Stability Of Two-phase Flow Research Articles

Experimental research on cooling performances of radial expanding-channeled heat sinks applied for multiple heat sources

Intrinsic two-phase flow instability using multiple parallel boiling channels hinders the actual implementation of flow boiling cooling techniques in high heat flux and multiple heat source applications. Recently, we proposed a radial expanding-channeled heat sink (ECHS) to realize a high two-phase flow stability under extreme conditions. In the present work, the cooling performances of two heat sinks connected in series or in parallel were investigated with the heat load adjusted individually. The results indicated that when two ECHS were connected in parallel, the inlet flowrate was evenly distributed into the two heat sinks, and their heat transfer performance was identical. At a sudden change in the heat load of one path, the flowrate of the other heat sink may change by 10% to 30% without inducing flow instability or heat deterioration. The thermal resistance for both heat sinks was lower than 0.025 °C/W, and no instability or dry-out was observed as long as the initial flowrate was larger than 0.28 kg/min. With the proposed ECHS, multiple heat sources such as central processing unit, server, rack, etc. can be connected in series or in parallel to construct an integrated cooling system for large-scale data center applications.

Effect of a soluble surfactant on the linear stability of two-phase flows in a finite-length channel

We study numerically the effect of a soluble surfactant on the stability of two-phase flow in a finite-length microchannel. The two streams drag the surfactant molecules toward the downstream end of the interface against the action of the Marangoni stress. The sharp reduction of the interfacial tension at that end enhances the interface deformation and considerably destabilizes the system, translating into a sharp reduction of the critical capillary number even for very small surfactant volume concentrations.

Hydrodynamics in bubble columns with helically-finned tube Internals: Experiments and CFD-PBM simulation

Experiments and CFD-PBM simulation were performed in this study to investigate the effects of helically-finned tube (HFT) internals on the gas-liquid flow in a laboratory-scale bubble column. Hydrodynamic parameters, including gas holdup, bubble size and flow field as well as turbulence properties, were compared for an empty column and other two columns with bare-tube (BT) or HFT internals. We found that bare rods promote the stability of two-phase flow, whereas helical fins destabilize the flow. Meanwhile, the spatial distribution of gas holdup or bubble size becomes more uniform in the presence of HFT internals, whereas liquid velocity or turbulent dissipation rate evidently decreases. The fin-induced spiral movement, flow resistance and bubble accumulation are the mechanisms underlying these experimental observations. Integration of rod and helical fin may be promising for process intensification in bubble columns.

Control of Boiling Instabilities in a Two-Phase Pumpless Loop Using Water-Alcohol Mixtures

AbstractThe mitigation of boiling instabilities by using water-alcohol mixtures as coolants in a two-phase pumpless loop is experimentally investigated. Water-miscible alcohol, namely, 2-propanol, is added to water in quantities of 0.05, 0.1, and 0.2 mol fractions to reduce the bubble departure diameters. Boiling experiments are carried out in the pumpless loop involving a copper-based split-flow microchannel evaporator. Two-phase flow stability and heat transfer performance are compared between pure water and 2-propanol-water mixtures. Temperature measurements at the evaporator inlet, evaporator outlet, and condenser inlet show a considerable reduction in fluctuations when the binary mixtures are used as coolants. Therefore, the addition of a small amount of alcohol to water is proposed as a simple technique to control two-phase flow instabilities as against complex geometry modification to the evaporator, often studied in the literature. The pumpless loop with water-alcohol mixtures is a viable thermal management solution for power electronic devices that involve time-varying workloads.

Instability of stratified two-phase flows in rectangular ducts

The linear stability of stratified two-phase flows in rectangular ducts is studied numerically. The linear stability analysis takes into account all possible infinitesimal three-dimensional disturbances and is carried out by solution of the associated eigenproblem. The neutral stability boundary and the corresponding critical wave number are obtained for liquid – liquid and air – water systems. Depending on the problem parameters, the instability sets in owing to short, intermediate, or long wave most unstable perturbations. Patterns of the most unstable disturbances are reported and discussed. It is shown that the instability arises due to shear, or interfacial mechanisms. Effects of the surface tension and of width/height aspect ratio are also studied. The results support the premise that the stability analysis of stratified two-phase flow in the simpler geometry of two-infinite plates can provide a reasonable estimation of the conditions for which this flow pattern can be considered to be linearly stable.

Some numerical achievements on Na boiling dynamics and next technical route

The paper proposes an overview of an R&D program for Gen4 Sodium Fast Reactors that was launched in 2012 at CEA to investigate Na boiling dynamics under an Unprotected Loss of Flow scenario. Part of the program motivation was an incident periodic two-phase flow pattern (limit cycle) along the transient that was calculated at first. If proven, such a stable phenomenology could indeed attractively improve reactor safety by providing a time-window for recovering pumping power. The paper is restricted to the thermal–hydraulic side of the R&D initiative that targeted to credit prediction of the boiling flow dynamics. First, the 1-D numerical achievements obtained by simulating with CATHARE3 thermal hydraulic system code, some out-of-pile programs from the 70–80 s, are detailed. The latter were indeed already very informative on two-phase flow stability scenarios, by so offering a first sound qualification basis. Related benefits allowed by a recent air–water experimental program that supported some modifications of wall and interfacial closure laws, are pointed out. The second part of the paper outlooks the next technical route and its early advances. Part of it, is 3-D simulation by subchannel code TrioMC which first application shows its added-value to perform a detailed analysis of a scaled experiment. Predictive capabilities of 3-D approach will be also central to credit a full-scale transposition. Progress on flow stability analysis, including subcooled condensation as well as connected channel reflooding and pressure wave aspects, could be as such expected from recent developments on CFD. Indeed, advances for an all flow regimes representation opportunely suit Na boiling physics. Bifurcation and stability analysis with the developed BACCARAT model is finally identified as providing a tailored mathematical approach to connect the periodic flow pattern with a Hopf bifurcation. Its ability to provide a mechanistic view of the scenarios shift on a natural convection test-case, is reported. On the experimental side, the scope of a new program planed with IPPE on a wire-spaced 19 pins bundle, is briefly introduced. HARIBO program suits some Gen4 hydraulic design specifics and targets CATHARE3 and TrioMC qualification.

Frequency domain analysis of two-phase flow instabilities in a helical tube once through steam generator for HTGR

Two-phase flow instabilities will disturb the control systems and cause mechanical vibrations and failure of components, which should be avoided. Two-phase flow instabilities in a helical tube Once Through Steam Generator (OTSG) for High Temperature Gas-cooled Reactor (HTGR) were investigated. First, a linear frequency domain model of the helical tube OTSG was developed. The convective evaporation process was divided into single phase water region, two-phase flow boiling region and superheated steam region. Incompressible assumptions were adopted in the single phase water and superheated steam regions. Homogeneous flow model was used in the two-phase region. Small perturbation linearization and Laplace transform were performed to analyze the governing equations in the three regions. The corresponding frequency domain transfer functions describing the flow pulsation and pressure drop pulsation of the two-phase flow systems were derived. By solving the Nyquist curve of the transfer function with matlab software, the two-phase flow instabilities and the stable boundary of the evaporation process in helical tubes can be determined. The new model in this paper was verified using previous experimental results. The two-phase flow stability in the helical tube OTSG of High Temperature Gas-cooled Reactor Pebble-bed Module (HTR-PM) was analyzed using the developed frequency domain model at the design power levels. The stability boundary of the two-phase flow and effects of various parameters on flow instabilities were also investigated. Then an accumulator was added to this linear frequency domain model, which was used to study the effect of compressible volume or accumulator on DWO and PDO. Using this model, the effects of throttle valves at different locations on the stability of the system with an accumulator were also studied. The results show that the helical tube OTSG for HTR-PM can operate stably under the designed working conditions with enough margin.

Dynamics and modelling of bubble formation in asymmetric parallel microchannels

Dynamics of gas-liquid two-phase flow in parallel microchannels is the bottleneck to be solved during numbering-up. This paper focuses on the stability and distribution of gas-liquid two-phase flow in asymmetric parallel microchannels. Nitrogen and glycerol-water solutions are used as the gas and liquid phases respectively. The stability of bubble formation is better under higher flow rates of gas and liquid phases. When the flow rate ratio of gas and liquid phases is smaller, the uniformity of bubbles is better. The design of cavities in the microchannel degrades stability, while results in better uniformity. The uniformity of the bubble size is the best in the microchannel configuration with both cavities in the front and the rear. The predictive models of the relative deviation E(L), representing the uniformity of bubbles, are proposed according to the principle of conservation of pressure drop and the resistance relationship in the parallel microchannels.

Effect of throttling on the two-phase flow stability in an open natural circulation system

Low pressure natural circulation systems fit the concept of passive safety systems as useful feature to bring a nuclear power plant to a safe and stable state during a postulated accident without active systems. The operation is characterized by low mass fluxes and passing the known unstable two-phase region from activation to a continuous heat removal in the stable two-phase region. The occurring mass flow oscillations and the condensation induced water hammer phenomenon limited the performance of heat removal. These effects are highly undesirable because they may effect destruction of entire apparatus. The GENEVA test facility as an open natural circulation system represents the containment cooling condenser of the KERENATM reactor concept in the main dimensions. Initiated by the steam supply, the mass flow results from induced heat flux and is limited only by the internal pressure drops: the local, the frictional, the acceleration and the gravitational pressure drop across the flow path. The influence of the flow resistance on its unstable two-phase region and their oscillatory behavior are experimentally studied in detail. Its variation is summarized in stability maps plotted in the parameter plan of the heat flux and the subcooling number at the inlet of the heated section. Thereby, the dynamical evolution of the boiling boundary indicates the initiation of two-phase flow oscillations and the condensation induced water hammer phenomenon. The increasing local pressure drop at the inlet of the heated section reduces the velocity amplitudes but a clear reduction of the pressure surges could not be determined although these should be more pronounced due to decreasing buoyancy and increasing back flow.

On modeling fluidelastic instability in tube arrays subjected to two-phase cross-flow

An experimental program was undertaken in order to improve our understanding of the mechanism of fluidelastic instability in two-phase flows and to provide practical design guidance to avoid vibration problems in heat exchangers. Different ways of defining the traditional parameters, such as void fraction, damping, velocity, and 2-phase fluid density, were examined. It was found that the damping in quiescent fluid combined with the gamma densitometer measured void fraction and the interfacial velocity, collapses the available data well and provides the expected trend of two-phase flow stability data over the void fraction range from liquid to gas flows. The resulting stability maps represent a significant improvement over existing maps for predicting fluidelastic instability of tube bundles in two-phase flows. This result also tends to confirm the hypothesis that the basic mechanism of fluidelastic instability is the same for single and two-phase flows, providing that the parameters are properly defined.

Experimental investigation on the thermal and entropic behavior of a vertical helical tube with none-boiling upward air-water two-phase flow

Helical tubes have great compact structure, two-phase flow stability and well thermal performance. The increment of applications which use two-phase flow in helical tubes, necessitates the investigation on thermal and entropic characteristics of two-phase flow in helically coiled tubes. The present study provides experimental results of heat transfer characteristics of upward air-water none-boiling two-phase flow in a vertical helically coiled tube. The helical tube was put under constant heat flux. The inlet, outlet, wall and ambient temperature of helical tube were measured for calculating the heat transfer coefficient and entropy generation. The water superficial Dean number (Desl) was between 1000 and 4800. Also, the gas (air) superficial Dean number (Desg) was between 45 and 235. The VF (volume fraction) fraction was between 0.11 and 0.55. Almost all investigated cases were at the range of slug and plug flow regimes. Results revealed that the air-water two-phase flow could increase the heat transfer coefficient. Increment of air superficial Dean Number increased the heat transfer rate and entropy generation. A maximum increment of 35% and %26 were observed for heat transfer coefficient and entropy generation respectively. Also it was revealed that increment of VF increases the Witte-Shamsundar efficiency.

Reduced order modeling of a natural circulation system by proper orthogonal decomposition

Nonlinear dynamical systems like thermal hydraulic natural circulation systems can show a very complex behavior under specific conditions. Hence an in-depth analysis of the stability landscape of such systems is necessary. In the field of nuclear reactor technology, the application of advanced system codes, which numerically solve the set of nonlinear partial differential equations describing the underlying system dynamics, is common practice. However, this method is cumbersome and requires large computational effort in order to get an overview of the whole stability landscape of the system. Therefore we are looking for an efficient method which is suitable to reliably provide the solution manifold of the model equations with less computational effort compared to system codes. Here, the application of bifurcation theory together with advanced modeling techniques provides a powerful approach. However, system codes cannot be directly coupled with bifurcation codes due to restrictions of memory and available computation time. Thus an alternative methodology which delivers reliable results with less effort is desirable. An efficient method broadly used in physical and technical fields is the reduction of the system model order by both physically intuitive methods as well as rigorous mathematical optimization principles (Model Order Reduction Theory, MOR). In the paper at hand we demonstrate the application of this technique in the framework of (high-pressure) natural two-phase flow stability analysis. To this end, a well-investigated ROM method, the proper orthogonal decomposition (POD) approach, is selected and the stability of the two-phase flow is investigated by a system code (ATHLET) and a POD-ROM. The objective is to show that the stability behavior for the chosen system states can be calculated by the POD-ROM in a proper agreement with the (validated) system code. These results are the basis to develop a reduced thermal hydraulic model for low pressure conditions, which are not content of this article.

Principles for the application of bifurcation theory for the systematic analysis of nuclear reactor stability, Part2: Application

This paper is regarded as a continuation of the paper “Principles for the application of bifurcation theory for the systematic analysis of nuclear reactor stability, Part1” with the intention to provide examples demonstrating the application of the bifurcation analysis method in the framework of reactor stability analysis. Hence, we continue with chapter 5 which is devoted to three examples: (1) two-phase flow stability analysis, (2) occurrence of a generalized Hopf bifurcation (GHB) during a real nuclear reactor stability test and (3) existence of a complex stability behaviour in an environment of a double Hopf bifurcation point (Hopf-Hopf bifurcation, HHB). The efficiency of the RAM-ROM method is demonstrated for an operating point of NPP Leibstadt for which a sufficient experimental and system code database is available. These three examples of system dynamics demonstrate the partly very complex stability behaviour of nonlinear systems which cannot be explained by the application of linear stability analysis methods such as the estimation of the decay ratio (as a linear stability indicator). The consequences of the found bifurcation types in examples 2 and 3 on the particular solution structure of the underlying dynamic system will be discussed by using their respective normal forms in order to provide the reader a more clear access to the complex system behaviour around these bifurcation points. In case of the Hopf-Hopf bifurcation, we only present a selected part of solutions in this paper and refer the reader to a future paper, where more details of this bifurcation type are summarized and consequences to the full system are interpreted.

Stability of stratified flows through neo-Hookean soft-gel-coated walls

The linear stability of two-phase stratified flows through soft-gel-coated walls is studied in this work. A consistent nonlinear neo-Hookean model describes the soft-gel. The base state is characterised by a first normal stress difference in this model. This results in a significantly different effect on the stability of two-phase flows as compared to when a linear viscoelastic model is used for the soft-gel. Both fluids are assumed Newtonian and incompressible. We analyze the system in the absence and presence of a soluble surfactant. A linear stability analysis is carried out to identify different instability modes in the system. The linearised equations result in a generalised eigenvalue problem which is numerically solved employing a Chebyshev collocation technique. Three distinct instabilities are identified in the absence of a soluble surfactant. A long wave interfacial instability, a Tollmien-Schlichting wave type instability, and a gel-liquid instability arise in the system. Two distinct additional instabilities are identified in the presence of soluble surfactants in the system, a Marangoni driven long wave instability and a liquid-liquid short wave mode. It is shown that all instabilities except the Tollmien-Schlichting shear instability can be stabilised by the soft-gel layers for a suitable choice of parameters. Insights into the physical mechanism driving the different instabilities are discussed.

Stability of stratified two-phase channel flows of Newtonian/non-Newtonian shear-thinning fluids

Linear stability of horizontal and inclined stratified channel flows of Newtonian/non-Newtonian shear-thinning fluids is investigated with respect to all-wavelength perturbations. The Carreau model has been chosen for the modeling of the rheology of a shear-thinning fluid, owing to its capability to describe properly the constant viscosity limits (Newtonian behavior) at low and high shear rates. The results are presented in the form of stability boundaries on flow pattern maps (with the phases’ superficial velocities as coordinates) for several practically important gas–liquid and liquid–liquid systems. The stability maps are accompanied by spatial profiles of the critical perturbations, along with the distributions of the effective and tangent viscosities in the non-Newtonian layer, to show the influence of the complex rheological behavior of shear-thinning liquids on the mechanisms responsible for triggering instability. Due to the complexity of the considered problem, a working methodology is proposed to alleviate the search for the stability boundary. Implementation of the proposed methodology helps to reveal that in many practical cases the investigation of the simpler Newtonian problem is sufficient for the prediction of the exact (non-Newtonian) stability boundary of smooth stratified flow (i.e., in case of horizontal gas–liquid flow). Therefore, the knowledge gained from the stability analysis of Newtonian fluids is applicable to those (usually highly viscous) non-Newtonian systems. Since the stability of stratified flow involving highly viscous Newtonian liquids has not been researched in the literature, interesting findings on the viscosity effects are also obtained. The study highlights the limitations of applying the simpler and widely used power-law model for characterizing the shear-thinning behavior of the liquid. That model would predict a rigid layer (infinite viscosity) at the interface, where the shear rates in the viscous liquid are low, and thereby unphysical representation of the interaction between the phases and its impact on the stability boundary.

Closure relations effects on the prediction of the stratified two-phase flow stability via the two-fluid model

The possibility of predicting the exact long wave linear stability boundary via the two-fluid (TF) model for horizontal and inclined stratified two-phase flow is examined. The application of the TF model requires the introduction of empirical closure relations for the velocity profile shape factors and for the wave induced wall and interfacial shear stresses. The latter are recognized as the problematic closure laws. In order to explore the closure relations effects and to suggest the necessary modifications that can improve the stability predictions of the TF model, the results are compared with the exact long wave solution of the Orr–Sommerfeld equations for the two-plate geometry. It is demonstrated that with the shape factors corrections and the inclusion of wave induced stresses effects, the TF model is able to fully reproduce the exact long wave neutral stability curves. The wave induced shear stresses in phase with the wave slope, which give rise to the so called “sheltering force”, were found to have a remarkable destabilizing effect in many cases of horizontal and inclined flows. In such cases, the sheltering effects must be included in the TF model, otherwise the region of smooth stratified flow would be significantly over predicted. Based on the results of the exact analysis, a simple closure relation for the sheltering term in the TF model is provided.

Bifurcation analysis of density wave oscillations in natural circulation loop

The present study explores the linear stability and bifurcation analysis of two-phase flow in the single heated channel for the natural circulation loop. The bifurcation analysis of this natural circulation system is limited to the detection of Hopf bifurcation, generalized Hopf bifurcation and turning point or limit point bifurcation of limit cycles. These natural circulation loops have several engineering applications and prone to thermal-hydraulic instability, which may lead to an operationally inefficient system. Stability boundary (linearly) obtained on a Npch- Nsub plane predicts the existence of limit cycles (a nonlinear phenomenon) for these loops and gives motivation for the further study on the detection of nonlinear phenomena. The subsequent results show a novel case of stability boundary (listed as Type A in the present work) emerging for higher subcooling numbers on Npch- Nsub parameter plane, which has not been noted in literature earlier and lies near to the Ledinegg stability boundary, which is a static instability. The characteristics of points nearby Type A stability boundary predicts that this boundary is different from the Ledinegg stability boundary. This Type A stability boundary divided by subcritical and supercritical Hopf bifurcation, which is separated by a generalized Hopf bifurcation (GH) point. Sustained oscillations in flow velocity can be observed near to the Hopf points on this boundary. These subcritical (hard and dangerous) and supercritical (soft and safe) Hopf bifurcations have been examined with time series graphs for all types (Type A-C) of stability boundaries present for the loops and occurring on the Npch- Nsub parameter plane. The two other type of boundaries known earlier is characterized here as Type B and C, respectively, which is basically Type I and Type II density wave oscillations. The transition of stability boundaries is discussed on the Npch- Nsub parameter plane, which shows that Type A stability boundary contains GH bifurcation coming from Type B stability boundary. The bifurcation of limit cycles originating from the Hopf points in the system has been examined and it is found that limit point bifurcation of limit cycles is occurring in the system, originating from the generalized Hopf points.

Effect of dispersed phase fraction on the drag coefficient of a droplet or a bubble in an idealized two-phase flow

In the present work, the drag force on a fluid sphere surrounded by a second is determined. This information is essential to characterize dispersed flows. Most of two-phase flow models are empirically obtained from experiments involving isolated bubbles or droplets. Their applicability to flows with a high number of bubbles or droplets is questionable. Moreover, correlations that take into account the effect of local dispersed phase fraction, the ratio of occupied volume by the dispersed phase, are sparse.This paper explores the drag on a fluid sphere inside an idealized flow. All spheres are assumed identical having the same velocity and equidistributed in space. We propose a new relation for the drag coefficient of a sphere (bubbles or droplets) depending on Reynolds number Re, dispersed phase fraction ε, viscosity ratio μ∗ and density ratio ρ∗. Analytical and numerical results are compared with previous studies including experimental measurements. The results lead to a proposal for a general relation of the drag coefficient for a sphere inside a cloud of spheres.In the idealized bubbly/droplet flow considered, the slip ratio is very small, and the flow around a sphere can often be characterized by Stokes approximation. In addition, dispersed phase fraction, ε, has a strong effect on drag essentially through confinement. The wake and hydrodynamic interactions between spheres are comparatively small. The proposed relation can be used to elaborate a two-phase flow model for bubbly or annular flows. This work proposes an improvement on the closure relations for the drag coefficient (CD).The dependence of the drag coefficient (CD) with void fraction is of utmost importance to evaluate the stability of two-phase flows. The drag coefficient is a source term of the averaged Navier–Stokes equation.The main conclusion is that dispersed phase fraction extends the range for which Stokes flow represents accurately the flow around bubbles/droplets.

Predicting two-phase flow distribution and stability in systems with many parallel heated channels

Two-phase heat exchangers are used in a variety of industrial processes in which the boiling fluid flows through a network of parallel channels. In some situations, the fluid may not be uniformly distributed through all the channels, causing a degradation in the thermal performance of the system. A methodology for modeling two-phase flow distributions in parallel-channel systems is developed. The methodology combines a pressure-drop model for individual parallel channels with a pump curve into a system flow network. Due to the non-monotonicity of the pressure drop as a function of flow rate for boiling channels, many steady-state solutions exist for the system flow equations. A new numerical approach is proposed to analyze the stability of these solutions, based on a generalized eigenvalue problem. The method is specifically designed for analyzing systems with hundreds of identical parallel channels.The method is first applied to analyze the flow distribution and stability behavior in two-channel and five-channel systems. The asymptotic behavior of the flow stability is then analyzed for increasing numbers of channels, and it is shown that the stability behavior of a system with a constant flow-rate pump curve simplifies to the stability behavior for a constant pressure-drop pump curve. A parametric study is conducted to assess the influence of inlet temperature, heat flux, and flow rate on the stability of the uniform flow distribution solution as well as on the severity of flow maldistribution. Below some critical inlet subcooling, uniform flow distribution is always stable and maldistribution does not occur, regardless of heat flux and flow rate. Above this critical inlet subcooling, there is a range of operating parameters for which uniform flow distribution is unstable. With increasing inlet subcooling, this range broadens and the severity of the associated maldistribution increases.

Flow regime identification for upward two-phase flow in helically coiled tubes

Abstract Helically coiled tubes have excellent compactness, heat transfer efficiency, thermal stress adaptability and two-phase flow stability. The increasing application of two-phase flow in helically coiled tube devices necessitates a deep knowledge of phase distribution features for different flow regimes as well as flow regime transitions characteristics. This paper identified the flow regimes of air–water adiabatic two-phase flow in three helically coiled tubes using a double-sensor conductivity probe and high speed camera. Similar to the inclined straight tube, a total of six flow regimes were found in the helically coiled tubes. The influence of centrifugal force on the phase distribution was analyzed for each regime. A set of straight-forward flow regime transition criteria were proposed and then validated by benchmark experiments conducted in an inclined straight tube. These criteria were then used to generate flow regime maps based on the experiments with liquid superficial velocities ranging from 0.03 to 4.12 m/s and gas superficial velocities ranging from 0.03 to 16 m/s. Compared to the horizontal and inclined straight tubes, the helically coiled tube experienced shrinkage in the bubbly flow region; also, wavy and annular flows appeared at slower gas superficial velocities.