling an estimate to be made of the size and distribution of precipitates necessary to prevent this form of embrittlement. It is also suggested that a similar mechanism is responsible for the inhibition, by precipitates, of fission gas swelling in irradiated uranium.     

Fission gas release from UO2 fuel during low-temperature irradiation

A new mathematical interpretation is presented of fission gas release from UO₂ fuel during low-temperature irradiation in terms of a defect trap model and the knock-out process. In the present model it is assumed that gas in an intermediate state exists side by side with the dissolved fission gas and that trapped in bubbles. The present model gives a satisfactory interpretation of the relative proportion of isotopes in the steady state fission-gas release. The dependence of the fission-gas release rate on the fission rate is also interpreted; regimes either proportional to the square of fission rate or proportional to fission rate are predicted, depending on the fission rate interval considered.     

Computer modeling of steady state fission gas behavior in carbide fuels

A model for the simulation of long-term, steady-state fission gas behavior in carbide fuels is formulated. It is assumed that fission gas release occurs entirely through gas atom diffusion to grain boundaries and cracks. Fission gas bubbles are assumed to remain stationary and to grow as the net result of gas atom precipitation into the bubbles from the matrix solid and gas atom re-solution from the bubbles into the matrix. Furthermore, assuming that local gas atom redistribution process in the immediate neighborhood of a bubble is very rapid, the bubble size is assumed to correspond to the equilibrium size that maintains exact balance between the rate of gas atom re-solution and that of gas atom precipitation.The model also treats the effect of attachment between bubbles and second-phase precipitates; the experimentally observed faster growth rate of precipitate bubbles is simulated using a reduced re-solution parameter for precipitate bubbles. With the grain matrix assumed to be spherical, the model allows the computation of the radial distribution of the intragranular bubbles and the gas atom concentration in the matrix.The flux of gas atoms arriving at the grain boundary is computed. The continual growth of grain boundary bubbles, resulting from the accumulation of gas atoms on the grain boundary, leads to grain boundary interlinkage and all gas atoms that subsequently reach the grain boundary are assumed to be released. Similarly, all gas atoms generated following the interlinkage of intragranular bubbles are also assumed to be immediately released.Application of the model indicates that fission gas swelling is largely due to intragranular bubbles. Grain boundary bubbles, although very large in size, contribute little to fission gas swelling and the contribution from gas atoms in solid solution in the matrix is even less significant.Physical parameters entering the model were assigned numerical values that closely represent the physical characteristics of the irradiation samples. Careful comparisons between the results of sensitivity studies and the experimental data readily identify the re-solution parameter to have the strongest influence on the results predicted by the code and that the grain size, and not the temperature, is the dominant factor affecting gas release.When allowance is made for the uncertainties of the experimental data, the predicted fission gas swelling also correlates well with experiment. The spread in the fuel swelling data, however, indicates that fuel cracking, and not fission gas swelling alone, very often contributes significantly to the fuel external dimensional changes. The linear fission gas swelling rate prediceted by the model exhibits almost a linear variation with temperature. This result correlates well with the linear swelling rate obtained from experimental swelling data if immersion density data alone are used, in order to eliminate the sources of uncertainties associated with fuel cracking.     

On the rate theory model for fission-gas behaviour in nuclear fuels

The rate theory model of the homogeneous nucleation and subsequent growth of intragranular fission gas bubbles has been extended to allow for the inclusion of very much larger bubbles. A simple treatment of the behaviour of grain-boundary gas is included in the present model, which allows for the re-solution of gas from intergranular bubbles. The effect of varying the model parameters and of including bubble mobility in the theory have been considered and it is concluded that the dominant parameters for gas release are temperature, grain size, re-solution rate and bubble migration and coalescence.     

Effects of Annealing under High Static Pressure on Precipitation Behavior of Fission Gas Bubbles in Irradiated UO2

Annealing experiments were carried out on irradiated UO₂ in argon gas under high pressure (600 and 1,000 kg/cm²) as well as atmospheric, at temperatures of 1,400°–1,600°C. The effects of high external pressure on the behavior of fission gas bubbles in the irradiated UO₂ were studied by comparing replica electron micrographs of fractured surfaces of specimens annealed under different temperatures and pressures. The results indicate that high pressures such as above 600 kg/cm² can be effective in surpressing the growth of fission gas bubbles in both intergranular and intragranular zones, and in inhibiting the joining together of intergranular bubbles to form direct passages for fission gas release.     

Swelling and gas release in oxide fuels during fast temperature transients

A previously reported intergranular swelling and gas release model for oxide fuels has been modified to predict fission gas behavior during fast temperature transients. Under steady state or slowly varying conditions it has been assumed in the previous model that the pressure caused by the fission gas within the gas bubbles is in equilibrium with the surface tension of the bubbles. During a fast transient, however, net vacancy migration to the bubbles may be insufficient to maintain this equilibrium. In order to ascertain the net vacancy flow, it is necessary to model the point defect behavior in the fuel. Knowing the net flow of vacancies to the bubble and the bubble size, the bubble diffusivity can be determined and the long range migration of the gas out of the fuel can be calculated. The model has also been modified to allow release of all the gas on the grain boundaries during a fast temperature transient.The gas release predicted by the revised model shows good agreement to fast transient gas release data from an EBR-II TREAT H-3 (Transient Reactor Test Facility) test. Agreement has also been obtained between predictions using the model and gas release data obtained by Argonne National Laboratory from out-of-reactor transient heating experiments on irradiated UO₂. It was found necessary to increase the gas bubble diffusivity used in the model by a factor of thirty during the transient to provide agreement between calculations and measurements. Other workers have also found that such an increase is necessary for agreement and attribute the increased diffusivity to yielding at the bubble surface due to the increased pressure.     

Modelling intragranular fission gas release in irradiation of sintered LWR UO2 fuel

A model for the release of stable fission gases by diffusion from sintered LWR UO₂ fuel grains is presented. The model takes into account intragranular gas bubble behaviour as a function of grain radius. The bubbles are assumed to be immobile and the gas migrates to grain boundaries by diffusion of single gas atoms. The intragranular bubble population in the model at low burn-ups or temperatures consists of numerous small bubbles. The presence of the bubbles attenuates the effective gas atom diffusion coefficient. Rapid coarsening of the bubble population in increased burn-up at elevated temperatures weakens significantly the attenuation of the effective diffusion coefficient. The solution method introduced in earlier papers, locally accurate method, is enhanced to allow accurate calculation of the intragranular gas behaviour in time varying conditions without excessive computing time. Qualitatively the detailed model can predict the gas retention in the grain better than a more simple model.     

弥散型燃料的裂变气体行为研究

探讨了弥散型燃料中对辐照肿胀有重要影响的裂变气体的行为机理。裂变气体原子聚集成气泡引起燃料相肿胀，气泡的尺寸分布是影响辐照肿胀的重要因素。决定气泡生长的裂变气体的行为机理主要有：裂变气体原子的产生和热扩散迁移，气泡的成核和聚合长大，气泡内气体原子的重溶，燃料相的辐照亚晶化等过程。燃料中各种尺寸的气泡浓度随时间的变化率可用气泡生长的动力学速率方程组来描述。当裂变密度较高时，辐照产生的缺陷引起燃料相的     

Investigations on swelling and fission gas behaviour in uranium dioxide

UO₂ irradiated at temperatures between 1000 and 2100 K was investigated with respect to fission gas behaviour and swelling. The amount of fission gas was measured in three steps as released fission gas, fission gas retained in bubbles and pores, and fission gas in the fuel matrix. The retained fission gas reaches concentrations up to $\text{1.6 × 10}{}^{\mathrm{?2}}$ gas atoms per uranium atom at temperatures below 1250 K and decreases with increasing temperature. The swelling was evaluated by measuring the volume changes and by immersion density measurements. The maximum fission gas swelling without extensive bubble migration is about 20% at 2000 K. It diminishes to about 5% at 1250 K.     

The fission gas bubble distribution in a mixed oxide fast reactor fuel pin

The fission gas bubble distribution has been studied in a mixed oxide fast reactor fuel pin irradiated in DIDO MTR to 2.8% burn-up at centre and surface temperatures of 2000 and 1000°C. The intragranular fission gas bubbles are very small (<6 nm diameter) and this is a consequence of the high re-solution rate at fast reactor ratings. The bubbles nucleate heterogeneously and linear arrays of bubbles, due to nucleation on fission tracks, are observed up to irradiation temperatures of 1900°C. At 1980°C ~4% of the fission gas produced is present in intragranular bubbles. There is no definite evidence for gas bubble mobility or coalescence. Apart from any effects of columnar grain growth fission gas release in fast reactor fuel pins seems to occur predominantly by the diffusion of single gas atoms, at least up to irradiation temperatures of 2000°C.