A mathematical model for investigation of the rewetting of fuel rods

The computer model ZETHYF simulating the reflood phase after a loss-of-coolant accident with emphasis on the investigation of coolant channel is described. The thermal behaviour of the fuel rod is modeled based on a detailed representation of the heat transfer mechanisms and a moving mesh around the quench front. The flow conditions in the coolant channel are simulated as a one-dimensional transient one- or two-phase flow.     

A one-dimensional Lagrangian model for large-volume mixing

In this paper, we describe the main features of a new Lagrangian model for mixing in large stably stratified enclosures. This new modeling approach eliminates artificial diffusion in strongly convectively dominated flows. The hyperbolic behavior of the system of PDEs requires a numerical method with no artificial diffusion to preserve the very strong gradients that can be present. We present the rudiments of the model and discuss an important aspect of the discretization error analysis. The new BMIX code is first validated against an analytical model which has been shown to model experimental data very well. Finally, a comparison is presented against experimental data, gathered from a two-enclosure exchange flow set-up. Both comparisons show good agreement and verify the suitability of this new modeling approach and the correctness of the BMIX code. The computer code can simulate mixing in any stably stratified large enclosure containing a multi-component fluid.     

On the process of rewetting a hot surface by a falling liquid film

A two-dimensional transient heat conduction model for rewetting a hot surface by a falling liquid film predicts that for stainless steel, Inconel or Zircaloy only a wall thickness of some 0.020 in. takes part in the rewetting process in steam at 100–1000 psia. The rewetting rate is nearly independent of heat flux and thermal conductivity, but increases with pressure and decreases with volumetric heat capacity.     

A remedy for the ill-posedness of the one-dimensional two-fluid model

A one-dimensional two-fluid model with explicit momentum flux parameters is proposed as a remedy for the mathematical ill-posedness of the standard one-dimensional two-fluid model. By constructing a simplified two-phase flow using existing correlations for the distribution parameter C_o and velocity profile, the momentum flux parameter is determined as a function of void fraction and density ratio for the whole range of flow regime. By performing a linear stability analysis for a two-phase flow in a channel described by the proposed model, an analytical expression for the growth factor is derived as a function of wave number, void fraction, drag coefficient, and relative velocity. From the analytical expression for the growth factor, the stability criteria are derived as a function of void faction, density ratio, and momentum flux parameters. It is shown that the two-phase flow in a channel described by the proposed model is stable in the whole range of flow regime.     

A correlation of rewetting data

Following a loss-of-coolant accident in a water reactor the fuel pins dry out and overheat and it becomes necessary to rewet them to restore normal temperatures. A thermal conduction analysis of rewetting is presented in which it is shown that the heat transfer coefficient associated with rewetting may be taken as an arbitrary function of surface temperature, rather than a constant, without changing the dependency of rewetting velocity on the other variables. An effective heat transfer coefficient then replaces the constant value used in previous expressions for the rewetting velocity. Experiments at atmospheric pressure show that the rewetting rate increases with inlet water subcooling. The available rewetting data at both atmospheric and elevated pressure have been analysed using an existing theoretical model. Taking the effective heat transfer coefficient as proportional to the product of mass flow rate and inlet subcooling a data fit has been achieved to within a factor of two. Expressions are given which predict rewetting rates for a wide range of pressures, wall temperatures, subcoolings, clad materials and geometries.     

An analytical solution to fuel-and-cladding model of the rewetting of a nuclear fuel rod

An exact solution of the quasi-steady two-dimensional conduction equation for the rewetting of a nuclear fuel rod in water reactor emergency core cooling is obtained for a fuel-and-cladding model. A method of solving non-separable differential equations is presented, which is used in the present analysis. The recently developed theorem of orthogonality of piecewise continuous eigenfunctions is also used to handle the composite rod in the present model. The present analysis reveals that the wet front velocity increases with the increase of the gap resistance between the fuel and the cladding, and approaches a limiting value, which is equal to the wet front velocity of the tube of cladding alone, as the gap resistance becomes infinite. For convenience in practical application, the results of the present analysis are correlated in simple expressions.     

An analytical solution to a two-dimensional model of the rewetting of a hot dry rod

The model considers a hot dry rod of infinite length cooled by a film of liquid moving along its surface. The heat transfer coefficient is assumed to be constant on the wet side and zero on the dry side of the rewetting front, and the liquid film is assumed to move at constant speed. We derive an analytical formula relating the temperature difference in the rod, the temperature at the rewetting front, the wet side heat transfer coefficient, and the rewetting speed. The formula is thought to apply to the rewetting of a fuel rod during emergency cooling by flooding.     

On the stability of a one-dimensional two-fluid model

An analysis on the stability of the governing differential equations for area averaged one-dimensional two-fluid model is presented. The momentum flux parameters for gas and liquid are introduced to incorporate the effect of void fraction profiles and velocity profiles. The stability of the governing differential equations is determined in terms of gas and liquid momentum flux parameters. It is shown that the two-fluid model is well posed with certain restrictions on the liquid and gas momentum flux parameters. Simplified flow configurations for bubbly flow, slug flow, and annular flow are constructed to test the validity of proposed stability criteria. The momentum flux parameters are calculated for these flow configurations by assuming a power-law profile for both velocity and void fraction. Existing correlation for volumetric distribution parameter C_o is used. By employing simplified velocity profiles, the void fraction profile is determined from C_o correlation. It is found that the void fraction is wall-peaked at low void fraction and it becomes center-peaked as the void fraction increases. A simplified annular flow is also constructed. With these flow configurations, the momentum flux parameters are determined. It is shown that the calculated momentum flux parameters are located in the stable region above the analytically determined stability boundary. The analyses results indicate that the use of momentum flux parameter is promising, since they reflect flow structure and help to stabilize the governing differential equations.     

A bounding surface model for concrete

A rate-independent, three-dimensional bounding surface model for concrete is developed. The model adopts a scalar representation of the damage related to the strain and stress states of the material. The salient features of the model are: (1) a bounding surface in stress space that shrinks uniformly as damage accumulates, and (2) material parameters that depend on damage and the distance between the current stress point and the bounding surface. Satisfactory prediction is obtained of the multiaxial compression behavior of concrete in monotonic and cyclic loadings. The model is relatively simple and incrementally linear, and its finite element implementation appears promising.     

Analytical investigation of a one-dimensional homogenized model for a pressurized water reactor core

A one-dimensional homogenized model for dynamic fluid-structure interaction in a pressurized water reactor core is used to study the influence of the virtual density and spacer's stiffness. The model consists of a linear system of partial differential equations for fluid velocity, rod velocity and pressure. For these equations analytical solutions are deduced for boundary conditions prescribing either periodic wall oscillations or linearly growing wall accelerations from rest. The theoretical model for the virtual density is verified by comparison to an experiment. For zero spacer stiffness, purely acoustic oscillations appear. For positive spacer stiffness, additional oscillations arise with relative rod motions. The wavelengths of the latter oscillations are small for weak spacers. Large numerical effort would be required in a more complete three-dimensional core-model to resolve such short wave lengths. In fact in a typical core the spacer's stiffness c_s is small in comparison to the fluid bulk modulus K. For it might be appropriate to neglect the influence of the spacers.     

A method for determining rewetting velocity under generalized boiling conditions

A method is presented for the solution of the quasi-steady state two-dimensional rewetting conduction problem. Using the method of separation of variables, a solution is presented for the case of an arbitrary heat transfer coefficient profile. Numerical results are also presented for the particular case in which a sputting region exists immediately behind the wet front. These results show that value of the heat transfer coefficient in the sputtering region determines the rewetting temperature if the sputtering region length exceeds one quarter of the class thickness. The numerical results in general show that the rewetting phenomenon is very localized with respect to the wet front. Results are compared favorably with experimental data in the low to moderate flow rate range.     

A general model for predicting coolant activity behaviour for fuel-failure monitoring analysis

A mathematical treatment has been developed to predict the release of volatile fission products from operating defective nuclear fuel elements. The fission product activity in both the fuel-to-sheath gap and primary heat transport system as a function of time can be predicted during all reactor operating conditions, including: startup, steady-state, shutdown, and bundle-shifting manoeuvres. In addition, an improved ability to predict the coolant activity of the ¹³⁵Xe isotope in commercial reactors is discussed. A method is also proposed to estimate both the burnup and the amount of tramp uranium deposits in-core. The model has been validated against in-reactor experiments conducted with defective fuel elements containing natural and artificial failures at the Chalk River Laboratories. Lastly, the model has been benchmarked against a defective fuel occurrence in a commercial reactor.     

The effect of precursory cooling on rewetting of a slab

The Dua and Tien (1976) model for the rewetting of a slab with precursory cooling is solved exactly by separation of variables. The solution for the rewetting velocity is found to agree very well with a Wiener Hopf technique solution to this model by the author. Rewetting rates predicted by the approximate solution of Dua and Tien are found to agree with the present solution for small Peclet numbers, while underpredicting them for large Peclet numbers. Theoretical quench front velocities compare favorably with experimental data for copper quenched by liquid nitrogen. Precursory cooling is shown to be able to greatly increase the rewetting velocity, in particular for cases of high flow rates, while neglecting it in modelling may result in much too low quench velocities, as compared to experimental measurements.     

A heat transfer correlation based on a surface renewal model for molten core concrete interaction study

The prediction of heat transfer between corium pool and concrete basemat is of particular significance in the framework of the study of PWR's severe accident. Heat transfer directly governs the ablation velocity of concrete in case of molten core concrete interaction (MCCI) and, consequently, the time delay when the reactor cavity may fail. From a restricted hydrodynamic point of view, this issue is related to heat transfer between a heated bubbling pool and a porous wall with gas injection. Several experimental studies have been performed with simulant materials and many correlations have been provided to address this issue. The comparisons of the results of these correlations with the measurements and their extrapolation to reactor materials show that strong discrepancies between the results of these models are obtained which probably means that some phenomena are not well taken into account. The main purpose of this paper is to present an alternative heat transfer model which was originally developed for chemical engineering applications (bubble columns) by Deckwer. A part of this work is devoted to the presentation of this model, which is based on a surface renewal assumption. Comparison of the results of this model with available experimental data in different systems are presented and discussed. These comparisons clearly show that this model can be used to deal with the particular problem of MCCI. The analyses also lead to enrich the original model by taking into account the thermal resistance of the wall: a new formulation of the Deckwer's correlation is finally proposed.     

Damage-plastic loading surface model for concrete

The paper presents a plasticity-type model which is characterized by normality, continuity, convexity and associatedness. It utilizes two loading surfaces, of which one describes the behavior associated with damage due to inelastic volume change (expansion from micro-cracking, or contraction from pore collapse or closure) and the other plasticity at no inelastic volume change. Since only one of these two surfaces is active at the same time, the formulation is equivalent to a single loading surface. The deviatoric section of the damage loading surface is a rounded triangle, and the Rendulić section meridians have the shape of a slanted ellipse. The loading surfaces are smooth and have no corners. Although the model involves approximately the same number of material constants as the previous models of similar capability, identification of these constants from test data is much simpler. The properties of the model are also more advantageous for numerical finite element applications. The model is shown to be capable of describing the basic uniaxial and multiaxial test data for concrete except degradation of material stiffness.     

Analytical one-dimensional frequency response and stability model for PWR nuclear power plants

A dynamic model for PWR nuclear power plants is presented. The plant is assumed to consist of a one-dimensional single-channel core, a counterflow once-through steam generator (represented by two nodes according to the non-boiling and boiling region) and the necessary connecting coolant lines. The model describes analytically the frequency response behaviour of important parameters of such a plant with respect to perturbations in reactivity, subcooling or mass flow (both at the entrances to the reactor core and/or the secondary steam generator side), and perturbations in steam load or system pressure (on the secondary side of the steam generator). From corresponding ‘open’ loop considerations, it can then be concluded - by applying the Nyquist criterion - upon the degree of the stability behaviour of the underlying system. Based on this theoretical model, a computer code named ADYPMO has been established.From the knowledge of the frequency response behaviour of such a system, the corresponding transient behaviour with respect to a stepwise or any other perturbation signal can also be calculated by applying an appropriate retransformation method, e.g. by using the digital code FRETI. To demonstrate this procedure, a transient experimental curve measured during the pre-operational test period at the PWR nuclear power plant KKS Stade was recalculated using the combination ADYPMO-FRETI. Good agreement between theoretical calculations and experimental results give an insight into the validity and efficiency of the underlying theoretical model and the applied retransformation method.     

Parametric study on rewetting velocities obtained with a two-dimensional heat conduction code

A comprehensive parametric study has been performed to quantify the effect of different variables on the rewetting velocity in a light water reactor following a loss-of-coolant accident. To this purpose, a numerical solution of the general two-dimensional (axial and radial) heat conduction equation in cylindrical geometry has been obtained. The method used is the alternating-direction implicit procedure developed by Peaceman and Rachford. The model accounts for decay heat generation in the fuel, coolant subcooling, different wall temperatures and different heat transfer coefficients across the gap and at the clad surface. The two-dimensional model can be reduced to a one-dimensional model by setting the heat conduction in either the radial or axial direction to zero. Results with the new model agree with previous models and with experimental data.The variables studied were: axial and/or radial heat conduction, clad temperature, quench temperature, coolant temperature, temperature for the onset of nucleate boiling, heat transfer coefficients, stored and decay heats, clad material and clad thickness. The critical thickness (clad thickness for which the calculated rewetting velocity remains constant) was also determined and found to be larger than the clad thickness of light water reactor fuel pins under usual reflood conditions. According to these calculations, the stored and decay heats affect the rewetting velocity significantly.     

Experimental investigation of the rewetting process in a freon-113 vapour environment

An experimental facility to simulate two-phase high-pressure flow phenomena using low-pressure Freon-113 has been built, commissioned, and employed to investigate the rewetting process of hot surfaces in a Freon-113 environment at pressures up to 5.43 bar. The experimental results and the correlations derived show good agreement with previous results using water and confirm the feasibility of simulating rewetting phenomena in water by the use of Freon.