Characterization of a porous medium employing numerical tools: Permeability and pressure-drop from Darcy to turbulence

A large set of microscopic flow simulations in the Representative Elementary Volume (REV) of a porous medium formed by staggered square cylinders is presented. For each Reynolds number selected, 10 different porosities are simulated in the 5–95% range. The Reynolds number is varied from Re = 10⁻³ to Re = 10⁵, covering the Stokes flow regime, the laminar flow regime and the turbulence flow regime. Low and moderate Reynolds number flow solutions (Re  200) are achieved by numerically solving the 2D Navier–Stokes equations. Reynolds Averaged Navier–Stokes equations are employed to simulate the turbulence regime. Numerical results allow the investigation of the microscopic features of the flow as a function of the porosity and Reynolds number. Based on these microscopic results, the permeability of the porous medium is computed and a porosity-dependent correlation is developed for this macroscopic parameter. The Darcy–Forchheimer term or, equivalently, the friction factor, is also computed to characterize the porous medium for the complete range of porosity and Reynolds number simulated. The Forchheimer coefficient is found to be weakly dependent on the Reynolds number and strongly dependent on the porosity if the flow is fully turbulent. A porosity-dependent correlation is proposed for this quantity for high Reynolds numbers.     

A new turbulence model for porous media flows. Part II: Analysis and validation using microscopic simulations

A new model for turbulent flows in porous media developed in an accompanying paper [F.E. Teruel, Rizwan-uddin, A new turbulence model for porous media flows. Part I: Constitutive equations and model closure, Int. J. Heat Mass Transfer (2009), doi:10.1016/j.ijheatmasstransfer.2009.04.017.], in which new definitions of the macroscopic turbulence quantities are introduced, is analyzed and validated. The model is validated using a simple but often used porous medium consisting of a staggered arrangement of square cylinders. Theoretically predicted values of the newly defined turbulence variables, under fully developed conditions, are compared with corresponding variables used in existing turbulence models. Additionally, evolution of the macroscopic turbulence quantities obtained numerically using the model developed here are compared with reference results, obtained by averaging over space the microscopic level solution of the RANS equations. Comparison exercise for the 75% porosity case is carried out for a range of turbulence intensity at the entrance of the porous medium. Comparison of results shows very good agreement. The spatial evolution of the dispersive kinetic energy, which is included in the definition of the macroscopic turbulent kinetic energy introduced here, is computed using the microscopic solution. Its magnitude relative to the conventional turbulent kinetic energy shows the importance of this quantity in the representation of turbulence effects in porous media flows.     

Multi-scale simulations of plasma with iPIC3D

The implicit Particle-in-Cell method for the computer simulation of plasma, and its implementation in a three-dimensional parallel code, called iPIC3D, are presented. The implicit integration in time of the Vlasov–Maxwell system, removes the numerical stability constraints and it enables kinetic plasma simulations at magnetohydrodynamics time scales. Simulations of magnetic reconnection in plasma are presented to show the effectiveness of the algorithm.     

Finite integral transform method to solve asymmetric heat conduction in a multilayer annulus with time-dependent boundary conditions

The separation of variables (SOV) method has recently been applied to solve time-dependent heat conduction problem in a multilayer annulus. It is observed that the transverse (radial) eigenvalues for the solution in polar (r-θ) coordinate system are always real numbers (unlike in the case of similar multidimensional Cartesian problems where they may be imaginary) allowing one to obtain the solution analytically. However, the SOV method cannot be applied when the boundary conditions and/or the source terms are time-dependent, for example, in a nuclear fuel rod subjected to time-dependent boundaries or heat sources. In this paper, we present an alternative approach using the finite integral transform method to solve the asymmetric heat conduction problem in a multilayer annulus with time-dependent boundary conditions and/or heat sources. An eigenfunction expansion approach satisfying periodic boundary condition in the angular direction is used. After integral transformation and subsequent weighted summation over the radial layers, partial derivative with respect to r-variable is eliminated and, first order ordinary differential equations (ODEs) are formed for the transformed temperatures. Solutions of ODEs are then inverted to obtain the temperature distribution in each layer. Since the proposed solution requires the same eigenfunctions as those in the similar problem with time-independent sources and/or boundary conditions—a problem solved using the SOV method—it is also “free” from imaginary eigenvalues.     

Fission gas production in reactor fuels including the effects of ternary fission

An understanding of gas bubble formation and migration in nuclear fuel and its impacts on fuel and cladding materials requires knowledge of the isotopic composition of the gases and their generation rates. In this paper, we present results of simulations for the production of the dominant noble gases (helium, xenon, krypton) in nuclear fuels for different reactor core configurations and fuel compositions. The calculations were performed using detailed nuclear burn simulations with Monte Carlo nuclear transport, and included ternary fission to ensure an accurate treatment of helium production. For all reactor designs and fuels considered xenon was found to be the most dominant gas produced. Variation in the composition of fission gases is quantified for: (1) the burn time, (2) the composition of the fuel, and (3) the neutron energy spectrum. These three factors determine the relative fraction of each gas and its transmutation into or from stable gas by subsequent neutron capture.     

A web-based nuclear simulator using RELAP5 and LabVIEW

A web-based nuclear reactor simulator has been developed using the best-estimate nuclear system analysis code RELAP5 as its engine, and LabVIEW for graphical user interface and web-casting. Simulator retains the accuracy of the best-estimate code. Results are displayed in user friendly graphical format. Color-coded nominal values are displayed along with the current status of different variables in tab activated windows. Some variables of interest are also shown as a function of time. All graphical outputs are displayed in web browsers making the simulator's front end independent of the operating system. The interactive simulation feature allows the users to simulate specific reactor transients – such as LOCA, scram, etc. – using a single click. Simulator's graphical output can be web-casted and is thus available to anybody with access to the web. Moreover, if permitted, the simulator can be operated remotely from another site connected to the server via the World Wide Web.     

An innovative research reactor design

A new and innovative core design for a research reactor is presented. It is shown that while using the standard, low enriched uranium as fuel, the maximum thermal flux per MW of power for the core design suggested and analyzed here is greater than those found in existing state of the art facilities without detrimentally affecting the other design specs. A design optimization is also carried out to achieve the following characteristics of a pool type research reactor of 10 MW power: high thermal neutron fluxes; sufficient space to locate facilities in the reflector; and an acceptable life cycle. In addition, the design is limited to standard fuel material of low enriched uranium. More specifically, the goal is to maximize the maximum thermal flux to power ratio in a moderate power reactor design maintaining, or even enhancing, other design aspects that are desired in a modern state of the art multi-purpose facility. The multi-purpose reactor design should allow most of the applications generally carried out in existing multi-purpose research reactors. Starting from the design of the German research reactor, FRM-II, which delivers high thermal neutron fluxes, an azimuthally asymmetric cylindrical core design with an inner and outer reflector, is developed. More specifically, one half of the annular core (0 < θ < π) is thicker than the other half. Two variations of the design are analyzed using MCNP, ORIGEN2 and MONTEBURNS codes. Both lead to a high thermal flux zone, a moderate thermal flux zone, and a low thermal flux zone in the outer reflector. Moreover, it is shown that the inner reflector is suitable for fast flux irradiation positions. The first design leads to a life cycle of 41 days and high, moderate and low (non-perturbed) thermal neutron fluxes of 4.2 × 10¹⁴ n cm⁻² s⁻¹, 3.0 × 10¹⁴ n cm⁻² s⁻¹, and 2.0 × 10¹⁴ n cm⁻² s⁻¹, respectively. Heat deposition in the cladding, coolant and fuel material is also calculated to determine coolant flow rate, coolant outlet temperature and maximum fuel temperature under steady-state operating conditions. Finally, a more compact version of the asymmetric core is developed where a maximum (non-perturbed) thermal flux of 5.0 × 10¹⁴ n cm⁻² s⁻¹ is achieved. The core life of this more compact version is estimated to be about 23 days.     

Turning points and sub- and supercritical bifurcations in a simple BWR model

Stability and semi-analytical bifurcation analyses of BWRs have been performed using a dynamical system approach. A reduced order model of a BWR that includes simple neutronics as well as thermal hydraulics has been used. Analyses have been carried out using a bifurcation analysis code, BIFDD, that carries out analytic–numeric stability and bifurcation analyses of set of ordinary differential equations and ODEs with delays. A large segment of the parameter space has been investigated using this very efficient tool. Stability boundaries are obtained in several two-dimensional parameter spaces. In addition, the nature of bifurcation along these stability boundaries has also been determined. Results indicate that both subcritical as well as supercritical Poincaré–Andronov–Hopf bifurcations are likely to occur in regions of interest in parameter space.In addition to the semi-analytical bifurcation studies, the governing equations have also been integrated numerically. Results confirm the findings of the stability and bifurcation analyses. Numerical integrations, carried out for parameter values away from the stability boundary, further show that the bifurcation curves, in many cases of subcritical bifurcations, have a turning point. The bifurcation curve in these cases extends back into the unstable region. These results show that it is possible to experience large amplitude stable oscillations in the unstable region infinitesimally close to the stability boundary. Moreover, large amplitude stable oscillations are also possible, following large but finite perturbations, in the stable region of the parameter space near the stability boundary. These findings provide alternate explanation for the experimental and operational observations in BWRs that indicate the existence of stable limit cycle oscillations and the possibility of growing amplitude oscillations.Results obtained here using a simple model suggest that further work along these lines, with more detailed models, is needed to identify operating conditions and perturbation amplitudes that might lead to stable limit cycles or growing amplitude oscillations in current and next generation of BWRs.     

Non-linear dynamics of space-independent xenon oscillations

The space-independent xenon oscillation problem relevant to power nuclear reactors is studied. Xenon oscillations - both, without temperature feedback and in the presence of temperature feedback - have been analyzed semi-analytically and numerically. The effects of various parameters on the nature of bifurcation are studied. Bifurcation analysis shows that Hopf bifurcation occurs, and it is found that both sub- critical and super-critical Hopf bifurcation can occur in different regions of parameter space. Numerical experiments show that 'outsidey the unstable periodic solutions that exist for the sub-critical Hopf bifurcation case, there exist large amplitude, stable periodic solutions to which the initial conditions, outside the basin of attraction of the stable fixed point, evolve. Though the existence of sub-critical Hopf bifurcation indicates that large amplitude perturbations even in the stable region may lead to initially diverging oscillations, it is reassuring that the oscillation amplitude remains bounded     

BWR stability and bifurcation analysis using reduced order models and system codes: Identification of a subcritical Hopf bifurcation using RAMONA

The system code RAMONA, as well as a recently developed BWR reduced order model (ROM), are employed for the stability analysis of a specific operational point of the Leibstadt nuclear power plant. This has been done in order to assess the ROM’s applicability and limitations in a quantitative manner.