اهمیت تداخلات ورودی در عدم قطعیت و تجزیه و تحلیل حساسیت رفتار سوخت هسته ای

The propagation of uncertainties in a PWR fuel rod under steady-state irradiation is analyzed by computational means. A hypothetical steady-state scenario of the Three Mile Island 1 reactor fuel rod is modeled with the fuel performance FRAPCON, using realistic input uncertainties for the fabrication and model parameters, boundary conditions and material properties. The uncertainty and sensitivity analysis is performed by extensive Monte Carlo sampling of the inputs’ probability distribution and by applying correlation coefficient and Sobol’ variance decomposition analyses. The latter includes evaluation of the second order and total effect sensitivity indices, allowing the study of interactions between input variables. The results show that the interactions play a large role in the propagation of uncertainties, and first order methods such as the correlation coefficient analyses are in general insufficient for sensitivity analysis of the fuel rod. Significant improvement over the first order methods can be achieved by using higher order methods. The results also show that both the magnitude of the uncertainties and their propagation depends not only on the output in question, but also on burnup. The latter is due to onset of new phenomena (such as the fission gas release) and the gradual closure of the pellet-cladding gap with increasing burnup. Increasing burnup also affects the importance of input interactions. Interaction effects are typically highest in the moderate burnup (of the order of 10–40 MWd/kgU) regime, which covers a large portion of the operating regime of typical nuclear power plants. The results highlight the importance of using appropriate methods that can account for input interactions in the sensitivity analysis of the fuel rod behavior.

The nuclear fuel rod of a light-water reactor consists of an oxide fuel pellet stack enclosed inside a metallic cladding tube. The pellet stack is held in place by a spring and the rod is pressurized with heat-conducting gas, facilitating heat transfer across the gap between the pellets and the cladding. Analysis of the fuel rod's behavior under irradiation in a nuclear reactor involves solving the transfer of heat from the pellet into the surrounding coolant through the gap and the cladding, the mechanical response of the pellet and the cladding to thermal and mechanical stresses, the irradiation-induced changes in materials, and the release of gaseous fission products into the gas gap. All the phenomena are interconnected, constituting an extremely complex system with rich behavior in different operating regimes and strong dependency on burnup (Bailly et al., 1999 and Cacuci, 2010). The system can be modeled numerically with dedicated fuel performance codes, which traditionally focus either on the steady state irradiation or the simulation of transient scenarios. In both cases, one of the main purposes of numerical modeling is to provide understanding on how the individual phenomena interact and create the overall response of the rod to external conditions. An important aspect of such a study is the acquisition of detailed data that can be used to ensure that the rod performs within the safety and regulatory guidelines, and also to guide in revising the safety regulations (Rashid et al., 2011). There are several sources of uncertainty in fuel performance analysis. The rod's fabrication parameters, experimentally determined material properties and system parameters are never precise, but introduce various amounts of uncertainties into the system. These are propagated to model outputs such as the fuel centerline temperature, internal pressure and gap conductance. To ensure safe operation of the fuel rod, these uncertainties must be taken into account, either by conservative analysis or best estimate analysis accompanied by evaluation of the related uncertainties. In best estimate analysis quantifying both the magnitude and the source of the output uncertainties is necessary. The latter involves determining the contribution of the uncertainty of each model input to the overall output uncertainty, and is called sensitivity analysis. Sensitivity analysis in nuclear engineering has recently concentrated mostly on the reactor physics and neutronics and, on the other hand, on thermal hydraulic modeling. Sensitivity analysis of the fuel behavior models has received considerably less attention, although it is known that uncertainties related to fuel modeling can be significant and may also have broader impact on the thermal hydraulics and neutronics modeling (Christensen et al., 1981, Wilderman and Was, 1984, Syrjälahti, 2006 and Boulore et al., 2012). In addition, most studies focus on the direct (first-order) effects of the input on the model output. A common approach is to evaluate, e.g., the Spearman correlation coefficients or the first-order Sobol’ indices from the model output ( Boulore et al., 2012 and Glaeser, 2008), which neglect the higher order interactions between the input variables. However, for a complex system such as the fuel rod ( Rashid et al., 2011), these interactions can play a major role in the overall output uncertainties, and should not be neglected a priori ( Saltelli et al., 2008). In this paper, we investigate the role of input interactions in the uncertainty and sensitivity analysis of the nuclear fuel rod. For this purpose, we use the FRAPCON fuel performance code (Geelhood et al., 2011a and Geelhood et al., 2011b) and perform statistical analysis of the code's output by evaluating both the conventionally used Spearman correlation coefficients (Draper and Smith, 1998 and Kvam and Vidakovic, 2007) and the Sobol’ sensitivity indices (Saltelli et al., 2008 and Sobol’, 1993). We consider a steady state scenario, and focus on identifying the major sources of uncertainties, characterizing interactions between inputs and their dependencies on burnup. Since the initial states of transient calculations with non-fresh fuel are usually generated by such steady state simulations, our results have direct relevance for transient analyses also. The structure of the paper is as follows. In Section 2, we discuss the fuel performance code FRAPCON-3.4 used in the analysis of the scenario, and in Section 3 we describe the statistical analysis methods. The specifications of the modeled scenario and the input uncertainties are given in Section 4. In Section 5, we first show the best estimate plus uncertainty results of the scenario. Then, we illustrate how the data can be analyzed with the conventional methods and show that such an analysis remains incomplete in most cases. In Section 5.4, we repeat the analysis using the Sobol’ variance decomposition method. We show that the variance decomposition analysis gives a consistent and much more complete picture of the system's response to the input uncertainties. According to the analysis, the shortcomings of the Spearman correlation method are due to non-additive interactions between the input variables. These produce uncertainties, whose source can be identified only with higher order methods such as the variance decomposition. We also compare the quantitative effectiveness of the variance decomposition to the Spearman correlation method, showing significant improvement. Using the evaluated total effects, we rank the input uncertainties with respect to their overall importance. We summarize the results of the paper in Section 6.

A hypothetical scenario of steady state irradiation of a TMI-1 fuel rod has been modeled with the FRAPCON-3.4 code. The uncertainties in input variables have been taken into account by Monte Carlo sampling from probability distributions, and extensive modeling of the different realizations has been done. The propagation of uncertainties has been analyzed using both the Spearman correlation coefficient method, which is conventionally used in the field, and by evaluating the first order, second order and total effect Sobol’ sensitivity indices. The Sobol’ variance decomposition is shown to perform significantly better in cases where non-additive interactions between input variables are present. This includes practically all the analyzed cases, with the exception of maximum fuel temperature and average cladding temperature. For example, in identifying the sources of uncertainty for the gap conductance at moderate burnup (22 MWd/kgU), the variance decomposition method can identify practically 100% of the input's contributions, while the Spearman correlation method only explains 65% of the output variance in the analyzed scenario. The results suggests that first order sensitivity analysis methods should be used with caution in fuel performance modeling. With the possible exception of fuel centerline and cladding temperatures, the analysis should be complemented with higher order methods that take into account input interactions. The interactions of the inputs and the necessity for higher order methods stem from the complexity of the system. For the same reason, identifying a single dominant source of uncertainty is not possible. The relative importance of the inputs depends not only on the considered output, but also on burnup. For instance, out of the 21 considered inputs, 15 have a contribution larger than 10% to the uncertainty of some output at some point in the scenario. In addition to complicating the analysis, this fact also makes it difficult to rule out inputs as a source of output uncertainty. However, some fairly consistent trends can be identified. The inputs affecting the gap width, and thus gap conductance, rank among the most important ones. These are the as-fabricated cladding dimensions, cladding creep, fuel thermal expansion and, to some degree, the as-fabricated pellet diameter. Mostly the fabrication parameters influence the uncertainties at low burnup, with the other inputs becoming more important later in the rod's life, although for the cladding radial displacement the situation is reversed. Other important uncertainty sources are the diffusion coefficient in the FGR model, fuel thermal conductivity and linear power. The ranking of the inputs is very similar to that obtained in other sensitivity studies of the fuel rod (Christensen et al., 1981, Wilderman and Was, 1984 and Boulore et al., 2012), although the present study identifies the FGR-related uncertainties among the most important ones. However, because FGR is a highly non-additive function of its inputs, it is easy to underestimate its contribution with first-order methods. In addition to FGR, the evolution of the pellet-cladding gap with changing burnup is identified as one of the main sources involved in the burnup dependency of the sensitivity indices. Since the open gap and closed gap cases are governed by different physical models, the closure of the gap has a big influence on many of the sensitivity indices. In addition, the burnup regime where the gap is in the process of closing (i.e., has a significant probability for both the open and closed states) is observed to have increased interaction between the inputs, caused by the mixing of the two states of the gap. Since the intermediate burnup regime is also an important regime of operation of nuclear reactors, this highlights the importance of using appropriate, higher order methods in the sensitivity analysis of the nuclear fuel rod.