Two-phase thermal-hydraulics and heat transfer

In general, thermal hydraulic modelling of nuclear reactor systems is based on the one-dimensional approach. Thermal hydraulic modelling of the steady state, transient and stability behaviour of two-phase natural circulation systems is no exception to this general approach.

System codes have reached a highly developed modelling status and a wide acceptance. They

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can reproduce accurately enough most of the existing safety related steady state and transient experiments, so far as the dominant physical mechanisms are known, as adequate models are included in the codes, and so far as the dominant phenomena are also understood by the code users. Thus, these codes are an excellent tool to analyse in a parametric way the dynamics and interrelation of a larger number of components in complex systems with different physics involved. However, the use of multi-dimensional modelling of two-phase natural convection in large vessels (including the calandria vessel of pressure tube type heavy water reactors), sumps or plenums of nuclear reactors is essential as the flow in these cases are multi-dimensional in nature.

Some of the most commonly adopted thermal hydraulic models applicable for the 1-D and multi-dimensional analyses are briefly described below:

3.2.2.1. Models for 1-D thermalhydraulic analysis (a) Homogeneous equilibrium model:

This is the simplest model for analysing the thermal hydraulic phenomena in a two-phase system. This considers one conservation equation each for the mass, energy and momentum of the mixture. Hence, it assumes complete thermal and mechanical equilibrium between the phases. This model is also referred to as Equal Velocity and Equal Temperature (EVET) model. Here, the mixture is considered to behave as a single fluid. An equation of state takes into account the variation of fluid properties. To take into account the differences in velocity between the phases, empirical slip models are used. The success of the model depends on the accuracy of the empirical relationships used in the code for the wall friction and wall heat transfer. Use of computer codes based on the homogeneous equilibrium model for accident and safety analysis of nuclear reactor systems is perhaps becoming a thing of the past.

However, in the earlier design stages, codes based on this model are still extensively used.

Further, most analyses to generate the stability behaviour (basically codes used to generate stability maps) of natural circulation systems are based on the homogeneous equilibrium model.

(b) Partial non-equilibrium models:

These models consider either the thermodynamic or mechanical non-equilibrium between the phases. The number of conservation equations in this case are either four or five. One of the most popular models which considers the mechanical non-equilibrium is the drift flux model.

If thermal non-equilibrium between the phases is considered, constitutive laws for interfacial area and evaporation/condensation at the interface must be included. In this case, the number of conservation equations is five, and if thermodynamic equilibrium is assumed the number of equations can be four. Well-assessed models for drift velocity and distribution parameter depending on the flow regimes are required for this model in addition to the heat transfer and pressure drop relationships. The main advantage of the drift flux model is that it simplifies the numerical computation of the momentum equation in comparison to the multi-fluid models.

Computer codes based on the four or five equation models are still used for safety and accident analyses in many countries. These models are also found to be useful in the analysis of the stability behaviour of BWRs belonging to both forced and natural circulation type.

19 (c ) Two-fluid models:

The two-fluid models take into account the thermodynamic and mechanical non-equilibrium between the liquid and vapour phases. In this case, the mass, energy and momentum balance equations are written separately for both the phases. The six conservation equations in the two-fluid model represent the propagation of phasic void, velocity, enthalpy and pressure disturbances. For closure, the conservation equations require constitutive equations for the interfacial and wall transfer terms. The most difficult task is the modelling of these constitutive relationships. In spite of several years of research, the interfacial constitutive laws are not yet well established. A classical approach is to define them based on flow regime maps used for identifying the flow patterns (bubbly, slug, churn, annular, etc.). The interfacial transfer relationships (mass, momentum and energy) can be calculated based on the flow pattern dependent relationships, which are selected by suitable flow pattern transition criteria.

Most of the large system codes used for safety and accident analysis, for example RELAP5, CATHARE, ATHLET, etc. adopt the one-dimensional two-fluid model. The two-fluid model is not yet commonly used for the stability analysis of nuclear reactor systems.

3.2.2.2. Modelling of multi-dimensional thermalhydraulics

The increasing need for more accuracy of the codes and more detailed descriptions of new cooling concepts leads to a tendency to use at least for local detailed analyses more and more 3D and time-dependent computational fluid dynamic (CFD) codes. There are several motivations for using CFD codes in the nuclear field: Some problems can only be treated in 2D or 3D, like the flow-regime determination, thermal stratification, phase separation, and phase distributions in large pools. CFD codes are required if dynamical 2D or 3D phenomena have to be investigated, like the oscillations of velocities, pressures, temperatures, or phase distributions. The relevance of such investigations may come e.g. from ensuring safe heat transfer from all sub-assemblies under natural convection, from the thermal striping induced thermal fatigue, or flow instabilities due to geysering or sloshing, and from the phase and boron distribution dynamics coupled to neutronics.

Finally CFD codes have to be used for scaling up detailed results from model experiments to full-scale reactor conditions. They are also more and more used even for designing experiments and their instrumentation arrangement. They can also reliably be used to study in a more qualitative manner the relevance of certain phenomena in flow and heat transfer problems, so far as the governing physics is included in the equations or in the numerical modelling. Thus, they are also used to determine the dominant physical phenomena of nuclear flow problems, so that finally a reliable efficient numerical model configuration (input) can be formulated for a system code.

Most CFD codes for two-phase or phase component flows are based on the multi-field approach, assuming that all fluids and phases are defined everywhere in the flow multi-field;

this means, the fluids are interpenetrating. The spatial distributions of the different fluids or phases are determined by their relative local volume fractions. As a consequence of this method, one has not only a tremendous increase of the number of equations to be solved, but there are also no explicit phase boundaries existing. This kind of modelling is the basis for the typical working tools available. As such flows are physically and topologically complex, such codes give a valuable support to understand in more detail the macroscopic physical phenomena occurring in multi-phase flows.

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The physical models in such codes, e.g. to model interfacial phenomena, are rather simple.

Therefore, a quantitative use of such CFD codes for two-phase flows is currently limited to mainly homogeneous flows. Modelling improvements are related to bubble or particle diameters which have to be specified. New developments are going to include multi-group concepts of bubble diameter classes or to develop interfacial area concentration equation models which allow up to now for simple flow regimes to calculate the dynamics of the surface of the interface, which is an important parameter in determining momentum, heat, and mass transfer between the phases. This could also help to improve the modelling of bubble coalescence and fragmentation. For other flow regimes than bubbly or droplet flows, the modelling of interfacial phenomena needs serious improvements.

In going from 1D discretisation to 2D or 3D, one has to use turbulence models when turbulent flows have to be investigated. Some of the two- or multi-phase codes have turbulence models for two-phase flows, but our current knowledge and experimental data base on turbulence is very weak: we have only a few turbulence data for the liquid phase in bubbly flow. This basis is not sufficient to adapt, calibrate, and validate these models. Therefore, none of the existing models can reproduce all those data. For other flow regimes, data as well as models are missing; problematic flow regimes will be bubbly, churn, slug, and droplet flows. The turbulence in non-adiabatic flows is characterized by a rapid change of bubble diameters which will influence the turbulence characteristics. Another related problem occurs in the missing wall functions and in the boundary conditions for the interface phenomena at free surfaces. Thus, the turbulence modelling in two-phase flows can only be accepted as a first step. The models can be used strictly only in an interpolative manner, this means in a parameter range, in which extensive validation was done before. R&D is necessary to improve the models and to extend them for all flow regimes, so that more reliable prediction capabilities can be achieved.

Despite these model deficiencies such codes were rather successful, because many multi-phase problems are governed by large-scale interfacial phenomena. Therefore, most development in the past was based on developing adequate numerical methods to solve the set of equations of multi-field models. The challenge here comes from the weak compressibility of the fluids or phases engaged and from the strong density variations from liquids to gases across interfaces. Therefore, the development of more efficient and more robust solution schemes and solvers is always a target for further improvements.

For such cases, in which separated phases have to be considered, special numerical tools were developed to keep the interface sharp, like interface capturing, interface tracking, or some approximate numerical methods like surface sharpening. In case of dispersed flows special highly accurate numerical schemes would be required to avoid or at least minimize numerical diffusion, but most codes use standard lower order schemes; thus one still has to live with more or less strong numerical diffusion.

In the past, most codes for nuclear applications with multi-phase flows were developed mainly in the large national research centres or in the industry, like AFDM (LANL/FZK), ATHLET (GRS), CATHARE (CEA), COBRA-TF (Pacific NW Lab.), ESTET-ASTRID (EDF), IVA (Siemens), MATTINA (FZK), MC-3D (CEA), RELAP5-3D (INEEL,DOE), ERHRAC (NPIC), TRAC (SNL,USNRC)… They have the advantage that the required nuclear specific modelling is included, and the models and numerics were selected according to the special requirements of nuclear applications. Now, powerful, and comfortable commercial CFD codes are becoming available, which are increasingly suitable for nuclear applications, like CFX-4,

21 COMET, FIDAP, FLOW-3D, FLUENT, PHOENICS, and others. There are major differences

between the principal modelling capabilities available in the commercial codes. None of those has all the modelling required for certain flow phenomena in innovative reactor systems; thus it has to be decided from case to case which code should be used. This brings some concerns regarding the certification effort when one is progressing towards a licensing procedure of a reactor.

With increasing requirements on accuracy and details of codes, also requirements regarding the accuracy and details to be provided by experiments are growing. Whereas for design purposes and for system code improvement and validation mainly realistic large scale experiments are of interest, for the development and validation of CFD codes for two-phase flows one needs primarily single effect experiments with very detailed instrumentation, preferably providing not only local data, but also field data, like PIV and tomography. Topics should be the interfacial phenomena and turbulence near bubbles with rapid diameter change, near free surfaces, and near walls with boiling, the bubble formation, coalescence and fragmentation, data on the interfacial area, and the turbulence in the continuous phases;

similar investigations are also needed for all other flow regimes, and the flow regime transitions. Nevertheless, some large scale more complex experiments with a detailed instrumentation are also required to validate the correct interrelation between the different models engaged in the CFD codes.

3.2.2.3. Modelling of heat transfer in two-phase flow

In two-phase flow, local convective heat transfer coefficients are evaluated from the channel local thermal hydraulic conditions and the heater rod surface temperature. In theory all the main two-phase flow patterns such as bubble flow, slug flow, churn flow and annular flow are likely to exhibit different heat transfer characteristics. It is generally recognized that more accurate prediction of flow boiling can be obtained by adopting a flow pattern specific method for each individual flow regime. However, practically all of the existing correlations correlate flow boiling data using a single equation covering all the different flow patterns. General boiling curve is used to define the wall heat transfer regimes (convective, pool boiling, flow boiling, film boiling, etc.) and the related convective heat transfer coefficient. In addition, distinction must be made for condensation and evaporation. Further, well-established models for the critical heat flux (CHF) are required for the calculation of pre-CHF and post-CHF wall heat transfer.

3.2.2.4. Modelling of neutron kinetics

Normally, the equations for neutron kinetics are obtained from the general space and time dependent multi-group diffusion equations for a finite reactor of non-uniform properties. The 3-D neutron kinetic calculations are often made with fast and thermal energy groups and six groups of delayed neutrons. To reduce the sophistication, a modified one group diffusion equation can be considered for fast calculations. In this, only the dynamics of the thermal energy group is simulated in detail.

For simplicity of calculations, most of the large system codes consider the point kinetics model assuming the overall buckling to be almost constant during the transient. The parameters used in these equations such as delayed neutron fraction, prompt neutron life time, delayed neutron precursor concentration, etc. are defined taking into account some phenomena formally neglected during the derivation of the point kinetics equation. An improvement to the

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point kinetics model is reflected in the one-dimensional kinetics equation that is also adopted in BWR calculations. In this, the radial buckling is assumed not to change during the transients. In point kinetics, reactivity coefficients for the void and Doppler effect are provided as inputs for the whole core. However, 1D and 3D neutron kinetics models facilitate direct evaluation of these feed back effects.

3.2.2.5. Modelling of two-phase natural circulation instabilities

From the modelling point of view, two types of instabilities are important. These are the static and dynamic instability. Static instability can be fully described by the steady state governing equations. Examples of this type of instability are Ledinegg (flow excursion), flow pattern transition instability, flashing instability, etc. For the dynamic instability, the feedback effects are important and hence it requires the solution of the time dependent conservation equations of mass, momentum and energy. Typical example is the density wave oscillations observed in two-phase systems. Often, thermalhydraulic systems exhibit compound instability that is usually a combination of two mechanisms. Here, the instability may begin with one of the known mechanism for static instability that may trigger some secondary mechanisms resulting in an oscillatory behaviour. Typical examples are parallel channel instability and coupled neutronic-thermalhydraulic instability of BWRs.

In generally, a two-phase natural circulation system can be designed to avoid all the different types of instability discussed above. This is done by carrying out a stability analysis, which is usually phenomena-specific and therefore cannot be described in detail in a brief note as the present one. However, the objectives of most stability analyses can be listed as:

(a) To generate stability maps which delineate the stable and unstable zones of operation;

(b) To specify adequate stability margin for design purposes; and

(c) To predict the nature of the unstable oscillatory behaviour of flow, power and temperature (in time domain) if the system passes through an unstable operating zone during power/pressure raising.

Most static instability analyses are carried out with the prime objective of generating a stability map. For the dynamic analysis, however, all the three objectives become important.

Both the linear and nonlinear methods are used for the analysis of this type of instability.

Mathematically, the dynamic behaviour of any system can be considered to be linear for small perturbations around the steady state operating condition. An analytical solution is sought for the perturbed equations to obtain the characteristic equation for the stability of the system. The characteristic equation is solved numerically to generate the stability map (locus of all neutrally stable points) of the system. The computer codes based on the linear stability method (e.g. RAMONA, NUFREQ, etc.) are very useful to predict the stability maps without consuming much computer time. However, it is beyond its capability to predict the non-linear effects that come into play after the neutral threshold (to achieve the third objective). Hence, direct numerical solution of the non-linear governing equations is carried out using finite difference methods. An additional objective of the non-linear computer codes is to provide an understanding of the basic physical mechanisms involved in BWR behaviour beyond neutral stability. In principle, system codes like RELAP5, ATHLET, CATHARE, etc. can be used for this purpose, although this is not the current practice (generally). If the system codes are used, this will require the use of qualified fine nodalisation as those used for the normal transient analysis (normally a coarse nodalisation) may not even predict the existence of instability (as

23 the numerical solution technique adopted in these codes are very robust which introduce

numerical damping through artificial/numerical diffusion).

For simulating coupled neutronic-thermalhydraulic instability, usually a point kinetics model is used in addition to a fuel heat transfer model (which basically solves the conduction equation in the fuel, fuel-clad gap and clad to obtain the feedback effects of fuel). Point kinetics equations are first order ordinary differential equations that can be easily integrated by using a conventional numerical technique. However, the problem is the introduction of feedback constants in them, which are spatially averaged over the whole core. So uncertainty is therefore introduced at this level in addition to the inaccuracy in neglecting the changes in core power distribution. Similarly, the way in which the neutronics and the thermal hydraulics interact during the calculation has an important effect on the capability to simulate multi-dimensional core dynamics. Grouping of core channels in thermal hydraulic regions having similar geometric and operating parameters can have strong influence on the out-of-phase mode oscillations.

3.2.2.6. Influence of non-condensable gasses

Condensation is defined as the removal of heat from a system in such a manner that vapour is converted into liquid. This may happen when vapour is cooled sufficiently below the saturation temperature to induce the nucleation of vapour. This mode of heat transfer is often

Condensation is defined as the removal of heat from a system in such a manner that vapour is converted into liquid. This may happen when vapour is cooled sufficiently below the saturation temperature to induce the nucleation of vapour. This mode of heat transfer is often