Jacques Devooght

Probabilistic reactor dynamics. I: The theory of continuous event trees

The concept of probabilistic reactor dynamics is formalized in which deterministic reactor dynamics is supplemented by the fact that deterministic trajectories in phase-space switch to other trajectories because of stochastic changes in the structure of the reactor such as a change of state of components as a result of a malfunction, regulation feedback, or human error. A set of partial differential equations is obtained under a Markovian assumption from the Chapman-Kolmogorov equation giving the probability π(x,i,t) that the reactor is in a state x where vector x describes neutronic and thermohydraulic variables, and in a component state i at time t. The integral form is equivalent to an event tree where branching occurs continuously. A backward Kolmogorov equation allows evaluation of the probability and the average time for x(t) to escape from a given safety domain.

Probabilistic reactor dynamics. II: A Monte Carlo study of a fast reactor transient

The concept of how probabilistic reactor dynamics applies to a realistic problem, an accidental transient of the primary side of a fast reactor, is demonstrated. A full description of the reactor model, including physical variables, evolution laws, and failure rates with their dependence on physical variables, is given. Failure probabilities and failure and success time distributions are evaluated. Vectorized and nonvectorized versions of a Monte Carlo algorithm as well as biased and nonbiased versions of this algorithm are compared.

Probabilistic dynamics as a tool for dynamic PSA

Abstract The assumptions, scope and achievements of a probabilistic dynamics theory based on a Chapman-Kolmogorov formulation of mixed probabilistic and deterministic dynamics are reviewed. The formulation of the theory involves both physical (or process) variables and (semi-) Markovian states of the system under study allowing the inclusion of human error modelling. The problem of crossing a safety threshold is used to emphasize the role of timing in concurrent sequences. We show how the adjoint formulation can be used to obtain information on the outcomes of transients as a function of its starting characteristics. These outcomes may, for instance, be damage resulting from safety boundary crossing, or reliability functions. A comparison is made between a Monte-Carlo solution and a DYLAM analysis of a simple multicomponent benchmark problem which shows that for the same accuracy a Monte-Carlo method is much less sensitive to the size of the problem.

Model uncertainty and model inaccuracy

Abstract The problem of model uncertainty versus model inaccuracy is examined in the light of the concept of the ‘probability of correctness of a model under a given context’ introduced by Apostolakis. To avoid possible difficulties linked with this concept, a distinction is introduced between ‘predictive’ models and ‘constitutive’ models, the former being generic in the sense that they can host the latter as submodels. A metric or distance between linear models as well as an objective of the model are introduced, from which we can give an operational definition of ‘model uncertainty’ (with respect to distribution of parameters of the associated constitutive models) and of ‘model accuracy’ with respect to a reference model. Finally the choice of a predictive model is linked to a loss function and a cost of using or defining a model.

Relationship between probabilistic dynamics and event trees

In this paper we prove that classical event trees can be derived rigorously from the theory of Probabilistic Dynamics only if we assume setpoint transitions, i.e., transitions that depend only on algebraic combinations of instantaneous values of the process variables (setpoints). Approximate formulae are also given in the more general case where the time of actuation after a setpoint is reached is stochastic, extending the classical event tree approach. The paper also reviews the theory of Probabilistic Dynamics and shows some important simplifications that can be used under certain common situations.

Probabilistic Dynamics : The Mathematical and Computing Problems Ahead

The methodology of probabilistic dynamics viewed as a continuous event tree theory is reviewed and other existing methods are shown to be particular cases corresponding to definite assumptions. Prospects for improvement of related numerical algorithms are examined.

Probabilistic reactor dynamics. III: A framework for time-dependent interaction between operator and reactor during a transient involving human error

During an accident, components fail or evolve within operating states because of operator actions. Physical variables such as pressure and temperature vary, and alarms appear and disappear. Operators diagnose the situation and effect countermeasures to recover the accidental sequence in due time. In this paper, a mathematical modeling of the complex interaction process that takes place between the operating crew and the reactor during an accident is proposed. This modeling derives from a generalization of the theory of continuous event trees developed for hardware systems to a mixture of human and hardware systems. Such a generalization requires extension of the evolution equations built under the Markovian assumption to semi-Markovian processes because dead times as well as nonexponential distributions must be modeled.

Correlation of quadrupole splitting and isomeric shift by Mössbauer spectroscopy in some trialkyltin halides

A systematic study of the Mossbauer spectra of trialkyltin halides (methyl, ethyl, propyl, butyl, isobutyl) has been made with the object of correlating the quadrupole splitting and the isomeric shift with the molecular structure. Assuming four-coordination (except for methyl), a simple model is presented which takes into account s, p, d hybridization of the tin as well as the ionic characters of the tinhalogen and the tin-alkyl bonds. the possibility of back donation by a pπ-dπ or pπ-pπ bond is taken into acount. The importance of the s electron shielding is stressed. The correlation with experimental data is good and yields satisfactory values for the parameters. The analysis suggests that there is substantial back donation.

Transport of radionuclides in stochastic media: 1. The quasi-asymptotic approximation

Abstract A three-dimensional quasi-asymptotic approximate equation is developed for the transport of radionuclides in a stochastic velocity field. This approximation is derived from an integro-differential equation of transport in stochastic media, commonly encountered in hydrogeology. The quasi-asmptotic equation turns out to be a generalised Telegraphers equation as found by Williams in the particular context of fractured media. We obtain the Telegraphers equation without specifying the causes responsible for the random velocity field. Our model may thus be applied in porous media as well as in fractured media. We give the developments leading to the analytical solution of the three-dimensional Telegraphers equation for constant parameters. This solution is then visualised for a source in the form of a square wave.

Solving Markovian systems of O.D.E. for availability and reliability calculations

Abstract The computation of availability or reliability in a Markovian approach involves the solution of O.D.E.: d p d t = L(t)p , where the transition matrix is time dependent, at least when some aggregation of the states is introduced to reduce the size of the problem. Methods of solution like uniformization are in this case unapplicable and we investigate here four explicit and six implicit R.K. methods from the point of view of stability, amount of numerical work and accuracy. The test problem chosen allows an analytical solution, a uniformization method (when L(t) is constant) and a fortiori all R.K. methods. The implicit trapezoidal rule used with a variable step scheme appears to be the best compromise between accuracy and computational work.